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Interaction of Nanometric Clay Platelets Bo Jo¨nsson,*,† C. Labbez,‡ and B. Cabane§ Theoretical Chemistry, Chemical Center, POB 124, S-221 00 Lund, Sweden, Institut de Carnot Bourgogne, UMR 5209 CNRS, UniVersite de Bourgogne, Faculte de Sciences Mirande, 9 AV. Alain SaVary, B.P. 47870, 21078 Dijon Cedex, France, and PMMH, ESPCI, 10 rue Vauquelin, F-75231 Paris Cedex 05, France ReceiVed April 9, 2008. ReVised Manuscript ReceiVed July 10, 2008 The free energy of interaction between two nanometric clay platelets immersed in an electrolyte solution has been calculated using Monte Carlo simulations as well as direct integration of the configurational integral. Each platelet has been modeled as a collection of charged spheres carrying a unit chargesthe face of a platelet contains negative charges, and the edge, positive charges. The calculations predict that a configuration of “overlapping coins” is the global free energy minimum at intermediate salt concentrations (10-100 mM). A second weaker minimum, corresponding to the well-known “house of cards” configuration, also appears in this salt interval. At low salt concentrations the electrostatic repulsion dominates, while at intermediate concentrations electrostatic interactions alone can create a net attraction between the platelets. At sufficiently high salt content (>200 mM), the van der Waals interaction takes over and the net interaction becomes attractive at essentially all separations. From the calculated free energy and its derivative, we can derive a yield stress and elasticity modulus in fair agreement with experiment. The roughness of the platelets affects the quantitative behavior of the free energy of interaction but does not alter the results in a qualitative way. From the variation of the free energy of interaction, we would tentatively describe the phase behavior as follows: At low salt, the interaction is strongly repulsive and the dispersion should appear as a solid (“repulsive gel”). With increasing salt concentration, the repulsion is weakened and a liquid phase appears (“sol”). A further increase of the salt content leads a second solid phase (“attractive gel”) governed by attractive interactions between the platelets. Finally, at sufficiently high salinity, the clay precipitates due to van der Waals forces.
Introduction Many clay materials swell in water and yield aqueous dispersions with unusual flow properties. Particularly remarkable are synthetic clays of the hectorite type, such as laponite, as well as natural montmorillonite clay. These materials are made of nanometric platelets with a thickness of ∼1 nm and varying lateral dimension, which for laponite is about 30 nm1ssee Figure 1. When water is added to a laponite powder, the clay platelets become ionized and a rising osmotic pressure in the interstitial solution causes the grains of powder to swell with water. With appropriate procedures, a clear homogeneous mixture of clay in water can be obtained.2-5 “Semidilute” laponite dispersions with volume fractions above 0.5% behave as gels (they have a yield stress).6,7,3,8,9 In continuous shear, they have a high viscosity at low shear rate and low viscosity at high shear rate with an unequalled degree of shear thinning and progressive thixotropic restructuring after shear.6,9 Upon drying, they form films with excellent barrier properties. These dispersions are often used in † ‡ §
Chemical Center. Universite de Bourgogne. PMMH, ESPCI.
(1) Rockwood Additives Limited: Cheshire WA8 0JU, UK. (2) Mourchid, A.; Delville, A.; Lambard, J.; Lecolier, E.; Levitz, P. Langmuir 1995, 11, 1942. (3) Pignon, F.; Magnin, A; Piau, J.-M.; Cabane, B.; Lindner, P.; Diat, O. Phys. ReV. E 1997, 56, 3281. (4) Mourchid, A.; LeColier, E.; van Damme, E.; Levitz, P. Langmuir 1998, 14, 4718. (5) Mongondry, P.; Tassin, J. F.; Nicolai, T. J. Colloid Interface Sci. 2005, 283, 397. (6) Pignon, F.; Magnin, A.; Piau, J.-M. J. Rheol. 1996, 40, 573. (7) Pignon, F.; Piau, J.-M.; Magnin, A. Phys. ReV. Lett. 1996, 76, 4857. (8) Cocard, S.; Tassin, J. F.; Nicolai, T. J. Rheol. 2000, 44, 585. (9) Martin, C.; Pignon, F; Piau, J.-M.; Magnin, A.; Lindner, P.; Cabane, B. Phys. ReV. E 2002, 66, 021401.
Figure 1. CRYO-TEM image of a dilute aqueous dispersion of hectorite clay particles (laponite RD) adapted from the work of Negrete-Herrera et al.10 The platelets are embedded in a thin film of vitreous ice. Since the platelets are extremely thin (1 nm), only those that are oriented parallel to the electron beam and normal to the film surface appear as dark filaments. The scale bar is 100 nm. Note that Laponite RD contains inorganic pyrophosphate.
the formulation of surface coatings, paper and polymer films, and household and personal care products.1 In applications, it is important to understand how the rheological or film forming properties of the dispersions derive from microscopic features of the clay platelets. Generally, these unusual properties are said to be related to an odd distribution of ionic sites on the platelets.11-19 In water at the equilibrium pH of (10) Negrete-Herrera, N.; Letoffe, J.-M.; Putaux, J.-L.; David, L.; BourgeatLami, E. Langmuir 2004, 20, 1564. (11) Neumann, B. S.; Sansom, K. G. Israel J. Chem. 1970, 8, 315. (12) Neumann, B. S.; Sansom, K. G. Clay Miner. 1970, 8, 389.
10.1021/la801118v CCC: $40.75 2008 American Chemical Society Published on Web 09/19/2008
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Figure 2. Schematic picture of a clay platelet with 556 negative and 100 positive sites drawn with thick lines.
laponite, the faces of the platelets carry negative charges and the edges carry positive charges leading to a net negative charge on the platelets, which is compensated by counterions that have been released to the aqueous phasessee Figure 2. The charge distribution creates a possibility for the edges of one platelet to be attracted to the face of a neighboring particle. There is strong experimental evidence that such edge-face attractions are the cause of the unusual rheological properties of laponite dispersions. Indeed, the addition of pyrophosphate ions, which bind to the edges and compensate their positive charges, causes a dramatic loss of the yield stress and viscosity.3,9,20 The importance of edge-face interactions has led to the conclusion that the platelets must spontaneously organize in such a way as to optimize these interactions. It has been further proposed21,3,22,9 that what best optimizes these interactions is the so-called “house of cards” configuration, in which neighboring particles are oriented more or less at right angles to each other, in a T configurationssee Figure 3b (we will in the following use the word structure for an assembly of platelets and reserve the word configuration for the geometric arrangement of two particles). This model has been called upon to explain the gel state and other properties of the dispersions in so many instances that it has become some kind of a reference model for dispersions of small clay particles in water. Variations of this structure have also been proposed, where small groups of particles, arranged as “stacked plates”, are in contact through their edges with another group of particles that have a nearly perpendicular orientation. The experimental evidence regarding these structures is surprisingly limited. A number of X-ray and neutron scattering experiments have been performed with semidilute laponite dispersions.24,6,3,25,9 These experiments confirm that the particles are aggregated. However, they do not discriminate between (13) Neumann, B. S.; Sansom, K. G. Clay Miner. 1971, 9, 231. (14) Fripiat, J.; Cases, J.; Francois, M.; Letellier, M. J. Colloid Interface Sci. 1982, 89, 378. (15) Ramsay, J. D. F.; Avery, R. G.; Benest, L. Faraday Disc. Chem. Soc. 1983, 76, 53. (16) Ramsay, J. D. F.; Swanton, S. W.; Bunce, J. J. Chem. Soc. Faraday Trans. 1990, 86, 3919. (17) Rosta, L.; von Gunten, H. R. J. Colloid Interface Sci. 1990, 134, 397. (18) Thomson, D. W.; Butterworth, J. T. J. Colloid Interface Sci. 1992, 151, 236. (19) Tawari, S. L.; Koch, D. L.; Cohen, C. J. Colloid Interface Sci. 2001, 240, 54. (20) Mongondry, P.; Nicolai, T.; Tassin, J. F. J. Coll. Inteface Sci. 2004, 275, 191. (21) van Olphen, H. An Introduction to Clay Colloid Chemistry, 2nd ed.; John Wiley and Sons Inc.: New York, 1977. (22) Nicolai, T.; Cocard, S. Eur. Phys. J. E 2001, 5, 221. (23) Martin, C.; Pignon, F.; Magnin, A.; Meireles, M.; Lelievre, V.; Lindner, P.; Cabane, B. Langmuir 2006, 22, 4065. (24) Shalkevich, A.; Stradner, A.; Bhat, S. K.; Muller, F.; Schurtenberger, P. Langmuir 2007, 23, 3570. (25) Levitz, P.; Lecolier, E.; Mourchid, A.; Delville, A.; Lyonnard, S. Europhys. Lett. 2000, 49, 672.
Figure 3. Two platelets at different separations and orientations. (a) “Overlapping coins” configuration. (b) House of cards configuration. (c) Configuration of stacked platelets. The center-center separations are 300, 200 and 50 Å, respectively. The negatively charged faces are drawn in light red, while the positively charged edges are drawn as dark blue.
different configurations of neighboring particles. This is because the experimental spectra result from an average over all configurations in the irradiated volume and the orientational averaging makes it difficult to distinguish between a structure with dominant T configurations and a structure with parallel plates and a distribution of spacings of the platelets. Unambiguous structural information is available only for concentrated dispersions, where macroscopic alignment of the platelets has been obtained.23 In this case, the interference patterns show that the structure is made of parallel layers, with a layer thickness that is close to the thickness of a particle (Figure 4a). However, there
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a pseudosite with appropriate charge and size, have been studied by several investigators.2,27-29 The crucial question is how to perform this coarse graining. It is obvious that quadrupoles are not good representatives of platelets that have oppositely charged faces and edgessthe quadrupole-quadrupole interaction is only the leading term, and due to geometry, particle size, and salt content, it is far from enough. Indeed, the combinations of edge-face and edge-edge interactions of platelets can be much more diverse than what is obtained with the interactions of quadrupoles. Since this balance between the two types of charges is essential, it has not been possible to make accurate predictions for the phase diagram of the dispersions nor of their rheological properties. We believe that the representation of charged sites has to be accurate down to a length scale corresponding to the screening length of the salt or better. The crucial point is that the particle size is larger or comparable to the Debye-Hu¨ckel screening length. On the basis of this observation we have chosen to approach a model clay suspension by a detailed study of the interaction between a pair of platelets. This allows us to calculate the potential of mean force between the pair, i.e. integrate over the angles describing the relative orientation. Various order parameters as well as angular averages are also obtained. We use these results to address the following questions: • What are the preferred configurations of a system of two disks with negative charges on the faces and positive charges on the edges? • Are the configurations relevant to the organization of nanometric clay particles in semidilute aqueous dispersions? • Do they make it possible to explain the rheological properties of the dispersions according to their structures and to the surface chemistry of the particles? Figure 4. Interference patterns obtained from neutron scattering by laponite dispersions that have been concentrated by uniaxial compression: 23 (a) Plane of scattering vectors parallel to the axis of compression and normal to the laponite layers. The two bright spots originate from diffraction of the neutrons by laponite layers. (b) Plane of scattering vectors normal to the axis of compression and parallel to the laponite layers. The absence of diffraction spots implies that there is no regular repetition within the directions of a layer.
is no regular repetition in the directions of the layers (Figure 4b), which appears to preclude a structure made of stacked plates arranged in parallel columns. Therefore the organization of concentrated dispersions neither matches the house of cards nor the stacked plates structure. Moreover, the structural experiments on less concentrated or semidilute dispersions do not provide supporting evidence for either of these structures. There have also been a number of attempts to predict the organization of such dispersions through theoretical approaches, and several electrostatical models of the clay platelets have been investigated. In order to obtain equilibrium structure and equilibrium properties, most authors have used a system made of a large number of particles and calculated average quantities that characterize the organization of these particles. The difficulty is, of course, the enormous complexity that originates from the need to describe not only the positions of particles but also their orientations. This has been tackled by simplifying the description of the orientations. Dijkstra et al.26 modeled laponite clay particles as quadrupoles and found structures corresponding to the socalled house of cards. Slightly more detailed models, where a collection of charged sites on the clay platelet were replaced by (26) Dijkstra, M.; Hansen, J. P.; Madden, P. A. Phys. ReV. Lett. 1995, 75, 2236.
Model and Methods We attempt to model the interaction between two clay particles in a salt solution, where each clay particle is modeled as a collection of discrete charged spheres forming a planar structure. The separation between two neighboring sites is 1.2 nm giving an area/charge of 1.44 nm2 or surface charge density of 0.11 C/m2. The hard core radius of the charged sites has been set to 0.75 nm. This choice has been guided by the article of Martin et al.9 A schematic picture of such a clay platelet is shown in Figure 2. The number of charged groups on a laponite clay particle is of the order of 700-1000, and in the calculations, we have used 656, which corresponds to a platelet radius of Rp ≈ 18 nm. Preliminary calculations with smaller platelets produced the same qualitative results as the particles with 656 sites. The face of a platelet is negatively charged while the edge is positively charged. In most calculations, the number of positive charges has been 100 giving a line charge density for the edge of 0.9 e/nm, which is slightly lower than the value given by the manufacturer.30 Calculations with higher line charge densities on the edges show the same qualitative features and in fact emphasize the results to be presented in the next section. We have used the dielectric continuum model, the so-called primitiVe model, where the solvent is described as a structureless medium solely characterized by its relative dielectric permittivity, εr. In the primitive model, one treats all charged species as charged hard spheres and the interaction is purely Coulombic. Preliminary calculations using this Hamiltonian turned out to be rather timeconsuming, and we decided to use an even simpler description. By only treating the salt implicitly, one can reduce the number of particles (27) Meyer, S.; Levitz, P.; Delville, A. J. Phys. Chem. B 2001, 105, 9595. (28) Odriozola, G.; Romero-Bastida, M.; Guevara-Rodriguez, F. de J. Phys. ReV. E 2004, 70, 021405. (29) Mossa, S.; deMichele, C.; Sciortino, F. J. Chem. Phys. 2007, 126, 014905. (30) Laponite Technical Bulletin, L104/90/A; Laporte Industries Ltd., 1990; 1.
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Figure 5. Schematic representation of two platelets defining the relevant angles. Θ1 and Θ2 are the angles between the normals to the platelets and the connection vector between their center of mass. The angle φ is the projected angle between the two normals.
and speed up the calculations. Thus, in the final model, the sites on two clay particles interact with a screened Coulomb potential,
lB exp(-κr) βu (r) ) ZiZj r
r > dhc
SC
βuSC(r) ) ∞
r < dhc
(1) (2)
where Zi is the ion valency, e the elementary charge, lB ) e2/4πε0εrkT the Bjerrum length, β ) 1/kT, and dhc is the hard sphere diameter of the ion. The effect of salt comes via the inverse screening length κ. After inclusion of the attractive van der Waals (vdW) term, the total interaction between two sites becomes,
( )
βutot(r) ) βuSC(r) - CvdW
dhc r
6
r > dhc
(3)
The value of CvdW has been in the range 0.1-0.2. In order to obtain the free energy of interaction, we have to average over all possible orientations, collectively denoted by Ω, of the two clay platelets. The center of mass for the two particles are kept fixed at a separation R and we define a connection Vector as the vector between the center of massssee Figure 5. Thus, we can define the free energy of interaction as,
βA(R) ) -ln
∫ dΩ exp[-βU(R, Ω)] + const
(4)
where U(R, Ω) ) The evaluation of the integral can be done in several ways, for example, by direct integration or by a Monte Carlo simulation.31 In order to partly characterize the orientational behavior of a pair of platelets, we have introduced an order parameter defined as, Σi,jutot(rij).
1 Ξ ) [3 cos2 Θ12 - 1] 2
(5)
where Θ12 is the angle between the two normalsssee Figure 5. The house of cards configuration in Figure 3b will have an order parameter Ξ < 0, while a stacked configuration like that in Figure 3c will have Ξ ≈ 1. A configuration with the two platelets in the same plane as in Figure 3a will also have an order parameter close to unity. It will, however, be distinguished from the stacked configuration by different average angles for and . We will in the following refer to a configuration with Ξ ≈ 1 and Θ1 ≈ Θ2 ≈ 90° as the overlapping coins configuration. Direct Integration. Since the platelets are approximately circular, we may limit the integration of eq 4 to three angles. These are the following: (i) Θ1, the angle between the normal to platelet 1 and the connection vector, (ii) Θ2, the angle between the normal to platelet 2 and the connection vector, and (iii) φ, the “dihedral” angle between the two normal vectors as shown in Figure 5. The limitation to these (31) Metropolis, N. A.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A.; Teller, E. J. Chem. Phys. 1953, 21, 1087.
Figure 6. Potential of mean force between two clay particles with 556 negative and 100 positive sites. The thick solid line is from an MC simulation, while the thin line with symbols is obtained via the direct integration procedure. The salt concentration is 40 mM and CvdW ) 0.1.
three angles means that the fine details of charge distribution, in particularly on the edge, are partially averaged out. In this integration procedure, with a typical step length of one degree, we can of course calculate the average value of any physical or structural quantity. For example, the distance dependent but orientationally averaged energy is,
)
∫ dΩ U(R, Ω)exp[-βU(R, Ω)] ∫ dΩ exp[-βU(R, Ω)]
(6)
The corresponding R-dependent angular entropy is then S(R) ) [ - A(R)]/T. Monte Carlo Simulation. The direct integration is a very convenient way from a numerical point of view to calculate the free energy and other thermodynamic and structural quantities. Unfortunately, the number of dimensions has to be rather small and in the present system we assume that the platelets have circular symmetry, which is not strictly correct. Monte Carlo (MC) simulations on the other hand can give an exact answer, but they have a tendency to be rather complicated and time-consuming when minima in the free energy are deeper than a few kT. In the present study, we have used the standard Metropolis algorithm31 with random displacement of the interparticle separation, R, together with random rotations of the two platelets. The rotations have been performed using the three Euler angles for each particle. Between 10 and 100 million attempted translations/rotations have been used with an acceptance ratio of approximately one-third. The simulated free energy curves with deep minima have been obtained from several simulations using a window technique in R. That is, the free energy curves presented below usually consist of several MC runs, each performed in a limited interval in R, “a window”. The different windows overlap and the set of free energy curves can be connected into one single curve. The range in free energy that can be covered with acceptable accuracy in a single run is about 5-8kT. MC simulations and direct integration produces qualitatively the same results, although the fine details may differ. Figure 6 shows a typical comparison between the two techniques.
Results Free Energy of Interaction. Some of the results obtained here for the system of two platelets are similar to what one would expect for any system of two identically charged particles. Indeed, when the edges of the platelets are negatively charged, as the faces, then the platelets repel each other at all separations. At low ionic strength, this repulsion is strong and long-range (this range is the Debye screening length). At high ionic strength, the repulsion still has a long-range tail that extends much beyond the Debye screening length, up to the particle diameter as seen in Figure 7. The screened Coulomb potential between two point
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Figure 7. Potential of mean force between two clay particles with all sites negatively charged giving a net charge of -656 e for three different salt concentrations (mM) as indicatedsfrom a direct integration of eq 4 with CvdW ) 0.0. The dashed lines show the corresponding screened Coulomb potential for two point particles, each with a charge of -656 e.
particles, each carrying the same net charge as the clay platelet, is also included for comparison. The point charge model is a rather poor approximation for essentially all separations as long as the platelet size Rp > κ-1. In the other limit, Rp < κ-1, the long-range part of the repulsion is of course well-described by the point charge model, while the short-range repulsion is underestimated. More interesting results are obtained when the edges carry a charge opposite to that of the faces of the platelets. However, for this feature to have significant effects, the salt concentration must be neither too low nor too high. At low ionic strengths (approximately 200 mM), most electrocstatic interactions are screened, and the interaction is dominated by van der Waals and excluded volume forcesssee below. In the following, we examine the interactions of platelets with positively charged edges, at salt concentrations (10-100 mM) such that electrical interactions are still important, but the screening length is short compared with the particle dimension. The most interesting result is the existence of configurations that lower the free energy for a particular center-center distance of the platelets. This occurs for platelets that have edges with a positive charge, at intermediate ionic strengths. Figure 8 shows that the potential of mean force may have two well-separated minima. The deeper, outer minimum is located at about 300 Å, which is slightly below the diameter of the platelets (360 Å). The weaker, inner minimum is found at ≈190 Å, which is the center-center separation for a T-shaped configuration. Of course, in order to identify the configurations that yield these minima in the potential of mean force, we need to examine the orientional correlations of the platelets. An increase of the salt concentration will eventually smear out both minima and the net interaction becomes determined by van der Waals attraction. Orientational Order Parameter. The existence of preferred orientations of the platelets corresponding to the minima in Figure 8 is evident in the variations of the orientational order parameter Ξssee Figure 9. The order parameter itself is not enough to deduce the configuration of the platelets in the minima, but we also need information about the average values of the two angles Θ1 and Θ2. Due to the symmetry of the system, it is, however, more informative to calculate the average of max[Θ1, Θ2] and min[Θ1, Θ2]. These average angles are displayed in Figure 10.
Figure 8. Free energy of interaction between two clay particles with 556 negatively and 100 positively charged sitessfrom MC simulations. Salt concentrations are indicated in the figure and CvdW ) 0.1.
Figure 9. Simulated order parameter, Ξ, as a function of separation and salt concentration. The platelets carry 556 negatively and 100 positively charged sites and CvdW ) 0.1.
Figure 10. Average max and min of the two angles (Θ1, Θ2) as a function of separation. The platelets contain 656 charged sites whereof 100 are positively charged, and the arrows indicate the position of minima in the corresponding potential of mean force. The vdW parameter is zero, the salt concentration is 40 mM, and the results are obtained via a direct integration, eq 4.
Let us follow the behavior of the order parameter as a function of platelet separation in a 40 mM solution: At very short distances,
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Figure 11. Free energy of interaction between two platelets at high salt concentration and a van der Waals coefficient, CvdW ) 0.1 as obtained from the direct integration procedure, eq 4. Each platelet consists of 556 negative and 100 positive sites. The solid line corresponds to infinite salt concentration, and the line with squares, to 300 mM of salt, respectively. The platelets have in both cases been perfectly planar. The curve with circles corresponds to a salt concentration of 300 mM, but the charged sites in the platelets have been randomly displaced, (2.5 Å, perpendicular to the platelet.
Ξ ≈ 1, meaning that the platelets are parallel. Since the distance is much shorter than the radius and the average max and min of Θ1 and Θ2 are close to zero (see Figure 10), this corresponds to the configuration of stacked plates. When the center-center distance exceeds the radius, the order parameter switches abruptly to negative values, indicating that the platelets have flipped into a nearly perpendicular configuration. This is the T configuration, which matches the configuration of platelets in the much publicized house of cards configuration. Since the potential of mean force has a weak minimum at this distance (Figure 8), this may be a favored configuration at intermediate ionic strengths (between 50 and 100 mM). Then, at a center-center distance of about one diameter, Ξ switches back to unity, indicating that the platelets have flipped to another configuration in which they are nearly parallel. Since the distance is slightly less than one diameter and the average max and min of Θ1 and Θ2 are close to 90° (see Figure 10), this configuration must be that of two overlapping coins as shown in Figure 3a. Moreover, since this configuration corresponds to the deepest minimum in free energy, it is, in a certain salt concentration range, the preferred configuration of the platelets. Finally, when the center-center distance becomes equal to the diameter or larger, the orientational order parameter switches back to zero. In this range, the platelets are far apart and the potential of mean force is very small, hence their orientations are uncorrelated. Effect of van der Waals Interaction. The van der Waals interaction plays only a minor role as long as the salt concentration is not too high. Above 100 mM, however, it is an important attractive contribution, and for even higher salt concentrations or in the presence of divalent ions, it dominates and gives rise to a strong net attractive interaction. The van der Waals term will favor a stacked configuration as can be seen from the very short center-to-center separation in Figure 11. If the number of positive charges on the edge is increased and the amount of negative charges on the face is reduced, then the minimum for the overlapping coins configuration becomes deeper. Conversely, if the amount of positive edge charge is reduced then both the minima corresponding to the overlapping coins and the house of cards configurations will disappear. This balance between repulsive and attractive electrostatic interactions is modulated by the van der Waals interaction. Figure 12 shows how the vdW interaction strengthens the attraction in general and in particular for the overlapping coins configuration. A consequence of the amplifying effect of the vdW interaction is that the amount of positive edge charge can be reduced, while still maintaining a
Figure 12. Comparison of the free energy of interaction between two platelets with and without a vdW interaction as obtained from the direct integration procedure, eq 4. (a) Salt concentration of 60 mM and the platelets carrying 556 negatively and 100 positively charged sites. (b) Salt concentration of 40 mM and the edge charge, Z+, and the vdW parameter varied.
Figure 13. Free energy, thermal energy, and entropy, A ) - TS, as a function of separation between two platelets with 556 negative and 100 positive sites each. The results are obtained from the direct integration procedure, eqs 4 and 6. The salt concentration is 40 mM and CvdW ) 0.1.
free energy minimum for the overlapping coins configuration. Figure 12b shows how a reduction of the edge charge from +100 to +80 e extinguishes both the overlapping coins and the house of cards configurations. With a vdW minimum of 0.2kT, the overlapping coins but not the house of cards configuration is regained. Figure 13 shows the free energy and its energy and entropy components as a function of the center-center separation. The free energy is a balance between an attractive energy term and a repulsive entropy term. The energy of interaction for a pair of platelets arranged as in the overlapping coins configuration and with fixed angles has been reported by Odriozola et al.28 They found a deep energy minimum of the order of 15kT, but an even deeper minmimum for the T configuration of about 20kT. These results were obtained for fixed orientations of the platelets and are not directly comparable to the present results, but nonetheless, they point to the possible importance of the overlapping coins configuration. The results presented so far has been for perfectly
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flat platelets, except one of the curves in Figure 11. This is of course one out of many deficiencies in our model, and one can easily imagine that a slightly irregular platelet could alter the relative importance of different free energy minima. In an attempt to mimick a particle roughness, we modified the previous platelet structure by allowing a fixed random displacement of the sites perpendicular to the plateletsthe maximum displacement was (2.5 Å. The simulations show, not unexpectedly, that the depth of the free energy minimum is reduced, but the final conclusion is still that the overlapping coins configuration is the global free energy minimum.
Discussion The aim of this discussion is to reexamine the questions that were raised in the Introduction, regarding the preferred configurations of a pair of platelets, the relation of these configurations to the structures of laponite and other hectorite clay dispersions, and the consequences for the rheological behavior of these dispersions. Configurations of Two Disks. The central result of this work is the discovery of configurations of the two-disks system for which the free energy (potential of mean force) is below that at infinite separation. This occurs for platelets that have positively charged edges at intermediate ionic strengths. The configurations correspond to specific center-center distances of the particles. Hence, if these distances are accessible, i.e. at appropriate particle concentration, the particles will spontaneously aggregate in one of these configurations. The most favorable configuration, from the free energy point of view, is that of overlapping coins, in which the platelets are aligned and slightly overlapping. For platelets with edges that have a positive charge, this configuration can lower the free energy by several kT in the absence of any vdW attractions, and even more in presence of vdW attractions. Moreover, the free energy minimum that results from the overlap of the platelets is quite narrow. This narrow potential well results from the competition of edge-face attraction, that grows according to the length of edges that overlap the faces, and face-face repulsion, that grows according to the area of overlap. The range of ionic strengths for which this configuration is favorable is rather broad, extending from approximately 10 to 60 mM in absence of vdW attractions (at lower ionic strength the face-face repulsion is too strong, and at higher ionic strength the edge-face attraction vanishes). Another local free energy minimum is the T configuration that corresponds to the house of cards configuration. For platelets with edges that have a positive charge, this configuration can lower the free energy by a few kT. It corresponds to a rather broad minimum in the variation of free energy with centercenter distance, presumably because the angle of the platelets and the point of contact can vary without causing much change in the interaction. The range of ionic strengths for which this minimum exists is 40-100 mM independent of vdW attractions. It is interesting to compare the relative energy of this configuration with that of the overlapping coins. At ionic strengths between 10 and 50 mM, the system will settle as overlapping coins; at higher ionic strengths, between 50 and 100 mM, the two configurations will compete to some extent. Finally, there are configurations for which the free energy is not favorable with respect to infinite separation, but which may still be preferred if the particle distances or orientations are constrained. This is the case for center-center distances that are much less than the particle radius (R < 150 Å). In this range, the free energy rises quite fast with decreasing center-center
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separation. However, the orientational order parameter remains near unity over this whole range of distances, indicating that the configuration is that of stacked plates. Thus, the stacked plates is not a preferred configuration with respect to distances, but it may be the least unfavorable configuration if the platelet separation is forced to be short (high concentration). Structures of Aqueous Dispersions. The question is whether these favorable configurations are relevant to the “organization” of nanosized clay particles in semidilute aqueous dispersions. By “organization” we mean (a) where the particles are located (if they phase separate or not) and (b) how neighboring particles are arranged with respect to each other. Concerning the first point, particles will aggregate only if the salt concentration is larger than 5-10 mM. The actual value can vary with the magnitude of the vdW parameter and the amount of edge charge. Reasonable choices for these model parameters do not significantly change the aggregation threshold. In the literature, there is an apparent conflict between (a) osmotic pressure measurements,2 which yield a positive osmotic pressure when laponite dispersions are equilibrated with an aqueous phase of ionic strengths 0.1, 5, and 10 mM, indicating that the particles repel each other under these conditions, and (b) observations from scattering studies,5,9,32 which reveal a slow aggregation process in dispersions made by mixing laponite powder with aqueous solutions containing 1 mM of NaCl. However, it is important to note that the ionic strength in the dispersions is not 1 mM. Indeed, when such dispersions were centrifuged,9 it was found that the concentration of Na+ in the supernatant was 6.5 mM, and therefore, the total ionic strength must be in excess of 10 mM. Thus, our prediction that the particles repel at ionic strengths below 5-10 mM is in agreement with the osmotic stress experiments, and our prediction that the particles attract a higher ionic strengths matches the observations made with scattering techniques, where the ionic strength was actually higher than 10 mM. Concerning the second point, the free energy profiles and also the orientational order parameter profiles show that the particles will either be in the house of cards or in the overlapping coins configuration. At ionic strengths below 50 mM, the overlapping coins configuration is more favorable, whereas above 50 mM the two free energy minima are of more comparable magnitudessee Figure 8. This is an important finding, since until now it has been postulated that semidilute dispersions would always settle in the house of cards structure. Thus, we need to examine the consequences of either configuration for the structure and properties of the dispersions. On the basis of the variation of the free energy of interaction, we may tentatively describe a phase diagram for clay platelets as follows: For a given, not too high, clay concentration and low ionic strength, the interaction between the platelets is strongly repulsive giving rise to a “repulsive gel” or Wigner glass. With increasing salt concentration, the repulsion is diminished and a liquid phase (“sol”) appears. A further salt increase brings the system into a regime where the overlapping coins configuration is a free energy minimum. This phase can be described as an “attractive gel” in contrast to the Wigner glass. At very high salt concentrations, the electrostatic interactions become unimportant and the system is governed by attractive van der Waals interactions leading to a precipitation of the clay platelets. At sufficiently high clay concentrations, then the screening from the counterions is enough to bring the system directly into a solid phase with net (32) Michot, L. J.; Bihannic, I.; Maddi, S.; Baravian, C.; Levitz, P.; Davidson, P. Langmuir 2008, 24, 3127.
Interaction of Nanometric Clay Platelets
attractive electrostatic interactions and a liquid phase will never appear. This scenario, with small modifications, is in agreement with experimental phase diagrams.2,33,24 The overlapping coins configuration imposes a nearly complete alignment of the disks (see Figures 9 and 10), leaving very little orientational freedom. The angular constraint can also be seen from the entropy profile in Figure 13. For a dispersion containing many platelets, this will inevitably lead to the formation of layers. Within a layer, neighboring platelets will be in the overlapping coins configuration. Adjacent layers will pack at a separation set by the overall volume fraction of platelets. This is exactly the structure that was found through neutron scattering experiments of concentrated laponite dispersions, with a regular repetition of layers that have the thickness of 1 particle and no peridodicity in the plane of the layers.23 These findings raise a serious question concerning the range of existence of the house of cards structure. According to the results presented here, the T configuration is only a free energy minimum at intermediate salt content, 40-100 mM, and it is never the global free energy minimum. Flow Behavior of the Dispersions. From the previous discussion, we infer that most particles will be in the overlapping coins configuration in all dispersions with a volume fraction exceeding 0.1. Accordingly, the mechanical properties of these dispersions must be related to the interaction free energy of two particles in this configuration. For example, consider the case with 556 negatively and 100 positively charged sites and a vdW parameter CvdW ) 0.1 dispersed at a volume fraction φ ) 0.016 in an aqueous solution of 1 mM. The total ionic strength, including the counterions, is approximately 10 mM. Under these conditions, the depth of the free energy minimum for the overlapping coins configuration is 7kT corresponding to an energy density/particle of 450 J/m3. In order to produce a flow in this dispersion, it will be necessary to apply a stress that is comparable to this energy density. Hence, the yield stress must be of the order of 450 Pa, which compares favorably with the yield stress measured by Pignon et al.6 of 200 Pa. The house of cards configuration would, under the same conditions, be repulsive meaning zero yield stress (33) Tanaka, H.; Meunier, J.; Bonn, D. Phys. ReV. E 2004, E69, 031404.
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as can be seen from the free energy curve in Figure 8a. Thus, the strong yield stress of laponite dispersions is nicely explained by the deep free energy minimum for the overlapping coins configuration. The elastic modulus of laponite gels have been measured by several authors. Martin et al.9 report an elastic modulus, G′, between 1000 and 2000 Pa for a volume fraction of 1.6% and a salt concentration of 6.5 mM. We can estimate the force constant in the overlapping coins configuration in Figure 8a for a salt concentration of 10 mM to approximately 0.01 N/m. From the approximate relation
∂2A 2 R ∂R2 G′ ≈ 3V
(7)
where R is the center-to-center separation in the coin configuration and V is the volume per particle. We get an average modulus of 4000 Pa, which is in fair agreement with experiment. In eq 7, we have assumed an idealized situation with all particles joined into some sort of “living polymer” and that all chains contribute to the elasticity.
Conclusions We have calculated the free energy of interaction for a pair of model clay platelets, where the faces are negatively and the edges positively charged. At low,
