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Controlling the Cohesion of Cement Paste Bo Jo¨nsson,*,† A. Nonat,‡ C. Labbez,‡ B. Cabane,§ and H. Wennerstro¨m| Theoretical Chemistry, Chemical Center, POB 124, S-221 00 Lund, Sweden, LRRS, UMR CNRS 5313, Universite de Bourgogne, F-21078 Dijon Cedex, France, PMMH, ESPCI, 10 rue Vauquelin, F-75231 Paris Cedex 05, France, and Physical Chemistry 1, Chemical Center, POB 124, S-221 00 Lund, Sweden Received April 19, 2005. In Final Form: July 8, 2005 The main source of cohesion in cement paste is the nanoparticles of calcium silicate hydrate (C-S-H), which are formed upon the dissolution of the original tricalcium silicate (C3S). The interaction between highly charged C-S-H particles in the presence of divalent calcium counterions is strongly attractive because of ion-ion correlations and a negligible entropic repulsion. Traditional double-layer theory based on the Poisson-Boltzmann equation becomes qualitatively incorrect in these systems. Monte Carlo (MC) simulations in the framework of the primitive model of electrolyte solution is then an alternative, where ion-ion correlations are properly included. In addition to divalent calcium counterions, commercial Portland cement contains a variety of other ions (sodium, potassium, sulfate, etc.). The influence of high concentrations of these ionic additives as well as pH on the stability of the final concrete construction is investigated through MC simulations in a grand canonical ensemble. The results show that calcium ions have a strong physical affinity (in opposition to specific chemical adsorption) to the negatively charged silicate particles of interest (C-S-H, C3S). This gives concrete surprisingly robust properties, and the cement cohesion is unaffected by the addition of a large variety of additives provided that the calcium concentration and the C-S-H surface charge are high enough. This general phenomenon is also semiquantitatively reproduced from a simple analytical model. The simulations also predict that the affinity of divalent counterions for a highly and oppositely charged surface sometimes is high enough to cause a “charge reversal” of the apparent surface charge in agreement with electrophoretic measurements on both C3S and C-S-H particles.
Introduction Cement is one of the most widely used building materials in the world, and it is amazingly variable in its composition. The earliest known cements were made in Roman times from lime, volcanic ash, and clay, so-called Pozzolanic cements.1,2 Modern Portland cement is made from limestone and clay, which are heated to a high temperature, 1500 °C, producing a cement clinker. The clinker product is a polyphasic material including calcium silicates (Ca3SiO5dC3S, 50-70 wt % and Ca2SiO4, 10-30%), calcium aluminate, and aluminoferrite (5-20%). Because of the natural origin of the raw materials including possible pollutants and the different fuels used to burn the clinker, it may also contain a few percent of alkali oxides or sulfates. Finally, a variety of waste materials, including urban sludge3-5 and industrial waste,6-8 are often burnt together with the minerals as a convenient way to dispose of them. The volumes of waste that are incorporated in this way can be substantial, amounting to about 5% secondary material and 80% alternative fuel. The final cement product is obtained by grinding the clinkers with about 5% gypsum (CaSO4). At an early stage, after mixing cement with water and sand, concrete is a fluid cement paste with embedded †
Theoretical Chemistry, Chemical Center. Universite de Bourgogne. § PMMH. | Physical Chemistry 1, Chemical Center. ‡
(1) Vitruvius: Ten Books on Architecture; Cambridge University Press: Cambridge, U.K., 1999. (2) MacLaren, D. C.; White, M. A. J. Chem. Educ. 2003, 80, 623. (3) Romeu, J. M. Cemento-Hormigon 1994, 65, 756. (4) Ochsenreiter, C.; Kuyumcu, H. Cem. Int. 2004, 2, 58. (5) Klaska, R.; Baetzner, S.; Moeller, H.; Paul, M.; Roppelt, T. Cem. Int. 2003, 1, 88. (6) Scheur, A. Cem. Int. 2003, 1, 48. (7) Trezza, M. A.; Scian, A. N. Cem. Concr. Res. 2000, 30, 137. (8) Wanzura, F.; Wendt, B. ZKG Int. 2003, 56, 53.
gravels. The paste is a concentrated suspension of 10100 µm grains immersed in an aqueous solution; the volume fraction is typically around 40%. As soon as the cement constituents are in contact with water, they begin to dissolve, and the surface of the grains, mainly C3S, becomes negatively charged. The solution contains from the very beginning calcium, potassium, and/or sodium, sulfate, hydroxide, silicate, and aluminate ions. The C3S particles form a network in the first minutes after mixing because of attractive forces between them;9,10 see Figure 1a. The solution is quickly supersaturated with respect to calcium silicate hydrate (CaO-SiO2-H2O ) C-S-H) and calcium sulfoaluminate hydrate (ettringite), which precipitate. The dissolution-precipitation continues, while maintaining low concentrations of silicates and aluminates in the solution. Initially, pH is increasing until the precipitation of calcium hydroxide (portlandite) occurs. Among the precipitated hydrates, the main component is C-S-H, which constitutes at least 60% of the fully hydrated cement paste. It is generally recognized as responsible for the setting and hardening of cement because it precipitates at the surface of the anhydrous calcium silicate grains, preferentially first at the contact between them. AFM experiments11 reveal that C-S-H particles, immersed in a Ca(OH)2 solution, attract each other at short range (2 nm) provided that pH exceeds 12. The strength of the paste increases during the hydration process because of the augmentation of the number of contact points between the cement grains created by C-S-H particles; see Figure 1b. Mechanical experiments10 show that the intrinsic properties of the network remain the same throughout the hydration process. A fully (9) Jiang, S. P.; Mutin, J.-C.; Nonat, A. Cem. Concr. Res. 1996, 26, 491. (10) Nachbaur, L.; Mutin, J.-C.; Nonat, A.; Choplin, L. Cem. Concr. Res. 2001, 31, 183. (11) Lesko, S.; Lesniewska, E.; Nonat, A.; Mutin, J.-C.; Goudonnet, J.-P. Ultramicroscopy 2001, 86, 11.
10.1021/la051048z CCC: $30.25 © 2005 American Chemical Society Published on Web 09/01/2005
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Figure 1. Schematic drawing of early cement paste with sand, C3S particles drawn as hatched, approximately spherical objects, and C-S-H platelets drawn as black bars. (Left) At an early stage, water has been added, and the C3S particles have formed a weak network. C3S has also started to go into solution, and the precipitation threshold of C-S-H has been reached. The C-S-H particles form preferentially close to the contact points of the C3S particles. (Right) At a later stage, a significant portion of C3S has dissolved, and a corresponding amount of C-S-H has been created. The contact points between the C3S grains mediated by the C-S-H particles has increased in number, and so has the cohesion of the paste.
hydrated cement paste can exhibit a high compressive strength, >100 MPa, whereas its tensile strength is low (∼2 MPa).12 This is probably due to the fact that the elastic limit of the material is small: the critical strain that cement paste can support is dhc
(3)
r < dhc
(4)
where e and 0 are the elementary charge and dielectric permittivity of vacuum, respectively. These interactions are applied to all ions in a “box” defined by the charged walls separated a distance of h. The size of the simulation box is LLh, where the lateral dimensions (L) are determined by the net ionic charge of ions in the Monte Carlo (MC) box. Besides their mutual interactions, the ions also interact with the charged surfaces in the simulation box. The potential generated by the two squared surfaces of dimension L2 is
Φ(z) )
xL /2 + (z + (sh/2))
σ
∑ 4L ln 4π s)1,-1 0 r
2
2
x(L/2)
2
+ L/2
+ (z + (sh/2))2
(5)
-2|z + (sh/2)| × (L/2)4 - (z + (sh/2))4 - (L2/2)(z + (sh/2))2 π arcsin + 2 {(L/2)2 + (z + (sh/2))2}2
[
]
where Φ(z) is the electrostatic potential in position z and the two charged surfaces are placed at (h/2. Electrostatic interactions are long-ranged, and a correction term approximating interactions outside the MC box has also been added.18,32 Thus, the total energy is a sum of ionion, ion-wall, and wall-wall interactions plus a longrange correction term
Utot ) Uii + Uiw + Uww + Ulr
(6)
The numerical procedure we follow here has been tested in a number of previous double-layer simulations, and we feel confident about its stability17,18 Osmotic Pressure. The osmotic pressure of the confined solution, pconf osm , may be calculated according to either of the following expressions17,33
σ2
∑i ci(wall) - 2
pconf osm ) kBT
r 0
∑i ci(mp) + p
pconf osm ) kBT
corr
+ phc
(7)
where mp stands for midplane. These relations are exact (31) Lund, M.; Jo¨nsson, B.; Pedersen, T. Mar. Chem. 2003, 80, 95. (32) Jo¨nsson, B.; Wennerstro¨m, H.; Halle, B. J. Phys. Chem. 1980, 84, 2179.
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within the primitive model. The term pcorr comes from the fact that ions on either side of the midplane correlate and give an attractive contribution to the pressure. In the mean field description, this term disappears because of electroneutrality (i.e., there are no correlations across the midplane). Here we have used the second relation for the evaluation of the pressure because it tends to give the best accuracy. If the ions have a finite size dhc, then one also gets a hard core term, phc, that describes the additional pressure due to the finite volume of the ions. In some fortuitous cases, ion-ion correlation and ionic size effects compensate, and the net pressure is surprisingly well described by a mean field approximation.34 The hard core radius has a clear physical origin, that is, ions cannot overlap because of the quantum mechanical exchange repulsion. In simulations or other theories of electrolyte solutions, however, it tends to achieve the character of a fitting parameter. There is not a clear-cut choice of dhc. Should it be the bare ion diameter or that of a hydrated ion? Fortunately, for the present study of cement paste there is a relatively broad range for dhc where it has only a weak effect on the interparticle forces. An interesting discussion of the importance of the ionic size can be found in a combined experimental and theoretical study by Kekicheff et al.35 Equation 7 gives the osmotic pressure in the confined region, but the experimentally interesting quantity is the net osmotic pressure, bulk posm ) pconf osm - posm
(8)
where the bulk pressure is calculated for a bulk with the same chemical potential(s) as the double layer. The osmotic pressure in the bulk was calculated as36
pbulk osm )
[
〈Utot〉
1+ xbulk + 4φ∑∑gij(dhc)xbulk ∑i cbulk i i j 3k T i j
kBT
B
]
(9)
where φ is the volume fraction of ions, xi is the fraction of ions of type i in the bulk, and gij(dhc) is the contact value of the ion-ion correlation function. Monte Carlo Simulations. The electrostatic interaction described above defines the Hamiltonian (i.e., forms the basis for a MC simulation of the solution phase of cement paste). We use the standard Metropolis algorithm37 extended to a grand canonical ensemble.38 That is, in addition to the random movements of ions, there will also be the creation and anhilation of salt pairs. A random displacement of an already existing ion will be accepted if the new energy fulfills
exp(-∆U/kBT) > ξ
(10)
where ξ is a random number between zero and 1 and ∆U old ) Unew tot - Utot . A similar criterion applies for the creation of a salt pair at random positions in the box. For example, the creation of a sodium sulfate triplet is accepted if (33) Wennerstro¨m, H.; Jo¨nsson, B.; Linse, P. J. Chem. Phys. 1982, 76, 4665. (34) Kjellander, R.; Marcˇelja, S. J. Phys. (France) 1988, 49, 1009. (35) Kekiche, P.; Marcˇelja, S.; Senden, T. J.; Shubin, V. E. J. Chem. Phys. 1993, 99, 6098. (36) McQuarrie, D. A. Statistical Mechanics; Harper Collins: New York, 1976. (37) Metropolis, N. A.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A.; Teller, E. J. Chem. Phys. 1953, 21, 1087. (38) Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: San Diego, CA, 1996.
V3 × (NNa + 1)(NNa + 2)(NSO4 + 1) exp(µtot/kBT - ∆U/kBT) > ξ (11) where µtot is the total chemical potential for the sodium sulfate salt and ∆U is the change in electrostatic energy upon the insertions. A similar relation holds for the anhilation of a sodium sulfate triplet,
NNa(NNa - 1)NSO4 V3
exp(-µtot/kBT - ∆U/kBT) > ξ (12)
The numbers of co- and counterions vary, but the total number of ions was always larger than approximately 200, and in some high-salt cases, it increased to more than 1000 ions. Size convergence was ensured by performing simulations with double the number of ions. Each particle was subject to 105 attempted moves. Simulations with 106 moves/ion confirmed that convergence was achieved with respect to number of configuration. In the simulations of the confined solution, we apply periodic boundary conditions in two dimensions, that is, the directions parallel to the charged surfaces. In the third dimension, the system is limited by the charged walls, which are impenetrable to the ions. The results are equilibrium distributions of all ions in the confined solution, their average concentration, and the direct force acting between the two halves of the system shown in Figure 2. From this knowledge, we can calculate the osmotic pressure, and we can also analyze the different components of the pressure as defined in eq 7. Because these simulations are performed in the grand canonical ensemble, a necessary input is the total chemical potentials of the salt, µtot/kBT, in the corresponding bulk solution. Thus, initially we have to perform a set of simulations for the relevant bulk conditions. Here we used the ordinary canonical ensemble with the interactions following eq 3. Periodic boundary conditions were applied in three dimensions, and the number of configurations generated in the simulations was approximately an order of magnitude smaller than in the simulations of the confined solution. The chemical potential was calculated with the Widom insertion technique,39 where an ion is inserted at a random position, r, in the simulation box and the excess chemical potential is obtained from
µex ) -kBT ln〈exp[-∆U(r)/kBT]〉0
(13)
where ∆U(r) is the interaction energy between the inserted ion and all other ions in the box. 〈...〉 0 symbolizes an average over the unperturbed system, that is, the inserted ion is removed after calculating ∆U(r). The Coulomb interaction is long-ranged, and we have added a correction term to handle correlations between the inserted ion and ions outside the MC box.40 This modified insertion technique allows us to obtain single-ion activities. The salt chemical potential is then simply a sum of the relevant single-ion activities. An alternative route to avoid the complications of the long-ranged Coulomb interaction is to insert a neutral salt into the Widom procedure. This is, however, numerically inferior to the insertion of single ions, in particular, for asymmetric salts such as Na3PO4 consisting of several ions. The accuracy in µex/kBT is better than (39) Widom, B. J. Chem. Phys. 1963, 39, 2808. (40) Svensson, B. R.; Woodward, C. E. Mol. Phys. 1988, 64, 247.
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(0.005, and the accuracy in osmotic pressure is better than (2 mM. The latter is needed to calculate the net osmotic pressure; see eq 9. It is also a check of the consistency of the simulations because we know that the osmotic pressure in the confined solution should approach the bulk value when h f ∞. Simple Model. As a complement to the Monte Carlo simulations of the model in Figure 2, we first introduce a simple free-energy function for an even simpler model system. We argue that this system captures the essential physics of the cohesion in cement paste. It has the proper response to changes in counterion valency, added salt, solvent quality, and surface charge density. As a starting point, consider the model system in Figure 2. If the interaction between the counterions is strong, then they will become strongly correlated, and we can imagine a situation where each counterion is confined to its own little cylinder with the ends of the cylinder being the charged surfaces. That is, each cylinder contains only one counterion, and we assume that there is no interaction between two cylinders. The extreme of this is of course if the counterion, of charge Ze, moves only along the cylinder axis. To simplify the system further, we also replace the charged circular surfaces with a point charge equal to Ze/2, and finally we obtain the model system depicted in Figure 3. The distance δ is chosen such that the potential at the end points from the nearest surface is the same with a point charge as with a smeared out surface charge density on a circular area with radius R. That is, we have the condition
(Ze/2)Ze ) 4π0rδ
dr ∫0R Zeσ2πr 4π0rr
(14)
Noting that σ ) Ze/2πR2, we get
δ)
R 2
and
σ)
Ze 8πδ2
(15)
Thus, for a given counterion valency, we can use the distance δ as a measure of the surface charge density. Let us now estimate the energy by placing the counterion in the middle between the “surfaces”. The energy for that particular configuration is not as low as the average energy of the system. Likewise, we can overestimate the entropy by assuming that the distribution is uniform. Using such a heuristic approach, we can write the free energy of the system in Figure 3 as
7Z2lB A(h) h )- ln 2 ) kBT 4(h + 2δ) Zl B
-
7 4(S1 + 1/x2πS2)
- ln S1 (16)
where we have used the dimensionless parameters introduced in eqs 1 and 2. A constant, kT ln Z2lB, has also been added to make the entropic term dimensionally consistent. A more extensive discussion of the model in Figure 3 can be found in refs 41 and 42. Here we merely note that eq 16 contains terms that depends in a qualitatively correct way on the same dimensionless parameters as those of the more elaborate model, Figure 2. There is (41) Jo¨nsson, B.; Wennerstro¨m, H. When Ion-Ion Corelations Are Important in Charged Colloidal Systems. In Electrostatic Effects in Soft Matter and Biophysics; Holm, C., Kekiche, P., Podgornik, R., Eds.; Kluwer Academic Publishers: Norwell, MA, 2001. (42) Jo¨nsson, B.; Wennerstro¨m, H. J. Adhes. 2004, 80, 339.
Figure 3. Simplified model with two fixed “surface” charges and one mobile counterion of valency Z confined to a line of length h. The displacement of the surface charges, δ, is supposed to mimic the surface charge density.
Figure 4. Force between the two fixed charges in the simplified “double layer”. The counterion valency has been varied, whereas σ ) 0.07 C/m2.
a competition between an entropic term that dominates for large S1 (i.e. large separations) and also for very small S1, whereas the interaction term can dominate at intermediate separations provided that S2 is sufficiently large. The force, F, acting on the fixed charges is determined by the derivative
F)-
∂(A(h)/kBT) 1 7 )+ (17) ∂S1 4(S1 + (1/x2πS2))2 S1
By solving for the condition of F ) 0 at a finite value of S1, we obtain the requirement that S2 g 128/49π for a nonmonotonic variation of the force. One advantage of expressing the free energy in terms of dimensionless quantities is that it clearly shows that there is no fundamental relation between the counterion valency and the nature of the force. Rather it is one of the oddities of nature that for monovalent counterions in an aqueous system it is essentially impossible to reach such a high surface charge density that S2 exceeds the critical value 128/49π ≈ 0.83, whereas this is readily achieved for counterions of higher valency. As an explicit example, choose σ ) 0.07 C/m2 (∼1e/200 Å2) in an aqueous system at room temperature (lB ) 7.1 Å). For monovalent ions S2 ) 0.22, which is well below the critical value, whereas for divalent counterions we have S2 ) 1.76, which yields a force curve with a primary minimum. For Z ) 3, S2 ≈ 12, and as shown in Figure 4, this results in a strongly attractive force curve. For room-temperature and aqueous conditions, the limiting charge density for a nonmonotonic force curve corresponds to σ ) 0.26 C/m2, h ) 6.2 Å, Z ) 1; σ ) 0.033 C/m2, h ) 25 Å, Z ) 2; and σ ) 0.010 C/m2, h ) 55 Å, Z ) 3. These values are for the simplified model system, but for an exact MC simulation of the model system in Figure 2, approximately the same numbers will result. When σ increases beyond these values, then the minimum in the interaction moves to shorter separations according to
S1 )
1 7 ( 8 x2πS 2
7 ( 8 x)
2
-
7 4x2πS2
(18)
The plus sign gives the position of the maximum of the force curve. At high surface charge densities, S2 . 1, and
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then eq 18 predicts that the minimum occurs for S1 ≈ (4/7)/(2πS2). For comparison with simulation results, it is instructive to analyze how the depth of the minimum varies with counterion valency at high charge density. Using eq 16 and neglecting logarithmic terms, we find that in the minimum
A(hmin) 7 ≈ - x2πS2 kBT 4
(19)
The minimum in free energy corresponds to the adhesion energy per counterion. The adhesion energy is typically measured per unit wall charge, which we obtain dividing by Z/2,
xπZσ 2e
Aadh ≈ -7lB kBT
(20)
Thus, in the regime of strong electrostatic coupling, the adhesion energy is relatively weakly dependent on the counterion valency as well as on the surface charge density. This simple model system captures the main physical features of a planar double layer and the cohesion in a cement paste. The addition of a second salt can also be interpreted in this simple fashion. For example, adding a monovalent counterion to a system already containing a divalent one means that the “average” counterion valency is reduced and that the attraction is weakened. We can use the simple model to estimate the effect of the addition of a Na+ salt to the bulk that already contains a Ca2+ salt (the concentration of anions in the confined region is low and will be neglected). Let us by cCa and cNa denote the calcium and sodium concentrations in the bulk solution, respectively, which for simplicity we will treat as being ideal. This means that the free-energy difference between a calcium and a sodium ion in the bulk is given by
cCa ∆Abulk ) -ln kBT cNa
(21)
The corresponding difference in the confined solution is given by eq 22,
∆Aconf )kBT
7 Ca
4(S1
+ (1/x2πS2Ca)) Na
4(S1
+ 7 + (1/x2πS2Na))
(22)
With these free-energy differences, we can calculate the probability for di- and monovalent counterions, respectively, in the confined solution as a function of the relative salt concentrations in the bulk. The probability, QCa, is given by
QCa )
exp[-(∆Aconf - ∆Abulk)/kBT] exp[-(∆Aconf - ∆Abulk)/kBT] + 1
(23)
and similarly for QNa ) 1 - QCa. We can now calculate an approximate net osmotic pressure for a given bulk salt ratio from Na posm ) QCapCa osm + QNaposm
(24)
Na where pCa osm and posm are calculated from eq 17. Figure 5 shows that the cohesion is very robust and that the robustness increases with increasing surface charge
Figure 5. Competition between calcium and sodium ions as approximately described by the simple model. The curves show how the attractive minimum is reduced for different surface charge densities (0.15-0.5 C/m2) upon increasing the sodium concentration in the bulk. cNa and cCa are the bulk concentrations of sodium and calcium, respectively. The pressure is calculated from eq 25.
Figure 6. (Left) Co- and counterion profiles for a confined solution in equilibrium with a bulk solution containing 20 mM CaX2 and 100 mM NaX. The surface charge density is 0.32 C/m2. The bold solid line shows the calcium concentration, and the thin solid line shows the sodium concentration. The anion concentration is represented by the dashed line. (Right) The fraction sodium charge (see text) in the confined solution as a function of surface separation. The bulk contains 20 mM CaX2 + 200 mM Na2SO4 and σ ) 0.32 C/m2 (lower solid curve), 20 mM CaX2 + 200 mM Na2SO4 and σ ) 0.16 C/m2 (upper solid curve), 20 mM CaX2 + 1000 mM NaX and σ ) 0.32 C/m2 (lower dashed curve), and 1 mM CaX2 + 1000 mM NaX and σ ) 0.32 C/m2 (upper dashed curve).
density. For a range of physically reasonable surface charge densities, the cohesion survives in the presence of large numbers of monovalent counterions in the bulk. We will return to this issue later on using more accurate data from the MC simulations. The qualitative picture of Figure 5 will, however, remain the same. Thus, we can, on the basis of the results from this simple model, conclude that two properties are important for the cohesion of cement: (i) the presence of divalent calcium counterions and (ii) a high surface charge density (i.e., a high pH). Simulation Results Ion Distributions. The general behavior of co- and counterions found already with the PB equation survives to a large extent even in highly coupled systems such as cement. A few exceptions appear, and it is our intention to start by describing these before discussing the resulting pressures. Figure 6a shows concentration profiles when the bulk contains 20 mM CaX2 and 100 mM NaX. Initially, at short a distance, there will hardly be any co-ions in the confined solution, but at distances larger than 10-20 Å, the concentration will be essentially the same as in the bulk. The calcium and sodium ions will compete for the charged surfaces, and at short separations, calcium will be the dominating counterion. Again, we can see that the bulk values are reached at a separation of a few nanometers. The competition between counterions of different valency is an important property, which has a profound
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Figure 7. (Left) Normalized co-ion distribution for four different surface separations: h ) 5, 10, 20, and 40 Å. The bulk solution contains 80 mM CaX2. (Right) Co-ion concentration as a function of surface separation. Solid lines are the average slit concentration, and dashed lines show the mid-plane concentration. The corresponding bulk concentrations are indicated with arrows to the right. The surface charge density is 0.32 C/m2.
influence on the cohesion of cement. Figure 6b demonstrates how the surface charge distribution and the bulk concentration ratio affect the competition between sodium and calcium. The fraction sodium charge in the confined solution,
ηNa )
〈cNa〉 2〈cCa〉 + 〈cNa〉
(25)
is low at short separations and then gradually approaches the bulk ratio (〈...〉 denotes an average over the slit). At low surface charge density and with 20 mM CaX2 and 200 mM Na2SO4 in the bulk, the limiting value is 10/11, whereas the value at h ) 60 Å is slightly less; see the upper solid curve in Figure 6b. By doubling the surface charge density, we favor the calcium ions, giving the lower solid curve in Figure 6b. The two dashed lines in Figure 6b correspond to the same charge density but a different bulk ratio of calcium and sodium ions. Thus, we can conclude that a high surface charge density favors calcium ions as counterions and that with the surface charge densities studied here, >0.1 C/m2, we need a high bulk sodium concentration in order to exchange the calcium ions in the confined solution. These are two important factors for the cohesion of cement. Charge Reversal. We can also allow the confined solution to be in equilibrium with a bulk solution, which contains only divalent counterions. Without monovalent counterions, we find that the addition of CaX2 to the bulk leads to a gradual build up of a second layer outside the charged surface. That is, the surface is “more than neutralized” by counterions ,and as a consequence, a layer of co-ions is formed about 5-10 Å from the surface. The accumulation of co-ions is limited to very short separations, z < 10 Å, and it increases with salt concentration; see Figure 7. It is not possible to go to very large separations in the simulations, but it is our belief that the two maxima shown for h ) 20 and 40 Å in Figure 7a will survive even at infinite separation. The co-ion layer is promoted by a high surface charge density; see Figure 8a. It also increases with increasing bulk salt concentration as can be seen from the ionic profiles in Figure 7b. This layered structure appears in double-layer systems when the energetic term dominates over the entropy. The accumulation of counterions may lead to a charge reversal near the surface. Let us formally define an apparent charge density seen at a position z as
σapp(z) ) -σ0 +
z dz′ F(z′) ∫-h/2
Figure 8. (Left) Co-ion (X-) distribution for a surface separation of 40 Å. The surface charge density is varied, whereas the bulk concentration is kept fixed at 80 mM CaX2. (Right) Apparent surface charge density, σapp(z), in arbitrary units. The surface charge density is 0.32 C/m2, h ) 40 Å, and the bulk contains CaX2 solution of varying concentration as indicated in the graph.
With only monovalent counterions or low surface charge density, σapp(z) is usually a monotonic function going from -σ0 to zero when z varies from -h/2 to 0. A nonmonotonic behavior (i.e., a charge reversal) is seen at high surface charge density and in the presence of divalent counterions, and it increases with increasing salt concentration in the bulk. Figure 8b shows the charge reversal for a system in equilibrium with the bulk, which has a varying calcium concentration. The charge reversal has been seen in a variety of electrophoretic experiments and is often interpreted as a consequence of specific ion binding.29,43 What we see here is that electrostatic interactions alone are enough to create an apparently changed surface charge. The charge reversal is another manifestation of ion-ion correlations and is sometimes seen under the same conditions at which the attractive double-layer forces appear. The two phenomena should not be confused. For example, the attraction exists over a much wider parameter space than the charge reversal. The fact that the dielectric continuum model predicts a charge reversal in excellent agreement with experiment is of course additional evidence that the model catches important and complex physical mechanisms in a highly charged suspension. In an aqueous solution with monovalent counterions, we do not see any charge reversal, and it is only with multivalent counterions or polyelectrolyte counterions that it is found.35,44 Thus, a consequence of the successive addition of NaX is that the layering of co-ions starts to disappear, but more important is the fact that the concentration decreases as shown in Figure 9. This is in line with the discussion above, where we stressed that the increase in the number of monovalent cations in the bulk leads to an “effective counterion” with a valency of less than 2. Under these circumstances, the system becomes dominated by the entropy, and canonical behavior both with respect to ion distribution and forces can be expected. We note that the “layering” of co-ions is also promoted by the co-ion valency. That is, divalent or trivalent coions more easily lead to a nonmonotonic profile in the slit. Figure 9b shows the total co-ion profiles for four different bulk conditions. Without any monovalent counterions, there is a strong buildup of a co-ion layer, and the co-ion concentration also readily exceeds the bulk value. With the addition of NaX, this pattern disappears, whereas it persists in the presence of multivalent co-ions (i.e., SO42and PO43-), although it does not lead to a charge reversal;
(26)
where F(z′) is the charge density of the mobile ions and -σ0 is the surface charge density of a wall placed at -h/2.
(43) Viallis-Terrisse, H.; Nonat, A.; Petit, J.-C. J. Colloid Interface Sci. 2001, 244, 58. (44) Sjo¨stro¨m, L.; A° kesson, T.; Jo¨nsson, B. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 889.
Controlling the Cohesion of Cement Paste
Figure 9. Distribution of co-ions in the confined solution for varying bulk conditions. The concentration curves have been normalized with the corresponding bulk concentration. The surface charge density is 0.32 C/m2, and the bulk always contains 20 mM CaX2 in addition to the sodium salts. (Left) Addition of NaX. (Right) Addition of sodium salts with different co-ions. The sodium concentration is always 200 mM.
Figure 10. Apparent surface charge density, σapp(z), in arbitrary units. The surface charge density is 0.32 C/m2, h ) 40 Å, and the bulk contains 20 mM CaX2 and added sodium salts with different co-ions. The sodium concentration is always 200 mM.
see Figure 10. Layering is not directly related to charge reversal. As a matter of fact, multivalent co-ions are expelled from the confined solution as compared to monovalent ones. Still, they display a more nonuniform ionic profile in the slit. The charge reversal is found in systems with divalent counterions. The addition of a salt with a monovalent counterion such as Na+ reduces the charge reversal, and eventually it will disappear. Sodium salts with multivalent anions are in this respect more effective, and even small amounts of Na3PO4 lead to the disappearance of charge reversal (Figure 10). However, charge reversal is also dependent on the surface charge density, and it can reappear upon increasing the surface charge density. Cohesion. Below we discuss for a number of technically important cases how the osmotic pressure and the cohesion vary with bulk conditions. The latter include pH and added salt of different valencies. The pH determines the surface charge density of the C-S-H nanoparticles and affects the balance between attractive and repulsive forces. It has previously been shown for the counterion-only case24,26,28 and it is also apparent from the simple model above that it is only at high surface charge density and in the presence of divalent or multivalent counterions that cohesion sets in. The general feature of adding salt is that nothing happens except under rather extreme conditions. That is, large amounts of a sodium salt are required to extinguish the cohesion, and at high pH, it is essentially impossible to “salt out” the attractive force. Note that the pressure is reported in concentration units (posm/RT ) 1 M corresponds to posm ) 2.5 MPa). Addition of CaX2. Figure 11a shows how the attractive minimum is reduced when pH and the surface charge density is reduced. The position of the minimum is also displaced to larger separation, which is in agreement with
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Figure 11. Net osmotic pressure as a function of surface separation. (Left) Varying salt and surface charge density. The bulk CaX2 concentrations used are 1, 20, 40, and 80 mM, and they give rise to four overlapping curves for each surface charge density. (Right) The same as in the left-hand plot but with an even lower surface charge density of 0.08 C/m2.
the predictions of the simple model. Note that the graph actually contains 12 curves grouped 4 and 4. That is, the variation of the concentration of a 2:1 salt (e.g. calcium chloride) has essentially no effect on the force curves up to a concentration of 80 mM. The highest surface charge density used, 0.57 C/m2, corresponds to a very high pH (>13) and gives a very short ranged, strong attraction. By changing the surface charge density approximately an order of magnitude, 0.08 C/m2, we still obtain an attractive force, but it is now reduced by 2 to 3 orders of magnitude. We can also note from Figure 11b that there seems to be a weak salt dependence in this case and that the addition of a calcium salt actually increases the attraction at all separations. These results are in excellent agreement with experiment, where it is found that the addition of a calcium salt does not lead to improved cohesion if the surface charge density is high. At low surface charge density, however, the cohesion is seen to increase up to about 20 mM calcium ions in the bulk.45 Competition between CaX2 and NaX. One set of experiment of interest is when calcium ions are gradually replaced by sodium ions, keeping the hydroxide concentration constant. This means that pH and consequently the surface charge density are kept constant, whereas the original divalent calcium counterion is replaced by the monovalent sodium ion. In general, it is not possible to go much below ∼1 mM Ca2+ in a cement paste because of the solubility of C-S-H. (This is not strictly correct because at very high pH the calcium concentration may be well below 1 mM; however, the surface charge density will then be very high, and cohesion is still maintained.) However, concrete may contain “reactive” silica aggregates, which under basic conditions form polysilicate ions in the solution and trap the calcium ions.46,47 In this extreme case, only monovalent counterions remain, and the interaction turns repulsive, promoting cracks in the structure. Figure 12 shows that replacing calcium hydroxide with sodium hydroxide does not alter the cohesion unless the calcium concentration drops to submillimolar concentrations. Note that the fraction of monovalent counterions is small at separations below 10-15 Å; see Figure 6b. One can also imagine a situation where the calcium concentration is kept constant and an increasing amount of NaX is added to the bulk solution. With the addition of small amounts of NaX and highly charged C-S-H particles, we should not expect any major changes in the (45) Plassard, C.; Lesniewska, E.; Pochard, I.; Nonat, A. Langmuir 2005, 21, 7263. (46) Gaboraud, F.; Nonat, A.; Chaumont, D.; Craievich, A.; Hanquet, B. J. Phys. Chem. B 1999, 103, 2091. (47) Gaboraud, F.; Nonat, A.; Chaumont, D.; Craievich, A. J. Phys. Chem. B 1999, 103, 5775.
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Figure 12. Pressure as a function of surface separation with constant hydroxide concentration equal to 40 mM but a varying Ca/Na ratio in the bulk solution. Note that the 20, 1, and 0.2 mM curves coincide. The surface charge density is 0.57 C/m2.
Figure 13. Pressure as a function of surface separation for varying NaX concentrations in the addition to a constant amount of CaX2. The surface charge density is 0.32 C/m2. (Left) 20 mM CaX2 and (Right) 1 mM CaX2.
surface forces. With large amounts, however, there ought to be a competition between calcium and sodium counterions for the surface, leading to a weakening of the cohesion. The interesting question is at which NaX concentration this becomes an important effect. In the first set of simulations, we used a calcium concentration of 20 mM, and the resulting pressure curves can be seen in Figure 13a. With the addition of 1000 mM NaX, there is a clear reduction of the attractive pressure and an even more pronounced effect on the cohesive energy. If the concentration of divalent counterions in the bulk is reduced but the high surface charge density is maintained, then we see a weakened cohesion and a primary maximum appearing; see Figure 13b. (We can imagine that the charge density is maintained by the addition of NaOH.) This reduction of the cohesion is a consequence of ion competition. That is, with increased bulk NaX concentration more monovalent counterions will enter the confined solution, and the repulsive entropic term will increase. Addition of Na2SO4 and Na3PO4. So far, we have studied only the effect of adding monovalent co-ions to the bulk solution. From a technical point of view, calcium sulfate is always added to the cement clinker. It is used to regulate the setting of cement and avoid a flash set due to the precipitation of calcium aluminates. On the basis of experience, we should expect that the addition of a 2:2 salt to the bulk would maintain the cohesion while causing a more pronounced build up of the co-ion layer (cf. Figure 7). In real life, 2:2 salts are scarce, but we see from Figure 14 that the addition of 10 mM CaSO4, which corresponds to the solubility of gypsum, has virtually no effect on the cohesion. However, increasing the sulfate concentration by adding sodium sulfate means that the original divalent calcium counterions are replaced and the cohesion will eventually be lost. With a low surface charge density, 0.16 C/m2, it is enough with 200 mM Na2SO4 to eradicate the attraction; see Figure 14b. This is in qualitative agreement
Jo¨ nsson et al.
Figure 14. Osmotic pressure as a function of separation for varying bulk conditions. The bulk always contains 20 mM CaX2. The addition of 10 mM CaSO4 to the bulk has no effect on the pressure, and the curve coincides with the results with only CaX2 in the bulk (“no add”). The surface charge density is (left) 0.32 C/m2 and (right) 0.16 C/m2.
Figure 15. (Left) Osmotic pressure as a function of separation upon addition of Na3PO4 to the bulk solution. The bulk always contains 20 mM CaX2, and the surface charge density is 0.32 C/m2. (Right) Reduction of the cohesion obtained with sodium salts with varying valency of the anion. The sodium concentration is in all cases 200 mM, 20 mM CaX2 has been used as the reference, and the surface charge density is 0.32 C/m2.
with rheological experiments by Garrault et al.,48 who found that the addition of K2SO4 decreased the viscosity in commercial cement paste as well as in pure calcium silicate paste. Thus, it cannot be attributed to the reaction of sulfate with aluminate phases. The addition of phosphate has the same effect as adding sulfate both in the simulations and on the rheological properties.49 The viscosity at short time is also reduced because the cohesion is gradually reduced by competition between the original divalent counterion and monovalent sodium from the phosphate salt. With 100 mM Na3PO4 and a high surface charge density, there is still a strong cohesion as is evident from Figure 15a. There does not seem to be any strong dependence on the valency of the co-ion, and equivalent concentrations of NaX, Na2SO4, and Na3PO4 have the same weak effect on the cohesion. Figure 15b give quantitative support to this conclusion. Conclusions The present simulations based on a dielectric continuum model describe the essential interactions in cement paste. They provide an understanding of many phenomena that have been observed but remained unexplained with respect to the behavior of cement pastes at both short and long times. Despite large variations in composition and environmental conditions, the cement cohesion is preserved, except under rare extreme conditions. The robustness is explained by the Coulomb interactions between the charged surfaces of the C-S-H particles and the ions confined between these surfaces. The two main factors controlling the cohesion are the surface charge density and the valency of the counterions. Figure 16 summarizes (48) Garrault, S.; Nachbaur, L.; Sauvaget, C. Ann. Chim. 2003, 28, S43. (49) Benard, P.; Garrault, S.; Nonat, A.; Cau-Dit-Coumes, C. J. Eur. Ceram. Soc., in press, 2004.
Controlling the Cohesion of Cement Paste
Figure 16. Stability diagram. The curves show the approximate boundaries for an attractive net osmotic pressure as a function of the composition in the bulk solution. That is, to the left of the curves for the different surface charge densities (in units of C/m2) there exists a range in separation where the pressure is negative. The charge densities are directly connected to different pH values in the sense that a high surface charge density corresponds to a high pH in the solution.
the results from the simulations in a stability diagram for cement paste. It shows how the cohesion varies with the concentration of monovalent (sodium or potassium) and divalent (calcium) counterions in the bulk solution. Calcium and sodium ions compete for the charged surfaces and the concentration ratio of Na+ to Ca2+ in the bulk has to be very high in order to destroy the cohesion. This ratio has to be all the higher as the charge density increases.
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In practice, in commercial cement the monovalent ions (Na+, K+) are present as alkali sulfates. Just after mixing with water, the surface charge density is low, and high monovalent counterion concentrations may erradicate the attraction between the C3S grains. This explains why a higher fluidity of the cement paste is observed during the mixing and moulding of concrete under such conditions. In the mature cement paste, sulfate ions have been replaced by OH- ions, and the cohesion is then always ensured because an increase in alkali concentration is always correlated with an increase in pH and consequently an increase in the surface charge density. The present type of simulations can probably also be extended to describe the interaction between C-S-H nanoparticles and charged polymers. These so-called superplasticizers are used to fluidify modern concretes, which in some cases leads to unexpected phenomena such as cement-additive incompatibility. Work along these lines is in progress. Acknowledgment. Valuable comments from Roland Kjellander and Torbjo¨rn Åkesson and a generous grant from The Swedish Foundation for Strategic Research are gratefully acknowledged. LA051048Z
