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Ion-Ion Correlation and Charge Reversal at Titrating Solid Interfaces Christophe Labbez,*,† Bo J€onsson,‡ Michal Skarba,§, and Michal Borkovec§ †
Institut Carnot de Bourgogne, UNR 5209 CNRS, Universit e de Bourgogne, F-21078 Dijon, France, Department of Theoretical Chemistry, Chemical Center, Lund University, POB 124, S-221 00 Lund, Sweden, and §Department of Inorganic, Analytical and Applied Chemistry, Geneva University, 1211 Geneva 4, Switzerland. Current address: Physical Chemistry I, Bayreuth University, 95440 Bayreuth, Germany )
‡
Received March 10, 2009. Revised Manuscript Received May 22, 2009 Confronting grand canonical titration Monte Carlo simulations (MC) with recently published titration and charge reversal (CR) experiments on silica surfaces by Dove and Craven (Dove, P. M.; Craven, C. M. Geochim. Cosmochim. Acta 2005, 69, 4963-4970) and van der Heyden et al. (van der Heyden, F. H. J.; Stein, D.; Besteman, K.; Lemay, S. G.; Dekker, C. Phys. Rev. Lett. 2006, 96, 224502), we show that ion-ion correlations quantitatively explain why divalent counterions strongly promote surface charge which, in turn, eventually causes a CR. Titration and CR results from simulations and experiments are in excellent agreement without any fitting parameters. This is the first unambiguous evidence that ion-ion correlations are instrumental in the creation of highly charged surfaces and responsible for their CR. Finally, we show that charge correlations result in “anomalous” charge regulation in strongly coupled conditions in qualitative disagreement with its classical treatment.
Introduction A key parameter for understanding electrostatic interactions in charged suspensions is the surface charge density, σ.3,4 This parameter may control phase separation and phase transition,5 promote dissolution,6 control particle growth,7 colloidal stability,8,9 and membrane selectivity,10 and play a significant role in catalysis and fluid transport through ionic channels.11,12 Typically a surface becomes charged through a titration process where surface groups ionize, for example, the titration of silanol groups -Si-OH h -Si-O- +H+. Ion-ion correlation in solutions of charged colloids has attracted much interest due to its ability to generate net attractive forces between equally charged colloids. For a planar electrical double layer, one can easily identify two contributions to this attraction. The first and most obvious is the appearance of an attractive term due to the correlation of an ion at one surface with another ion at the other surface. The magnitude of this interaction is only weakly dependent upon ion valency, and it gives an attractive contribution for all counterion valencies. In an electrical double layer with monovalent counterions, this attraction is usually much smaller than the entropic repulsion due to overlapping double layers and the net result is repulsive. With divalent counterions, however, strong correlations between *To whom correspondence should be addressed. E-mail: christophe.labbez@ u-bourgogne.fr.
counterions at the same surface cause a contraction of the diffuse double layer with a significant reduction of the entropic repulsion upon approach of the second surface. Under such circumstances, the net interaction becomes dominated by the attractive term. However, an equally important aspect of ion-ion correlation is its facilitation of surface titration processes. An example of vital importance is the surface titration of calcium silicate hydrates (C-S-H) in cement paste, which reach a very high surface charge density due to ion-ion correlation among the calcium counterions.13 The high surface charge density is then responsible for the setting of cement.14,15 An additional effect, due to ion-ion correlation, is when a charged surface immersed in a multivalent electrolyte more than compensates its charge, resulting in a charge reversal (CR)16,17 (see also ref 18). That is, the surface attracts counterions in excess of its own nominal charge, and the apparent charge seen a few a˚ngstr€oms from the surface appears to be of opposite sign to the bare surface. CR is experimentally characterized by several methods, for example, streaming current,2 and can induce repulsion between oppositely charged surfaces.19,20 Despite a vast number of studies, the origin of charge reversal is still controversial as highlighted in a recent review article,21 where Lyklema points out the lack of discrimination between the chemical (specific chemical adsorption) and physical (Coulombic interactions including charge correlations) origin of CR.
(1) Dove, P. M.; Craven, C. M. Geochim. Cosmochim. Acta 2005, 69, 4963–4970. (2) van der Heyden, F. H. J.; Stein, D.; Besteman, K.; Lemay, S. G.; Dekker, C. Phys. Rev. Lett. 2006, 96, 224502. (3) Honig, B.; Nicholls, A. Science 1995, 268, 1144–1149. (4) Hunter, K. A.; Liss, P. S. Nature 1979, 282, 823–825. (5) Chang, J.; Lesieur, P.; Delsanti, M.; Belloni, L.; Bonnet-Gonnet, C.; Cabane, B. J. Phys. Chem. 1995, 99, 15993–16001. (6) Dove, P. M.; Han, N.; Yoreo, J. J. D. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 15357–15362. (7) Elhadj, S.; Yereo, J. J. D.; Hoyer, J. R.; Dove, P. M. Proc. Natl. Acad. Sci. U. S.A. 2005, 103, 19237–19242. (8) Derjaguin, B. V.; Landau, L. Acta Physicochim. URSS 1941, 14, 633–662. (9) Verwey, E. J. W.; Overbeek, J. T. G. Theory of stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (10) Leung, K.; Rempe, S. B.; Lorenz, C. D. Phys. Rev. Lett. 2006, 96, 095504. (11) Stein, D.; Kruithof, M.; Dekker, C. Phys. Rev. Lett. 2004, 93, 035901. (12) Boda, D.; Valisko, M.; Eisenberg, B.; Nonner, W.; Henderson, D.; Gillespsie, D. Phys. Rev. Lett. 2007, 98, 168102.
(13) Labbez, C.; J€onsson, B.; Pochard, I.; Nonat, A.; Cabane, B. J. Phys. Chem. B 2006, 110, 9219–9230. (14) Delville, A.; Pellenq, R.; Caillol, J. J. Chem. Phys. 1997, 106, 7275. (15) J€onsson, B.; Nonat, A.; Labbez, C.; Cabane, B.; Wennerstr€om, H. Langmuir 2005, 21, 9221–9221. (16) Pashley, R. M. J. Colloid Interface Sci. 1984, 102, 22–35. (17) Sj€ostr€om; A˚kesson, T.; J€onsson, B. Ber. Bunsen-Ges. 1996, 100, 889–893. (18) Shklovskii, B. I. Phys. Rev. E 1999, 60, 5802–5811. (2) van der Heyden, F. H. J.; Stein, D.; Besteman, K.; Lemay, S. G.; Dekker, C. Phys. Rev. Lett. 2006, 96, 224502. (19) Besteman, K.; Zevenbergen, M. A. G.; Heering, H. A.; Lemay, S. G. Phys. Rev. Lett. 2004, 93, 170802. (20) Trulsson, M.; J€onsson, B.; A˚kesson, T.; Forsman, J.; Labbez, C. Phys. Rev. Lett. 2006, 97, 068302. (21) Lyklema, J. Colloids Surf., A 2006, 291, 3–12.
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The charging of interfaces (mineral or organic) has traditionally been described by the surface complexation model (SC),22-24 where titration of surface groups is described by an ensemble of mass balance equilibria with associated equilibrium constants. The SC is based on the Poisson-Boltzmann (PB) equation augmented with a so-called Stern layer. The PB equation, however, neglects ion-ion correlation and, for example, fails to predict the attraction between equally charged surfaces25,26 as well as the repulsion between oppositely charged ones.20 Despite its widespread use, the limitations of the SC model for particle dispersions in a multivalent salt solution have never been examined. Surprisingly, the same is true for concentrated particle dispersions, that is, when the interactions between the particles no longer can be neglected and give rise to so-called charge regulation. Moreover, although multivalent ions are ubiquitous in many systems, there are only few experimental studies with titration of metal oxide particles in the presence of such ions.1,27,28 Recently a grand canonical titration simulation method was developed29 and applied to a C-S-H dispersion.13 An excellent agreement was found between experiment and simulation for the titration process as well as for the CR. Unfortunately, due to the solubility range of C-S-H, the full titration curve is not accessible experimentally. In the present paper, we scrutinize the prediction from MC simulations for the charging process and CR by comparison to experimental data for silica particles dispersed in sodium and calcium chloride solutions. In particular, we focus on the contribution from ionic correlations. We also present a simulated charge regulation of two parallel silica surfaces in a Ca2+ salt solution when decreasing their separation. The experimental investigation of surface titration of silica fume particles by Dove and Craven1 and of CR by multivalent counterions from a streaming current analysis in silica nanochannels by van der Heyden et al.2 will serve as references.
Simulations and Method The simulations are based on the primitive model with the solvent as a structureless medium characterized by its relative dielectric permittivity, εr. All charged species are treated as charged hard spheres, and the interaction between two charges i and j separated by a distance r is Zi Zj e 2 uðrÞ ¼ 4πε0 εr r
if
r > dhc
ð1Þ
otherwise u(r) = ¥. Zi is the ion valency, e is the elementary charge, ε0 is the dielectric permittivity of vacuum, and dhc the hard sphere diameter of an ion always equal to 4 A˚. The particles are modeled as infinite planar walls with explicit titratable sites with the intrinsic dissociation constant K0 distributed on a square lattice. The titration process can be described as MOH h MO - þ H þ ,
K0 ¼
aM aH aMOH
ð2Þ
(22) Hiemstra, T.; VanRiemsdijk, W. H. J. Colloid Interface Sci. 1984, 133, 91–104. (23) Sverjensky, D. A. Nature 1993, 364, 776–780. (24) Borkovec, M. Langmuir 1997, 13, 2608–2613. (25) Guldbrand, L.; J€onsson, B.; Wennestr€om, H.; Linse, P. J. Chem. Phys. 1984, 80, 2221–2228. (26) Zohar, O.; Leizerson, I.; Sivan, U. Phys. Rev. Lett. 2006, 96, 177802. (27) Karlsson, M.; Craven, C. M.; Dove, P. M.; Casey, W. H. Aquat. Geochem. 2001, 7, 13–32. (28) B€ohmer, M. R.; Sofi, Y. E. A.; Foissy, A. J. Colloid Interface Sci. 1994, 164, 126–135. (29) Labbez, C.; J€onsson, B. Lect. Notes Comput. Sci. 2007, 4699, 66–72.
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Figure 1. Snapshot of a typical simulation box used in the Monte Carlo simulations showing two titrating silica surfaces separated by a salt solution. Titrating sites, modeling silanol groups of the silica surfaces, are treated as discrete and distributed on a square lattice of dimensions 4.56 4.56 A˚2. Sites are allowed to titrate according to the solution pH and composition.
where the a’s denote the activities of the species. The model, illustrated in Figure 1, is solved using a grand canonical Monte Carlo method,30 that is, at constant pH and chemical potential of the ions. In addition to ordinary MC moves, the titratable sites are allowed to change their charge status. We imagine the deprotonation of a surface site as a two-step process: the release of a proton from the surface followed by an exchange of an ion pair (H+, B-) with the bulk. The corresponding Boltzmann factor for the trial energy can be expressed as NB el exp -β μB þ ΔU þ ln10ðpH -pK0 Þ expð -βΔUÞ ¼ V ð3Þ where μ represents the chemical potential of a particular ion, V is the volume, NB is the number of ions B-, and ΔUel is the change in electrostatic energy. An analogous expression holds for protonation; for more details see ref 29.
Results and Discussion Titration. To establish the model parameters for a silica/ solution interface, we turn our attention to the case of silica in a 1:1 salt solution. The following set of parameters pK0 =7.7, Fs = 4.8 sites/nm2, the surface site density, and dis = 3.5 A˚, the minimum separation between an ion and a surface site, gives a perfect description of the charging process (see Figure 2a). The surface charge density, σ, increases, in absolute value, with pH but is always smaller than that in the ideal case due to the strong electrostatic repulsion between the deprotonated silanol groups. A change in the salt concentration modulates the screening of the electrostatic interactions which, in turn, results in a change in σ. The SC model is based on a mean field approximation with a smeared out surface charge density on the titrating particles. This means that neither the discrete nature of the titratable surface sites (30) Frenkel, D.; Smit, B. Understanding molecular simulation; Academic Press: London, 1996.
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Figure 2. (a) Simulated (MC) and experimental (symbols)1 surface charge density versus pH for silica particles dispersed in a sodium (1:1) salt solution at different concentrations. The ideal titration curve (ideal) is given as a reference. (b) Comparaison between the MC, assuming a smeared out surface charge density (MC smeared out), and SC calculations of the surface charge density in the same conditions as in (a). The 1000 mM Na+ experimental points are given for comparison. In the MC simulations, pK0 =7.7, Fs = 4.8 sites/nm2, and dis=3.5 A˚. In the SC model, a Stern layer is used with CS = 2.6 F/m2, the Stern layer capacitance. Otherwise, the values for pK0 and Fs are the same as those in the MC simulations.
nor the correlations between the mobile ions themselves as well as with the sites are taken into account. In an attempt to deal with the deficiencies of the SC model, a “Stern layer capacitance”, CS, has been used as a fitting parameter. The SC model is, however, unable to give an accurate description of the charging process of silica in 1:1 salt solution, while keeping the values for the surface parameters, that is, pK0 and Fs, equal to the previous values and using CS as a fitting parameter. Nonetheless, a perfect agreement between the SC model, with CS =2.6 F/m2, and MC simulations can be obtained (see Figure 2b), if one assumes a smeared out surface charge density even in the simulations. Thus, the discrete nature of the titrating surface sites and ion-site correlations are essential physical properties for an accurate description of the titration process. When the sodium counterions are replaced by divalent calcium ions, the simulations predict a strong increase in the surface charge density in perfect agreement with the titration data of Dove and Craven1 (see Figure 3a). Note that similar experimental results were obtained in the early work of Tadros et al.31 The same (31) Tadros, Th. F.; Lyklema, J. J. Electroanal. Chem. 1969, 22, 1–7.
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Figure 3. (a) Simulated (MC) and experimental (symbols)1 surface charge density versus pH for silica particles dispersed in a calcium (2:1) and sodium (1:1) salt solution at various concentrations. The corresponding simulated titration curves for particles having a smeared out surface charge density (MC smeared out) are also given for comparison. (b) MC and SC calculations of the surface charge density when a smeared out surface charge density is assumed (MC smeared out) and in the same conditions as in (a). The 1000 mM Ca2+ experimental points are plotted for comparison. The model parameters are the same as in Figure 2.
behavior has been observed from titration experiments on titania particles28 in multivalent electrolytes and, indirectly, from proton release measurements following adsorption of multivalent ions (or polyelectrolytes),32 emphasizing the universality of the phenomenon. As illustrated in Figure 3b, the classical SC approach predicts a much too small increase of the ionization fraction. This weakness of the model originates from the lack of energy gain associated to charge correlations. Here, we unambiguously show that the charge correlations (ion-ion and ion-site), without any additional fitting parameters, are enough to explain the increased of silanol group ionization in the presence of Ca2+ ions. Thus, the complexation constants often invoked in the SC have no physical basis. These constants do not represent a surface chemical complexation, but, instead, reflect the ion-ion and ion-site correlations neglected in the mean field treatment of the SC model. This means that even in cases where a surface chemical reaction exists, it becomes impossible to quantify it, and as a consequence the SC model fails to distinguish between ion specificity and pure electrostatic interactions. (32) Pittler, J.; Bu, W.; Vaknin, D.; McGillivray, D. J.; L€osche, M. Phys. Rev. Lett. 2006, 97, 046102.
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Figure 4. Contribution of the ion-ion and ion-site correlations among the effect of all correlations on the charging process of silica at two calcium salt concentrations, 67 and 1000 mM (symbols). The effect of all correlations is obtained from the difference between MC simulations with discrete charges and the SC model that neglects all correlations (σMC - σSC). The ion-ion correlation is evaluated from the difference between MC simulations with smeared out surface charge and the SC model (σMC,smearedout σSC). The ion-ion correlation contribution is thus given by (σMC - σSC)/(σMC,smeared out - σSC).
In an attempt to distinguish the effect of ion-ion from ion-site correlations, a MC calculation was performed using a smeared out surface charge density, that is, neglecting the ion-site correlations, and compared to the previous calculations (see Figure 3). When compared to the MC simulations including all charge correlations (Figure 3a), a significant drop in |σ| is observed, although the qualitative behavior of σ upon increasing the counterion valency remains the same. Indeed, it still leads to a much stronger surface ionization as compared to the SC model (see Figure 3b). Thus, the discrete nature of the surface groups has a quantitative but not qualitative effect. A quantification of the ion-ion and ion-site correlations influence is given in Figure 4. The ion-ion correlation influence can be deduced from the comparison between MC simulations of a smeared out surface charge and the SC model that neglects all correlations. The contribution from ion-site correlations on the other hand can be seen from the difference between MC simulations with a smeared out surface charge and with explicit sites. Two different regimes for the charging process can be found. That is, at low salt concentrations, the ion-site correlations are found to play a more important role than the ion-ion correlations. On the contrary, at high salt concentrations, the ion-ion correlations overcome the ion-site correlations. Note that, as one could have expected from Figure 2b, at low charge densities and thus weakly coupled systems, the MC and SC calculations agree. In any case, our results show that all the charge correlations are needed to accurately describe the experiments. Streaming Current. We can also compare our simulation data to the streaming current, Istr, experiments through silica channels performed by van der Heyden et al.2 in the presence of di- and trivalent counterions at pH 7.5. Istr is computed from the simulated local charge density, F(x). For a channel of height h and width w with w . h, Istr can be defined as Z h=2 3 Istr ¼ 2w FðXÞ vðXÞ dX with X ¼ x - dhc ð4Þ 2 0 where x is the distance normal to the silica surface and v(x) is the local fluid velocity, approximated as a Poiseuille flow with 7212 DOI: 10.1021/la900853e
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Figure 5. Comparison between simulated (line) and experimental (points)2 streaming currents of a silica channel as a function of electrolyte concentration of a CaCl2 and MgCl2 salt at pH 7.5. For Ca2+, the model parameters are the same as in Figure 2. For Mg2+, dis = 3.1 A˚ as obtained from the curve fitting of the silica titration experiments of Dove and Craven.1
boundary conditions: v(X = 0) = 0 and (X = h/2) = 0. In the former, we assume the no slip condition at the wall, which is strictly valid for large h and low ionic concentration.33 The zero velocity is arbitrarily set at x=3dhc/2, where by definition we also find the electrokinetic potential. The simulation uncertainty for the calculated streaming current is below 0.5 pA/bar, far below the experimental one. The simulations predict for both Mg2+ and Ca2+ a monotonic decrease in Istr upon increasing salt concentration as also found experimentally (see Figure 5). A sign reversal in the streaming potential is predicted at cCR ≈ 310 mM for Ca2+ and cCR ≈ 280 mM for Mg2+, in good agreement with numbers observed experimentally.2 What is more, the magnesium value is predicted to be below that of calcium in agreement with experiments2 due to a higher silica charge. Simulations were also performed for trivalent cations, where CR is predicted at submillimolar concentrations, cCR ≈ 400 μM, although 4 times higher than what was observed with cobalt(III)sepulchrate.2 It is worth noting that for nontitrating surfaces CR is found (not shown here) to be independent upon the charge description of the surface, that is, smeared out or discrete. This would tend to demonstrate that the ion-site correlations are negligible or at best a second order parameter in explaining the origin of CR in apparent contradiction with recent approximate studies which state that ion-site correlations prevail over ion-ion correlations.32,34 Note, however, that CR is found to occur at a salt concentration for which the contribution of ion-ion and ion-site correlations with respect to the surface site titration is equivalent. On the other hand, it is also clear that more work is needed to fully clarify the contribution of the different correlations. Here, we show that the inclusion of titratable groups is crucial for the prediction of CR. This brings us to the conclusion that the titration of the surface sites, that is, the proton transfer from the surface to the bulk, has a prominent role in the generation of CR for which both ion-site and ion-ion correlations come into play. Charge Regulation. We now investigate the surface charging process of two parallel planar silica surfaces upon decreasing their separation, that is, the charge regulation. The dependence of R on (33) Dufr^eche, J. F.; Malı´ kova, N.; Turq, P.; Marry, V. J. Mol. Liq. 2005, 118, 145–153. (34) Faraudo, J.; Travesset, A. J. Phys. Chem. C 2007, 111, 987–994.
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Figure 6. Degree of ionization, R, as a function of surface separation at various pH values. The surfaces are in contact with a 0.2 mM CaX2 solution. The model parameters are the same as in Figure 2. (a) Results from MC simulations (MC) at four different pH values; (b) results determined from MC simulations (MC) and SC model (SC) at two different pH values.
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potential of mean force (not shown). In this case, the energy (induced by the ion-ion correlations) dominates, which leads to an attraction between the particles and, subsequently, to an increase of the surface ionization for short h, which in turn strengthens the attraction. Close to contact the entropy regains in importance and eventually causes σ to collapse. However, when the Ca2+ concentration is increased to more than 10 mM (not shown), the rise in R is dramatically strengthened while the drop at close to contact is hardly measurable. In other words, the surfaces retain and even increase their charges all the way to contact. This charge regulation behavior may partly explain why Ca2+ promotes the experimentally observed dissolution of amorphous and crystalline silica surface under stress.37 Indeed, as recently rationalized by Dove et al.,6 the dissolution proceeds through a nucleation process following the same mechanistic theory as developed for growth and it is shown to be promoted by surface ionization. Ion-ion correlation also seems to be relevant for membrane selectivity. As an example, we have calculated the L-type Cachannel selectivity at various pH values in a reservoir containing always 100 mM Na+ to which is added either 1 mM or 100 μM Ca2+. The Ca-channel was modeled as an infinite slit pore with h=9 A˚, with walls decorated with titratable carboxylic groups, pK0=4.8, and Fs=3 sites/nm2. The selectivity, ξ, was calculated as the ratio between the average calcium and sodium concentrations. At pH 7 and 100 μM Ca2+, we found ξ=0.96 and R=0.70, while at 1 mM Ca2+ the values increased to ξ=3.7 and R=0.84. Boda et al.12 tried to mimic the selectivity with a constant ionization of the -COOH groups, but they were then forced to decrease the relative dielectric permittivity of the protein to 10 in order to fit the experimental data. When pH is reduced to 5, ξ drops to 0.10 and 0.15 at low and high calcium content, respectively, in agreement with the observed loss in membrane selectivity at low pH.38
Conclusions To summarize, we have shown by confronting Monte Carlo simulations with independent experiments that the dominating interaction which controls the charging process and charge reversion of silica in multivalent electrolyte is of purely electrostatic origin and strongly dependent on ion-ion correlations. In particular, we have demonstrated that the charging process and charge reversal are intimately related. We have also demonstrated that the accepted view on charge regulation as a monotonic drop in |σ| with decreasing separation is qualitatively wrong in highly coupled systems. Instead, |σ| is found to increase with decreasing h. This has profound influence for the stability of colloidal particles and for the early setting of normal Portland cement.
separation, h, in the presence of 0.2 mM Ca2+ at various pH is shown in Figure 6a. At low pH, a monotonic decrease of R with decreasing h, in conjunction with an increased osmotic repulsion (not shown), is found in qualitative agreement with previous mean-field studies35,36 (see also Figure 6b). This result can be rationalized as a balance between entropy and energy. That is, when pH is well below pK0, R is small and the free energy is dominated by the entropic term causing the particles to repel. As a result, the system minimizes its free energy by a reduction of the surface ionization. Conversely, atypical R versus h curves are predicted for surfaces immersed in divalent electrolytes at large pH values (pH g pK0), where R is found to increase when shortening h. This phenomenon is further illustrated in Figure 6b where, at pH 10.3, the SC model still predicts a monotonic decrease of R while an increase is found by the MC simulations. The degree of dissociation reaches a maximum at hmax ≈ 6 A˚, whereafter it decreases and annihilates at contact. The maximum in R corresponds exactly to the minimum of the
Acknowledgment. We thank R. Kjellander and S. G. Lemay for helpful discussions. M. Dove and F. H. J. van der Heyden are greatly acknowledged for the kind permission to use their experimental data. We also acknowledge the support of the CRI (Bourgogne University) and LUNARC (Lund University) for access to their computer facilities.
(35) Kirkwood, J. G.; Shumaker, J. B. Proc. Natl. Acad. Sci. U.S.A. 1952, 38, 863–871. (36) Ninham, B. W.; Overbeek, V. A. J. Theor. Biol. 1971, 31, 405.
(37) Anzalone, A.; Boles, J.; Greene, G.; Young, K.; Israelachvili, J.; Alcantar, N. Chem. Geol. 2006, 230, 230–231. (38) Delisle, B. P.; Satin, J. Biophys. J. 2000, 78, 1895–1905.
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