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Repulsion between Oppositely Charged Macromolecules or Particles M. Trulsson,* Bo Jo¨nsson, T. Åkesson, and J. Forsman Theoretical Chemistry, Chemical Center POB 124, S-221 00 Lund, Sweden
C. Labbez Institut Carnot de Bourgogne, UMR CNRS 5209, UniVersite´ de Bourgogne, F-21078 Dijon Cedex, France ReceiVed April 26, 2007. In Final Form: August 20, 2007 The interaction of two oppositely charged surfaces has been investigated using Monte Carlo simulations and approximate analytical methods. When immersed in an aqueous electrolyte containing only monovalent ions, two such surfaces will generally show an attraction at large and intermediate separations. However, if the electrolyte solution contains divalent or multivalent ions, then a repulsion can appear at intermediate separations. The repulsion increases with increasing concentration of the multivalent salt as well as with the valency of the multivalent ion. The addition of a second salt with only monovalent ions magnifies the effect. The repulsion between oppositely charged surfaces is an effect of ion-ion correlations, and it increases with increasing electrostatic coupling and, for example, a lowering of the dielectric permittivity enhances the effect. An apparent charge reversal of the surface neutralized by the multivalent ion is always observed together with a repulsion at large separation, whereas at intermediate separations a repulsion can appear without charge reversal. The effect is hardly observable for a symmetric multivalent salt (e.g., 2:2 or 3:3).
Introduction The interaction between charged surfaces in electrolyte solutions is important for essentially all colloidal and biological systems. Consequently, these interactions have been studied extensively during the last century, both experimentally and theoretically. The theoretical approach has for many years been based on a combination of the Poisson-Boltzmann (PB) equation and van der Waals forces. This so-called DLVO theory1,2 treats the solvent as a continuous medium described through a dielectric permittivity. The mean-field approximation in the PB equation means that ion-ion correlations are neglected, which becomes apparent when the electrostatic interactions increase.3,4 Thus, the DLVO theory can be expected to describe the behavior of colloidal systems at low electrostatic coupling. Perhaps the most interesting and well known behavior is the long-range screening of the electrostatic interaction between two charged surfaces in an electrolyte solution.5,6 The DLVO theory is not only qualitatively but also quantitatively in agreement with experiment in this case. For electrostatically highly coupled systems, the mean-field theories fail to account even qualitatively for a number of phenomena. Typical examples are charge reversal (or overcharging),7,8 where ions accumulate close to a surface and create an apparent surface charge with the opposite sign of the bare surface charge, and the counter-intuivative phenomenon of attraction between surfaces of like charge and its converse, repulsion * Corresponding author. E-mail:
[email protected] (1) Derjaguin, B. V.; Landau, L. Acta Phys. Chim. URSS 1941, 14, 633. (2) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (3) Guldbrand, L.; Jo¨nsson, B.; Wennerstro¨m, H.; Linse, P. J. Chem. Phys. 1984, 80, 2221. (4) Kjellander, R.; Marcˇelja, S. J. Chem. Phys. 1985, 82, 2122. (5) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531. (6) Pashley, R. M.; Israelachvili, J. N. J. Colloid Interface Sci. 1984, 101, 511. (7) Pashley, R. M. J. Colloid Interface Sci. 1984, 102, 23. (8) Sjo¨stro¨m, L.; Åkesson, T.; Jo¨nsson, B. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 889. (9) Besteman, K.; Zevenbergen, M. A. G.; Heering, H. A.; Lemay, S. G. Phys. ReV. Lett. 2004, 93, 170802.
between surfaces of opposite charge.9-11 More sophisticated methods, which include ion-ion correlations, such as the modified PB equation,12 the hypernetted chain theory,4 the correlationcorrected Poisson-Boltzmann,13 or Monte Carlo simulations (MC),3 are needed in order to describe these effects. The failure of the mean-field theory in the highly coupled regime is thus due to the lack of ion-ion correlations (electrostatic and/or hard core) and not to the use of a continuum model for the solvent. In a previous study, inspired by the experiments of Besteman et al.9 and by the fact that oppositely charged particles are common in cement paste, we investigated the origin of the repulsive interaction between two oppositely charged surfaces. It was found that if there is a charge reversal at one of the surfaces then there will also be a repulsive interaction at large separations. The repulsion will increase with decreasing separation and will also exist at intermediate separations, where the apparent surface charge reversal has disappeared. For example, the maximum in repulsion will never coincide with the charge reversal of one of the surfaces. The repulsion is not a direct consequence of the charge reversal. Thus, the former should not be seen as an effective electrostatic interaction between two apparently equally charged surfaces, except at large separations. At intermediate separations the repulsion is better described as an entropic effect due to the large amount of salt in the system. However, both phenomena have the same origin, namely ion-ion correlations, which lead to a larger accumulation of counterions close to a charge surface. If the bulk salt concentration is high, then a large number of extra counterions, and their co-ions, can be sucked in between the charged surfaces. If the salt consists of a multivalent counterion and monovalent co-ions, then the latter can make a significant contribution to the internal pressure. (10) Besteman, K.; Zevenbergen, M. A. G.; Lemay, S. G. Phys. ReV. E 2005, 72, 061501. (11) Trulsson, M.; Jo¨nsson, B.; Åkesson, T.; Labbez, C.; Forsman, J. Phys. ReV. Lett. 2006, 97, 068302. (12) Outhwaite, C. W.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 1983, 2 79, 707. (13) Forsman, J. J. Phys. Chem. B 2004, 108, 9236.
10.1021/la701222b CCC: $37.00 © 2007 American Chemical Society Published on Web 10/05/2007
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In this article, we extend our previous Monte Carlo simulations of oppositely charged surfaces to investigate the effect of (i) a symmetric salt, (ii) adding a second salt, (iii) varying the surface charge density of the nonovercharged surface, and (iv) varying the dielectric permittivity. All is done in order to describe the generality of the repulsion between the two oppositely charged surfaces. The results are also rationalized with an effectiVe meanfield theory.
Model The interaction between the oppositely charged surfaces is modeled as two planar surfaces with uniform surface charges densities, σ-1 (negative) and σ1 (positive), separated a distance h. The primitive model is used,14 with all ions explicitly considered, whereas the solvent is treated as a dielectric continuum characterized by a dielectric permittivity r. The temperature T is set to 298 K, and r is set to 78.7, corresponding to water at room temperature. In the simulations, the ion-ion interactions are calculated according to
qiqj u(rij) ) 4π0rrij u(rij) ) ∞
di + dj rij > 2
(1)
di + dj 2
(2)
rij
Φ-1. The condition for zero pressure, that is, the change from repulsion to attraction, will, however, occur when
cosh
( ) x( κh ) 2
2 2 1 1 (Φ-1 + Φ1 ) +1 22 Φ-1Φ1
)
(14)
The ratio between the surface potentials will determine the outcome. Apparently, zero pressure appears at shorter separation than charge reversal, which means that there will be a regime with oppositely charged surfaces and a repulsive interaction. In the PB approximation, this is an entropic repulsion due to the remaining counterions necessary for the neutralization of the system. The linear PB equation is surprisingly accurate for the surface potentials in our case, and using the nonlinear PB will change the pressure by at most 10-20% in the relevant regime. To use the above expressions, one needs to know the (effective) surface potentials Φ-1 and Φ1. One way to get these is through eff and Φeff were simulations. The effective potentials, Φ-1 1 extrapolated to contact by fitting an exponential function to the simulated mean-field potential profile. The function used was of the form
Φ(z) ) Φeff exp(-Rz)
(15)
where Φeff and R are two fitting parameters. From the fitting procedure, we find that R ≈ κ, where κ is the Debye-Hu¨ckel inverse screening length calculated from the salt concentration in the bulk solution. The effective potentials had to be determined at some separation (because the effective potentials are assumed to be constant). The ideal choice would be when the two surfaces do not interact, that is, one surface next to an infinite bulk solution. This situation was mimicked in a separate grand canonical simulation similar to the ones discussed above, the difference being that only one of the surfaces was charged (σ-1 or σ1) whereas the other was neutral and was placed at a separation large enough to ensure bulklike conditions. The other alternative, obtaining the effective potentials from an approximate theory, has been attempted by Zhang and Shklovskii.21,22 They proposed an expression for the effective surface potential based on a Wigner crystal approach neglecting the contribution of co-ions; for more details see the above references. We will discuss both routes in the following sections.
Results
p(h) ≈ 20rκ2[Φ-1Φ1 exp(-κh) (Φ-1 + Φ1 ) exp(-2κh)] (11) 2
(dΦ dz )
Equation 12 allows for an adjustment of the surface charge densities as the surfaces approach, and one can derive an expression describing at which separation charge reversal takes place:
D
where ΦD(h) is the Donnan potential,20 which depends on the separation h and ensures that the midplane potential goes to zero when h f ∞. The Donnan potential is defined as
[
σ ) -0r
|z - t|F(t, h) dt + σ-1 z + 1
p(h) )
as they approach and change their surface charge density. The surface potential can be related to the surface charge density:
2
Assuming constant potentials, the surfaces will adjust themselves (20) Donnan, F. G. Z. Electrochem. 1911, 17, 572. (21) Perel, V. O.; Shklovskii, B. I. Physica A 1999, 274, 446.
Figure 2 shows simulated pressure curves between two oppositely charged surfaces, with charge densities of -2 and 1 e/nm2, respectively. The slit is in equilibrium with 2:1 and 3:1 bulk electrolyte solutions, respectively. At low salt concentrations, (22) Shklovskii, B. I. Phys. ReV. E 1999, 60, 5802.
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Figure 3. Salt content between two oppositely charged surfaces in equilibrium with 1 and 5 mM 3:1 salt solutions. The surface charge densities are -2 and 1 e/nm2, respectively.
Figure 2. Net osmotic pressure between two oppositely charged surfaces in equilibrium with a bulk salt solution. The surface charge densities are σ-1 ) -2 and σ1 ) 1 e/nm2. The dashed curves are obtained from the linearized PB equation for two surfaces approaching at constant surface potentials; see eq 10. The potentials have the same sign but different magnitudes; see the text. X symbols delimit the region of charge reversal, which is found to the right of the symbols: (a) 2:1 salt solution in the bulk with black ) 100, red with circles ) 50, and blue with squares ) 10 mM and (b) 3:1 salt solution in the bulk with black ) 15, red with circles ) 5, and blue with squares ) 1 mM.
the surfaces attract each other as expected from the PB equation. At increasing concentration, however, a repulsion appears at intermediate separations. The repulsion is not seen in an aqueous solution with a 1:1 salt, indicating that it is a result of ion-ion correlations. It is interesting that the systems showing a repulsive interaction always experience an apparent surface charge reversal at large separations (i.e., the surface charges are (over)compensated for by multivalent counterions). However, the repulsive interaction and the charge reversal are only indirectly related. For example, it is possible to have a net repulsive interaction in a system that does not show a charge reversal. This is illustrated in Figure 2, where we have indicated the region of charge reversal by a cross. (Charge reversal is found to the right of the symbols.) It is clear that there is a repulsive interaction without charge reversal; however, both phenomena have the same origin: ion-ion correlations. The repulsion at intermediate separation is largely a consequence of the high salt concentration in the slit. That is, with two oppositely charged surfaces it is always energetically advantageous to increase the salt concentration in the slit significantly above the bulk value and ion-ion correlations enhance this effect. Thus, the repulsion should rather be viewed as an entropic effect from an “ideal” osmotic pressure rather than an electrostatic interaction between two surfaces of equal apparent charge; see also Figure 3. The repulsion and charge reversal are enhanced if the bulk salt concentration is increased. This simply reflects the fact that the entropic cost of increasing the slit concentration is reduced. There
Figure 4. Net osmotic pressure between two oppositely charged surfaces in equilibrium with a 4:1 salt solution. The surface charge densities are -2 and 1 e/nm2, respectively. The dashed curves are obtained from the linearized PB equation for two surfaces approaching at constant surface potentials; see eq 10. The bulk concentrations are black ) 0.5 and red with symbols ) 0.1 mM.
is also a dependency on the valency of the multivalent salt particle that cannot be explained by different ionic strengths. With increasing valency, the correlations become more important, and charge reversal and repulsive interactions appear more readily. With a divalent salt the repulsive interaction appears at a salt concentration of approximately 50 mM, whereas for a tri- or tetravalent salt repulsion sets in at 5 and 0.5 mM, respectively; see Figures 2 and 4. Ion-ion correlation increases the accumulation of counterions close to the charged walls and, with multivalent cations, it actually overcompensates for the bare surface charge at large separation; see Figure 5a. That is, normally the potential a few ångstro¨ms from a negatively charged surface would be negative, but for sufficiently high electrostatic coupling it changes sign and becomes positive! Table 1 shows the effective potentials extracted from simulations. Note that the effective potentials for the negatively charged surface (column 3) are all positive except for the lowest concentration of the 2:1 salt. The weak coupling between σ1 and the monovalent anions is not sufficient to create a charge reversal, which means that the effective potentials for the positively charged surface have the same sign as the surface charge density. Figure 6 shows how the net osmotic pressure changes with varying σ1 in a 3:1 salt solutions. If σ1 ) 0, the pressure is repulsive for all separations with some short-range oscillations due to packing effects of the counterions. Once the nonovercharged surface has a small positive charge, the pressure curve is essentially independent of σ1 at intermediate and large separations. This can be understood from PB theory, which predicts that the surface potential is a slowly varying function
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Figure 5. Electrostatic potential near the surfaces for a 3:1 salt at varying concentration as indicated. Charge reversal appears only at the negatively charged surface. (a) σ-1 ) -2 and (b) σ1 ) 1 e/nm2. Solid lines, simulated mean potential; dashed lines, extrapolation to contact with the potential assuming the form Φ(z) ) Φeff exp(-Rz). Table 1. Effective Surface Potentials from the Simulated Charge Profiles for Various Salts and Concentrationsa salt
cs (mM)
Φ-1 (mV)
Φ1 (mV)
2:1 2:1 2:1 3:1 3:1 3:1 4:1 4:1
10 50 100 1 5 15 0.1 0.5
-7 11 20 28 40 48 47 67
141 100 86 183 145 120 199 188
Figure 6. Net osmotic pressure as a function of separation for varying surface charge density (e/nm2) of the positively charged surface as indicated. σ-1 is kept constant at -2 e/nm2, and the bulk concentrations of a 3:1 salt are (a) 1 and (b) 5 mM.
a The numbers have been obtained from extrapolation as indicated in Figure 5. The surface charge densities are σ-1 ) -2 and σ1 ) 1 e/nm2.
of the surface charge density. A similar behavior is seen with 2:1 and 4:1 salt solutions. The addition of a second salt affects the charge reversal, and it can both increase and decrease depending on concentration.23-26 Figure 7 shows that the addition of a 1:1 or 2:1 salt to a solution containing a 3:1 salt leads to significantly increased repulsion at short separations. In disagreement with Martin-Molina et al. in their recent study,26 we see increased charge inversion when adding small amounts of a 1:1 or 2:1 salt. The discrepancy is due to the difference in the definition of charge reversal. MartinMolina et al. calculate the charge reversal by integrating only the multivalent ion profile, which means that an additional 1:1 salt will only indirectly affect the charge reversal through the multivalent ions. We believe, however, that our definition of charge inversion is more correct because it takes all ions into account. The increased repulsion can be understood from the (23) Zhang, R.; Shklovskii, B. I. Phys. ReV. E 2005, 72, 021405. (24) Martin-Molina, A.; Quesada-Perez, M.; Galisteo-Gonzalez, F.; HidalgoAlvarez, R. J. Phys.: Condens. Matter 2003, 15, S3475. (25) Jo¨nsson, B.; Nonat, A.; Labbez, C.; Cabane, B.; Wennerstro¨m, H. Langmuir 2005, 21, 9211. (26) Pianegonda, S.; Barbosa, M. C.; Levin, Y. Europhys. Lett. 2005, 71, 831.
Figure 7. Net osmotic pressure as a function of separation for a system in equilibrium with 5 mM 3:1 salt (black line). A second salt of 50 mM 1:1 (red with circles) and 25 mM 2:1 (blue with squares) has been added. The surface charge densities are σ-1 ) -2 and σ1 ) 1 e/nm2.
increased chemical potential of anions in the bulk, which naturally gives a higher concentration in the slit. At the same time, the electrostatic screening increases in the system, and at larger separations, the repulsion is reduced upon salt addition. Further salt addition will eventually screen out all of the electrostatic interactions, and the repulsion will of course disappear. Figure 8 shows the interaction between two oppositely charged surfaces in equilibrium with symmetrical divalent and trivalent salt solutions. Depending on the salt concentration and the surface charge densities, one can observe overcharging at one or both surfaces, where we focus only on the former case. The repulsive maxima seen in asymmetric electrolytes are strongly reduced, and the pressure is essentially attractive down to very small separations, where the trivial entropic repulsion sets in. (Note that the limit of accuracy in the simulations is approximately 1
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Figure 9. Net osmotic pressure between two oppositely charged surfaces -σ-1 ) -2 and σ1 ) 1 e/nm2. The bulk solution contains a 3:1 salt of 5 mM (lines) and 1 mM (lines with symbols). The dielectric permittivity has been varied: r ) 78 (black) and r ) 54 (red).
Figure 8. Net osmotic pressure between two oppositely charged surfaces from simulations (solid lines) and from the linearized PB equation with surface potentials from simulations (dashed lines). (a) The surface charge densities are σ-1 ) -3.5 and σ1 ) 1 e/nm2, and the bulk concentrations of a 2:2 salt are 20 mM (black) and 40 mM (red with symbols). (b) σ-1 ) -1 and σ1 ) 0.5, and the system is in equilibrium with 5 mM (black) and 1 mM (red with symbols) 3:3 salt solutions, respectively. Only the negatively charged surface is overcharged.
mM.) For a 3:3 salt, the repulsion at short separations is to a large extent due to the soft core repulsion, which explains the outward shift of the pressure minimum. The pressure curves are almost independent of salt concentration. The absence of a significant repulsive regime in the case of a symmetric salt is probably due to both a reduced number density and stronger correlations among the anions because they now are di- or trivalent. This also leads to an enhanced accumulation of anions at the positively charged wall with only a very small apparent surface charge density. That is, even if both surfaces have the same sign on their apparent surface charge densities/potentials, the interaction between these surfaces should be rather small. The previous results have shown that the repulsion between oppositely charged surfaces is a correlation effect, and it can therefore be expected in systems with strong electrostatic interactions. The easiest way, at least experimentally, to increase the coupling is by using a solvent with a dielectric permittivity lower than that of water.10 For example, a 1:1 mixture of ethanol and water has a dielectric permittivity of 54. Figure 9 shows the effect on lowering the dielectric constant in the 3:1 salt case with surface charge densities equal to -2 and 1 e/nm2. As expected, the repulsion increases, and charge reversal becomes easier, favored by an enhancement of salt in the slit.
Discussion The results presented above and supported by independent experimental studies7,9,10,27,28 clearly demonstrate the limitations of the traditional mean-field approximation. That is, in a strongly
coupled system we can expect ion-ion correlations to play an important role. This has earlier been shown to be the case for equally charged surfaces where the repulsion predicted by meanfield theory is incorrect and instead an attractive interaction dominates. This mechanism plays an important role in a number of technical (cement29,30) and biological (DNA31,32) systems. In the present study, we have focused on the interaction between two oppositely charged surfaces, where it turns out that ion-ion correlations are equally important. As for equally charged surfaces, ion-ion correlations play a key role in highly coupled system. Experimentally, this often turns out to be the case when multivalent counterions are present. However, we emphasize that other parameters such as the surface charge density and the dielectric permittivity of the solvent are also important and can be used to modify the importance of the correlations. It is sometimes, at least within the primitive model, fruitful to look upon the resulting forces as a result of the competition between entropic and energetic terms. That is, in an ideal system everything is entropic whereas when interaction comes into play the energetic term becomes increasingly important. For example, the exchange of monovalent to divalent ions not only increases the interaction but also reduces the entropy, halving the number of particles. What is given above is a qualitative picture. In the simulations, the counterions are treated as charged hard spheres, and when the charge density of a surface neutralized by monovalent counterions goes above approximately 1 e/nm2, then the collision term will contribute to the pressure and a more complicated picture emerges. We have in the present study tried to avoid this range of parameters, although a few cases reported above are clearly in this range. There have been different attempts to rationalize the repulsion and overcharging phenomena. One approximate way is to incorporate the charge distribution near the surface into a new “renormalized” surface charge density to be used in a PB calculation at constant surface charge density. This gives a reasonably good description of the force behavior at long range (27) Martin-Molina, A.; Maroto-Centeno, J. A.; Hidalgo-A Ä lvarez, R.; QuesadaPe´rez, M. J. Chem. Phys. 2006, 125, 144906. (28) Lesko, S.; Lesniewska, E.; Nonat, A.; Mutin, J.-C.; Goudonnet, J.-P. Ultramicroscopy 2001, 86, 11. (29) Plassard, C.; Lesniewska, E.; Pochard, I.; Nonat, A. Langmuir 2005, 21, 7263. (30) Delville, A.; Pellenq, R.; Caillot, J. J. Chem. Phys. 1997, 106, 7275. (31) Jo¨nsson, B.; Wennerstro¨m, H.; Nonat, A.; Cabane, B. Langmuir 2004, 20, 6702. (32) Rouzina, I.; Bloomfield, V. A. J. Phys. Chem. 1996, 100, 9977.
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Figure 10. Effective surface potentials from eq 15 (thick solid lines) and the corresponding WC potentials (thin lines with symbols) for a charged surface (σ ) -2 e/nm2) in equilibrium with a 2:1 (black curves), a 3:1 (red curves), or a 4:1 bulk solution (blue curves). The curves show how the potential varies as a function of salt concentration in the bulk.
for a system in equilibrium with an X:1 or 1:X salt, but it does not capture the transition to attractive pressure at short separation. However, if we estimate an effective potential at large separation and then perform a (linearized) PB calculation at constant surface potentials, then a non-monotonic curve will result. By keeping the surface potentials constant, we allow the charge density to respond to the change in separation mimicking the change in overcharging seen in the simulations; see Figure 2. This procedure seems to capture the coupling between the overcharged surface and the multivalent counterions quite well, and what remains is the contribution of monovalent ions to the overall pressure, which is well described by the PB equation. It is, however, only a part of the ion-ion correlations that is taken into account, and the weakness is apparent for a system with a 2:2 electrolyte (Figure 8), where the simulated pressure is virtually always attractive and the renormalized PB calculations predict a significant repulsion. In the same manner, the latter will never predict the attraction between two equally charged surfaces in equilibrium with a multivalent salt. It should be emphasized that this type of calculation cannot replace the simulations because it depends on the renormalized description of the surface. Unfortunately, there does not seem to be any experimental source from which one could obtain an effective potential. Thus, the linearized PB calculations can be used only to rationalize the simulated pressure curves. Shklovskii and Zhang21,22 have proposed an approximate method to calculate the effective potentials in the case of overcharging surfaces. The idea is based on a division of the system into two parts. One is a strongly correlated liquid at the charged surface and a more gaslike phase further away from the surface. The latter is described by the PB equation, whereas the free energy of the first part is approximated using a Wigner crystal (WC) approach. The partitioning of the system might seem appealing, but the numerical results are definitely not quantitative and in some cases not even qualitatively correct (even if the interaction parameter is high). As an example, in Figure 10 we have plotted Φ-1 calculated from simulations and the WC approach. The latter overestimates the effective surface potential and the overcharging of the surface for a large range of concentrations. The use of such potential values leads to an overestimate of the repulsion. For example, consider the pressure at 10 mM 2:1 salt concentration, where simulations predict a negative surface potential but the WC theory leads to overcharging and strongly repulsive pressure; see Figure 11. In practice, the observed repulsion can partially explain the degradations of hydrated cementitious materials, in particular,
Trulsson et al.
Figure 11. Net osmotic pressure between two oppositely charged surfaces calculated from simulation (black) and the WC approach (red with circles). The charges are σ-1 ) -2 and σ1 ) 1 e/nm2, respectively. The bulk solution contains a 10 mM 2:1 salt.
concrete constructions. Concrete subjected to an internal (gypsum) or external (seawater, lake, etc.) source of sulfate is known to potentially undergo a progressive and profound reorganization of its internal microstructure.34,35 In civil engineering jargon, this is called a sulfate attack. For example, concrete undergoing sulfate attack is often found to suffer from swelling, spalling, and cracking.33,34 There is evidence indicating that the degradation also causes a significant reduction of the mechanical properties of concrete,34,35 which in the most severe cases requires partial reconstruction. The observed electrostatic repulsion between oppositely charged particles could explain the observed expansion of concrete for two reasons. First, hydrated cement is composed of a mix of different mineral hydrate phases bearing opposite charges. The hydrated cement is formed through the dissolution/ precipitation process of an initial anydrous powder dispersed in water. The composition of the latter varies according to the type of cement, but it mainly contains tricalcium silicate and small amounts of tricalcium aluminate and gypsum. The precipitated hydrates are calcium silicate hydrate (C-S-H) and a large panel of aluminate hydrates (monosulfoaluminate, hydroxyaluminate, carboaluminate, and ettringite).36 C-S-H is negatively charged because of the titration of the surface silanol groups, and aluminate hydrates possess a structural positive charge due to the substitution of calcium ions by aluminum ions in the structure.36 Second, the expansion is known to be more pronounced for cement rich in aluminate and at high sulfate concentration. Further experimental and theoretical studies are performed to verify this hypothesis.
Conclusions From the simulations, we find that the addition of asymmetric salts to a system composed of oppositely charged particles may lead to a repulsion between the particles. This repulsion is strong for asymmetric salts and is virtually zero when the salt is symmetric. The repulsion is a result of ion-ion correlations, that allow a huge accumulation of salt in between the oppositely charged particles/surfaces, thereby generating a large osmotic pressure. The requirement for significant repulsion is that one surface has a high surface charge density and the multivalent ion of the asymmetric salt as the counterion. The surface charge density of the second surface is not crucial and can be close to neutral. Concomitant with the appearance of a repulsive pressure, one can also observe a charge reversal of the surface with (33) Khan, M. O.; Jo¨nsson, B. Biopolymers 1999, 49, 121. (34) St-John, D. A.; Poole, A. W.; Sims, I. Concrete Petrography: A Handbook of InVestigatiVe Techniques; Arnold: London, 1998. (35) Skalny, J.; Marchand, J.; Odler, I. Sulfate Attack on Concrete; EFN SPON: London, 2001. (36) Thorvaldson, T.; Vigfusson, V. A.; Larmour, R. K. Trans. R. Soc. Can. 1927, 21, 295.
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multivalent counterions. Note, however, that there is a region of repulsive interaction before the charge reversal takes place. That is, both phenomena are due to ion-ion correlations, but the repulsive pressure is not a consequence of the charge reversal per se. The repulsion can be enhanced by adding small amounts of monovalent salt. Correlations become more important in highly coupled systems, and, for example, a decrease in the dielectric permittivity will lead to a stronger repulsion. (37) Taylor, H. F. W. Cement Chemistry; Thomas Telford: London, 1997.
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In practice, the observed repulsion can explain the early swelling of hydrated cement rich in aluminate (positively charged) and calcium silicate (negatively charged) phases. The swelling is known to be particularly pronounced at high sulfate concentrations. Acknowledgment. Stimulating discussions on the setting of various cement types with Andre Nonat, Universite de Bourgogne, are gratefully acknowledged. This work was supported by the Foundation for Strategic Research, Sweden. LA701222B
