Langmuir 1992,8, 1796-1805
1796
Structure of Liquids at a Solid Interface: An Application to the Swelling of Clay by Water A. Delville Centre de Recherche sur la Mati2re Divisbe, CNRS, l b rue de la Fhollerie, 45071 Orlbans Cedex 02, France Received January 25,1992. In Final Form: March 25, 1992
A molecular description of the clay-water interface is used to model the organization of the solvent in a slit-shaped pore. Monte Carlo simulations determine the water content of the pore as a function of the interlamellar distance. The water molecules are layered in successive shells, whose number (1-4)depends on the available interlamellar space. Despite this marked solvent layering, the solvation forces display no oscillations. The energies and pressures give a clear quantification of the driving forces responsible for clay swelling. Introduction Near a solid interface, the structure of an adsorbed liquid differs totally from the structure of the bulk liquid. Layering’ of the solvent occurring in the vicinity of the solid surface may induce oscillations of the solvation forces.2 The characterization of the actual state of the adsorbed liquid is crucial for an understanding of interfacial phenomena like wetting, swelling,or adhesion. Much experimental3-sand theoretical+32 work has been devoted to these problems. The purpose of this paper is to provide (1) Snook, I.; van Megen, W. J. Chem. Phys. 1979,70,3099-3105. (2)Israelachvili, J. ‘ Intermolecular and Surfaces Forces;Academic Press: London, 1985;Chapter 13. (3)Fripiat, J.; Cases,J.; FranGois, M.; Letellier, M. J.Colloidlnterface Sci. 1982,89,37&400. (4)Skipper, N.T.;Soper, A. K.; McConnell, J. D. C.; Refson, K. Chem. Phys. Lett. 1990,166,141-145. (5)Hartman, H.; Sposito, G.; Yang, A.; Manne, S.; Gould, S. A. C.; Hansma, P. K. Clays Clay Minerals 1990,38,337-342. (6)Israelachvili, J. N.; Pashley, R. M. Nature 1983,306,249-250. (7)Viani, B. E.; Roth, C. B.; Low, P. F. Clays Clay Minerals 1985,33, 244-250. (8)Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991,353, 239-241. (9)Kjellander, R.; Marcelja, S.; Pashley, R. M.; Quirk, J. P. J. Phys. Chem. 1988,92,6489-6492. (IO) Svensson, B.; Jdnsson, B.; Woddward, C. E. J. Phys. Chem. 1990, 94,2105-2113. (11) Valleau, J. P.; Ivkov, R.; Torrie, G. M. J. Chem. Phys. 1991,95, 520-532. (12)Delville, A.; Laszlo, P. New J. Chem. 1989,13,481-491. (13)Abraham, F. F. J. Chem. Phys. 1978,68,3713-3716. (14)Sullivan, D. E.; Levesque, D.; Weis, J. J. J. Chem. Phys. 1980,72, 1170-1174. . (15)Snook, I. K.; van Megen, W. J. Chem. SOC.,Faraday Trans. 2 1981, 77,181-190. (16)Plischke, M.; Henderson, D. J.Chem. Phys. 1986,84,2846-2852. (17)Finn, J. E.; Monson, P. A. Phys. Rev. A 1989,39,6402-6408. (18)Schoen. M.: Cushman. J. H.: Diestler. D. J.: Rhvkerd. C. L. J. ~~
~~
~
(19) Sokolowski, S.:Fischer, J. Mol. Phys. 1990,71,393-412. (20)Sloth, P. J. Chem. Phys. 1990,93,1292-1298. (21)Nezbeda, I.; Reddy, M. R.; Smith, W. R. Mol. Phys. 1990, 71, 915-929. .-..-. . (22)Christou, N. I.; Whitehouse, J. S.; Nicholson, D.; Parsonage, N. G. Faraday Symp. Chem. SOC.1981,16,139-149. (23)JGnsson, B. Chem. Phys. Lett. 1981,82,520-525. (24)Lee, S.H.; Rasaiah, J. C.; Hubbard, J. B. J. Chem. Phys. 1986, 85,5232-5237. (25)Aloisi, G.; Foresti, M. L.; GuideYi, R.; Barnes, P. J.Chem. Phys. 1989,91,5592-5596. (26)Zhu, S . B.; Robinson, G. W. J. Chem. Phys. 1991,94,1403-1410. (27)Raghavan, K.;Foster, K.;Motakabbir, K.; Berkowitz, M. J. Chem. Phys. 1991,94,2110-2117. (28)Low, P. F.;Cushman,J. H.; Diestler, D. J. J.Colloid Interface Sci. 1984,100,576-580. (29)Kjellander, R.; Marcelja, S. Chem. Scr. 1985,25, 73-80. (30)Skipper, N.T.; Refson, K.; McConnell, J. D. C. J. Chem. Phys. 1991,94,7434-7445. (31)Delville, A. Langmuir 1991,7,547-555.
a theoretical model of the clay-water interface based on a molecular description of the solvent molecules and the solid surface. Such an approach is necessary to give an exact appraisal of the structure of the adsorbed liquid. We restrict our study to the clay-water interface because the clay surface is well characterized33 and because clay suspensions are used in many industrial applications: heterogeneous catalysis, cracking, drilling, gelling, decolorizing, etc. The same approach may, however, be used for other systems. The first section gives details of the model used for the Monte Carlo simulations of the clay--cation-water interface. A pore slit is formed by two clay sheets facing each other. A grand canonical procedure allows the determination of the water content of this pore as a function of interlamellar distance. It also determines the structure of the water molecule layers induced by the interlamellar Na+ counterions and by the clay surfaces. Finally, the solvation forces are estimated by a canonical simulation of the distribution of the water molecules in the pore.
Computational Methods (A) The Model. Many theoreticalstudieshave been published on this topic.*32 The MacMillan-Mayer approach is a first level of approximation. It treats only interactions among ‘solute” species (clay surfaces and Na+ counterions) and ignores the structure of the solvent molecules. In the framework of the primitive one only takes into account long range electrostatic interactions among the solute species with shortrange excluded volume effects. This is the basis of the DLV0%vM theory which perfectly reproduces the stability of colloidal suspensions at large interlamellar distances37(largerthan 20 A). The driving force responsible for the swelling of the charged surfaces is entr0pic;~8it is linearly proportional to the activity of the labile small ions present in the suspension. This ionic activity may be successfully approximated12 by the local ionic concentration midway between the charged surfaces. This explains the widespread use of Poisson-Boltzmann treatments10~3783* to calculate the distribution of labile ions in the interlamellar space. (32)Delville, A.; Sokolowski, S. To be submitted for publication. (33)Maegdefrau, E.; Hofmann, U. 2.Kristallogr., Kristallgeom. Kristallphys. Kristallchem. 1937,98,299-323. (34)Carley, D. D. J. Chem. Phys. 1967,46,3783-3788. (35)Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (36)Derjaguin, B.; Landau, L. D. Acta Physicochim. URSS 1941,14, 635. (37)Dubois, M.; Zemb, Th.; Belloni, L.; Delville,A.; Levitz, P.; Setton, R.J. Phys. Chem. 1992,96,2278-2286. (38)Delville, A.; Laszlo, P. Langmuir 1990,6,1289-1294.
0743-7463/92/2408-1796$03.0Q~Q 0 1992 American Chemical Society
Langmuir, Vol. 8, No.7, 1992 1797
Structure of Liquids a t a Solid Interface
Table I. MINDO Local Charge (e) of the Clay Atoms in the Unit Cell* Si Si Si Si 0 0
Figure 1. Model of montmorillonite used in the simulations. However, at small interlamellar distances (lessthan 20 8,)this approach is no longer valid since the structure of the solvent must be taken into account. Thiais the Born-Oppenheimer level of approximation. Many results have been published with solvent molecules treated as hard20P2lor soft spheresl3-lSin the presence of smooth hard or soft walls. Only few developments simulate real solvent molecules in the presence of real molecular surfaces, like metal2' or clay particle^.^"^^ The structure of a montmorillonite particle is similar to a stack of sandwichesconsisting of a layer of octahedral aluminum oxides, between two layers of silicium oxide.3s Figure 1 displays a top view of the two upper layers, showing the hexagonal cavities at the surface of the silicium oxides. These cavities are occupied by a hydroxide group with the hydrogen atom oriented parallel to the plane of the particle and pointing to the vacancy of the aluminum network. This is characteristic of montmorillonite dioctahedral clays.% For trioctahedral clays like he~torite,3~ there is no vacancy in the octahedral network and the hydrogen at the center of the hexagonal cavity is perpendicular to the plane of the particle. The ideal formula of clay montmorillonite is SiaAl&(OH)4,3s but this perfect clay does not exist. Real clays contain defects: some Al(II1) atoms of the octahedral network are replaced by Fe(I1)or Mg(I1)a t o m ~ ~ ~ *some ~ a nSi(1V) d atoms of the tetrahedral network are replaced by Al(II1) a t 0 m s , 3 ~ + ~ leading to a residual negative charge on the clay. This charge is compensated by exchangeable interlamellar cations (Na+,K+, Ca2+,,,,),3994'3 which may be solvated. The clay sheet may extend laterally to distances up to ca. 10 OOO 8, and may stack to form tactoids. The of these tactoids and their swelling ability are strongly correlated to the hydration e n t h a l p ~ ~ofl ,the ~ ~interlamellar counterions. Our model31932 simulates the organization of water molecules confined in a pore formed by two identical clay particles facing each other and including the neutralizing Na+ counterions. This molecular description of the clay-water interface requires the knowledge of the interaction potentials between each pair of particles: water-water, water-Na+, clay-water, and clay-Na+. The Na+-Na+ interaction is only electrostatic;the neglect of shortrange interactions is justified by the large separation between the cations. The water-water potential is the empirical potential TIP4P of Jorgensen,avu compatible with a Na+-water potential.4345 The clay-water and clay-Na+ potentials are obtained by a parametrization of MINDO quantum energies3l calculated for many configurations of one water molecule or Na+ cation in the vicinity of the fragment of montmorillonite shown in Figure 1. The coordinates of the atoms forming this clay particle were obtained by X-ray diffraction33 of Na+-montmorillonite, The dangling bonds are saturated by hydrogen atoms (not drawn in Figure 1). The partial tetrahedral substitution of Si(1V) atom by an Al(II1) atom is also taken into account.31 The water (39) Theng, B. K. G. The Chemistry of Clay-OrganicReactions;Adam Hilger: London, 1974. (40) Low, P. F. Soil Sci. SOC.Am. J. 1980, 44,667-676. (41) Cebula,D.J.;Thomas, R. K.;White, J. W. J.Chem. Soc.,Faraday Trans. I 1980, 76,314-321. (42) Schra", L. L.; Kwak,J. C. T. Clays Clay Miner. 1982,30,40-48. (43) Bopp, Ph. Software-Enturicklungin der Chemie;SpringerVerlag: Berlin, 1987; pp 69-98. (44) Jorgeneen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J . Chem. Phys. 1983, 79,926-935. (45) Bounds, D. G. Mol. Phys. 1984,54, 1335-1355.
0 0 0 0 0 0 0 0 0 0 A1 A1 A1 A1 0
z
X
Y
0.0 0.0 2.59 2.59 1.295 3.885 3.885 1.295 0.0 2.59 0.0 2.59 2.59 0.0 5.29 0.0 1.727 1.727 4.317 4.317 0.863 3.453 0.863 3.453 0.863 3.453 2.127 -0.463 0.863 3.453 0.863 3.453 0.863 3.453
2.99 5.98 1.495 7.475 2.2425 2.2425 6.7275 6.7275 4.485 8.97 2.99 1.495 7.475 5.98 4.485 8.97 8.97 2.99 4.485 7.475 1.495 2.99 4.485 5.98 7.475 8.97 5.285 9.77 1.495 2.99 4.485 5.98 7.475 8.97
0 0 0 0 0 H H H H H H H H a Coordinates are given in A.
0 0 0 0 0.59 0.59 0.59 0.59 0.59 0.59 -1.59 -1.59 -1.59 -1.59 -1.59 -1.59 -2.68 -2.68 -2.68 -2.68 -3.77 -3.77 -3.77 -3.77 -3.77 -3.77 -1.68 -1.68 -4.7 -4.7 -4.7 -4.7 -4.7 -4.7
P
-1.7 -1.7 -1.7 -1.7 0.93 0.93 0.93 0.93 0.93 0.642 0.93 0.93 0.93 0.93 0.642 0.642 -1.1 -1.1 -1.1 -1.1 0.29 0.642 0.29 0.642 0.29 0.29 -0.21 -0.14 -0.14 -0.21 -0.21 -0.21 -0.21 -0.21
molecule is rigid, with an H-0 distance of 0.9572 8, and an H0-H angle of 104.5°.43*u The clay-water potential is the sum of two interactions: electrostatic and van der W a a l ~ The . ~ ~electrostatic ~ ~ ~ energy results from the interaction between the atomic charges of the water molecules and clay particle, obtained by a Mulliken& population analysis of the MINDO molecular orbitals. The atomic charges of the clay particle are given in Table I. The MINDO charges of the atoms of the water molecule are 0.31 e and -0.155 e for 0 and H, respectively. The electrostatic energy is given by the relation a,
c4i4i
u,, = 437~0 r j
(1) rij where the running indices i and j describe the atoms of the water molecule and the atoms of the clay particle, respectively. The a, parameter is necessary to enable the description of the electrostatic quantum energy by a set of discrete charges. It takes into account the polarization of the MINDO orbitals of water contributing almost half the permanent dipole of water. The van der Waals contribution is calculated by the empirical formula
The parameters of eq 2 are displayed in Table 11. The aij parameters are not free: they must satisfy the constraint aoj = -2aw (3) in order to avoid a divergence of the long-range term of eq 2. The Morse formulation provides a better description of the repulsive term.31 component of the MINDO energy than the classical rl* In the same manner, the clay-sodium interaction is composed of an electrostatic and a van der Waals contribution. The
(46) Mulliken, R. S. J . Chem. Phys. 1955, 23, 1822.
1798 Langmuir, Vol. 8, No. 7, 1992 ij
j
Si
4 Albt
0 H
Table 11. Parameters of Equation 2 aii, A2kJ/mol bii, ABkJ/mol cii, kJ/mol 0.1122&+03 1.157e+16 0.84956e+02 2.539e+21 0.35097e+02 1.642e+17 2.789e+14 -0).11175e+03 4.456e+03 -0.80267e+02 1.299e+04 -0).5614Oe+02 2.623e+01 -0).42478e+02 -0.17549e+02 5.755e+03 4.810e+04 0.55875%+02 2.323e+02 0.40134e+02 Table 111. Parametere of Equation 5 aj, A6 kJ/mol bj, A9 kJ/mol cj, kJ/mol 0.2373e+05 0.4565e+05 0.2763e+05 -0.3607e+08 0.1682e+03 0.2106e+04 9.758e+03 0.2138e+02 0.4337e+05 0.1641e+02 0.1961e+05 1.102e+02
..,
Deluille dii, A-1
15.85 13.27 19.57 12.12
1.49 2.45 10.42 2.67 4.83 8.96
dj, A-1 X
1.651 0.1434
electrostatic energy is calculated by the relation uel =
ZJ Pj 'YN~
(4)
with CYN. set to 1.2845. The empirical van der Waals formula for the energy is given by
with the values given in Table I11 for the various parameters. A continuous field approximation is used in order to incorporate the long-range electrostatic ion-ion and ion-dipole interactions cut by the minimum-image method. For this purpose, we introduce an external field resulting from charges calculated by a preliminary Poisson-Boltzmann treatment of the diffuse layer.474s This classical procedure is equivalent to an Ewald summationa and ensures the convergence of the counterion concentration profile and energ~.~~-w The TIP4P model of bulk water'3sU assumes four interaction sites: the two hydrogen atoms, with a charge of -0.52e, the neutral oxygen atom, and a fourth neutralizing site of 1.04e located on the CZaxis at 0.15 A from the oxygen toward the H atoms. The water-water energy is made up of two contributions: the electrostatic energy due to the nine charged centers and the oxygen-oxygen van der Waals interaction, given by the empirical relation with Am = 2.511208 x 106 A12 kJ/mol and BOO= 2.55395 x 103 kJ/mol. The Na+-water i n t e r a c t i ~ n ~used ~ y ~is compatible with the TIP4P model.apU This energy is the sum of a threecenter electrostatic interaction and a van der Waals interaction between the Na+ ions and the atoms of the water molecule. The Na+-Oxygen van der Waals interaction is given by A6
exP(-b~.#) - C ~ ~ 0- DNadr6 /r~ (7) with A N ~ O 1.144 X lo5 kJ/mol, bN.0 = 3.5455 A-1, C N ~=O1.820 X 1@A4kJ/mol, and&, = -3.515 X l@A6kJ/mol. The empirical Na+-H van der Waals interaction is given by uvdw=
uvdw = AN, exP(-bN,r) (8) with AN, = 8.612 X lo3 kJ/mol and bN, = 3.3940 A-l. The clay particles used in the Monte Carlo simulations extends 31.08 and 35.88 A in the X and Y directions, respectively. This corresponds to 24 hexagonal cavities (see Figure 2). The charge density of the clay particles is set to 7.17 x 10-9 e/Az; it is
(47) Jdneeon, B.; WennerstrBm,H.; Halle, B. J.Phys. Chem. 1980,84, 2179-2185. (48)Torrie, G. M.; Valleau, J. P. J. Chem. Phys. 1980,73,5807-5816. (49) van Megen, W.; Snook, I. J. Chem. Phys. 1980, 73, 4656-4662. (50) Delville, A.; L a d o , P. New J. Chem. 1989, 13, 481-491.
Figure 2. Map of the clay-watbr potential at 3 A from the clay: -,E = -38 kJ/moV - -, E = -20 kJ/mol; - -, E = 0 kJ/mol; -, E = 30 kJ/mol; ..., E = 60 kJ/mol. Only Si ( 0 )and Al ( 0 )atoms of the tetrahedral network are drawn.
-.
-
+
characteristic of montmorilloniteBpMand is compensated by eight interlamellar Na+ counterions for each particle. Each fragment of clay contains two Al atoms substituting for Si atoms in the tetrahedral network, as is for montmorillonites.m@ The residual negative charge of the clay is uniformly distributed on all Al atoms of the octahedral network. Figure 2 shows the contour map of the clay-water energy, with one water molecule, located 3 A from the Si plane of the clay, pointing ita two hydrogens toward the clay surface and remaining parallel to the Y,Zplane. Minima occur at the center of the hexagonal cavities while the surface oxygen atoms are strongly repulsive. This low distance energy map is similar to the data obtained by atomic force measuremenk6 Of course, the probe of the atomic force apparatus is not a water molecule but a silicate tip. Nevertheless, the results are qualitatively equivalent: the hexagonal cavitiesappear attractive and the surface silicium oxide repulsive. (B)MonteCarlo Simulations. The Monte Carlo simulations use the classical Metropolis61 procedure, with minimum image truncation. At each step, a displacement of one water molecule or one Na+cation is tried. The displacement of the cation consists of a translation of the cation and ita first solvation sphere, considered as a rigid whole. The criterion used to decide if a water molecule belongs to the first solvation sphere of a specific cation results from the analysis of the cation-water radial distribution function: when the cation-water distance is less than the position of the first minimum of the radial distribution function, it belongs to the first hydration layer. This procedure is necessary because of the high solvating power of the Na+ cation (AH= -415 kJ/mol).b2 When some water molecules are available, the Na+ cation tends to saturate ita first hydration layer and motion implicating a break of the Na+-water bond will be accepted with a very low probability. The motion of a water molecule is composed of a triple translation plus a triple rotation, with the position of a water molecule specifiedby the Carteaian coordinates of ita oxygen atom and three Euler angles. All these motions are adapted in order to maintain an acceptation ratio about 1/3. The grand canonical simulations follow the procedure of Adams,m*M using the chemical potential of pure water ( p / k T = 18.37865). In the grand canonical simulations, each motion of a water molecule is followed by the trial of creation or annihilation of one water molecule. In order to improve the convergence of the Monte Carlo procedure, block average is used,M,b7with a blocksize equal to 5000,i.e. at least 10times the number ofwater molecules confined in the pore. (51) Metropolis,N.; h n b l u t h , A. W.; Rosenbluth,M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1963,21, 1087-1092. (52) Raahin, A. A.; Honig, B. J. Phys. Chem. 1985,89,5588-5593. (53) Adma, D. J. Mol. Phys. 1975,29, 307-311. (54) Nicholson, D.; Parsonnage,N. G.; Rowley, L. A. Mol. Phys. 1981,
44,629-651. (55) Prigogine, I.; Defays, R. Thermodynamique Chimique; Desoer: LiBge, 1950.
Structure of Liquids at a Solid Interface
Langmuir, Vol. 8, No.7, 1992 1799 was performed. This simulation reproduced the density and vaporization energy of bulk water, with a deviation less than 2 % Thus the grand canonical simulations seem reliable since the cell used for the simulation is larger (31.08 A X 35.88 A). Calculation of the pressure is performed in the canonical ensemble, starting from the last configuration of the grand canonical simulation. The pressure calculation requires between 200 OOO and 400 OOO iterations. The equilibration of the grand canonical step was started on a HP9000 minicomputer at the CRMD and finished on a Cray2 supercomputer (CCVR, Palaiseau). Final grand canonical and canonical calculations were performed on the Cray2. The pressure is calculated at a set of fixed points,z (between 20 and 40), distributed uniformly in the accessible interlamellar space. The total pressure is obtained by an average of the normal pressure estimated at each point z61
.
Figure 3. Snapshot obtained from one equilibrium configuration of water molecules and Na+ cation for an interlamellar distance of 15A. Only Si atoms of the clays particles are shown. The figure shows only one-fourth of the simulation box. The quality of Monte Carlo simulations is limited by the quality of the generator of pseudorandom numbers. For calculations on the Cray, we use the FORTRAN77 random number generator RANF, without any modification, but the random number generator of the minicomputer is not good enough. We therefore used a procedure suggested by Binder,m with two crossed chains of pseudorandom numbers generated by two different congruent procedures. The multiplicators were carefully selected according to the criterion of and the absence of statistical correlations in the sequence of random number was checked.m The purpose of the grand canonical simulations is the description of the organization of water molecules confined in a slit-shapedpore as a function of the distance between the parallel walls of the slit constituted from clay particles. Figure 3 shows a snapshot obtained from one equilibrium configuration of the water molecules and Na+ cations for an interlamellar distance of 15 A. However, because of solvent layering in the vicinity of the solid interface,lI2the number of confined water molecules cannot be simply estimated from the product of the density of bulk water and the available interlamellar volume. Thus the initial configurationwas always generated witha very low number (between 50 and 100) of randomly oriented water molecules plus the neutralizing Na+ counterions. This procedure avoids any artifact related to a possible initial configuration containing hindered water molecules that will refuse any motion because of their overlap. Of course the price to pay will be a long grand canonical equilibration step (between 3 X 106 and 5 X 106 iterations). During the first 1OOOO iterations, the number of water molecules sharply increases, and finally reaches a plateau. When the number of water molecules becomes steady, the calculation is restarted for a production step of 1X 106to 1.5 X 106 iterations. A critical problem with Monte Carlo simulations is the convergenceof the calculation: the results must be independent of the size of the simulation box, if they are to describe the properties of a macroscopic system correctly. Minimum image truncation of the potential may prevent this convergence. For that purpose, we used an external field contributionto compensate for the long range contribution of the ion-ion and dipole-ion interactions. This procedure was successful for the modeling the ion distribution between two charged ~ u r f a c e s . ~One ~ - ~may thus expect a satisfactorycorrelation for the cut of the long range dipoleion interaction, which has a shorter range than the ionion interaction. No correlations are used for the cut of the long range dipole-dipole interaction. To check the validity of this approximation, a preliminary grand canonical Monte Carlo simulation of bulk water in a cubic box with sides of only 20 A (66)Mehrotra, P. K.; Mezei, M.;Beveridge,D. L. J. Chem.Phys. 1983, 78,3156-3166. (57) Bishop, M.; kinks, S. J. Chem. Phys. 1987,87,3675-3676. (58) Binder, K.;Stauffer,D. Topics in Current Physics: Applications of the Monte Carlo Method in Statistical Physics; Binder, K., Ed.; Springer Verlag: Berlin, 19W, Vol. 36, Chapter 1. (59)James, F. Rep. R o g . Phys. 1980,43, Chapter 1. (60)Hammemley, J. M.; Handscomb, D. C. Monte Carlo Methods; Wiley New York, 1961, pp 30-31.
abs(zjj) du(rjj) - 1z ( & T d rSjj(z))
(9b)
where A is the surface of the clay-water interface. The normal pressure is the sum of two contributions: first (eq 9a),the transfer of momentum through a fictitious plane located at point z; second (eq 9b), the virial contributionsresulting from the force between any pair of interacting particles (ion, water molecules, and clay particles). However, a given pair of interacting particles does not necessarily contributeto the normal pressure at the position z; the center of mass of the particles must be located on either sides of a fictitious plane located at 2. It is reproduced by the function S i j ( Z ) , which may be viewed as the absolute value of the difference between two Heaviside step functions Sij(z)= abs(O(z, - Z ) - O(zj - 2))
(10)
For polyatomic molecules (water,clay), eq 9b is oversimplified because the center of mass of the particle (R&may not coincide with the interacting site (rd). The contribution of a pair of interacting particles ij to the normal pressure is given b y 2
where the coordinate Zj gives the position of the center of mass y defines the relative position of the of particle i, the vector R center of mass of the pair of particle i j , and the vector rddjgives the relative position of two interacting sites CY and /3 of particles i and j , respectively. The scalar product in eq 11 induces an orientational effect which modulates the magnitude of the solvation force.
Results and Discussion (A) Structure of Interlamellar Water. We performed grand canonical Monte Carlo (GCMC) simulations of the water content of a slit pore at interparticle distances (measured from Si to Si) varying between 8 and 17A. The corresponding available interlamellar space varies between 5 and 14A, covering the range between one and four layers of water (cf. Figures 4, 7, and 8). The structure of the interlamellar water is characterized by three different sources of information: (i) The local water concentration (noted c&)) expressed in moles per liter (M). (The mean density of bulk water is 55.6 M.) The local density of water is a powerful function in the study of the layering of water induced by the clay surfaces. (ii)The mean radial distribution (notedg(r))of the water molecules distributed around the 16Na+counterions. This (61) b o , M.; Beme, B. J. Mol. Phys. 1979,37,455-461. (62) Nezbeda, I. Mol. Phys. 1977,33,1287-1299.
1800 Langmuir, Vol. 8,No. 7, 1992
Delville
'-
1
IC) I
Y
500 400
i
A
P I
1 %
[
D=lO%,
--
Y
30
/I
L
loo 600
1 I
I
i
10
3
E
0
10 5
0 2
1
4
3
0 -5
z-D/2
r
5
0
(A)
Figure 4. Local concentration of water c,(z) in mol/L, versus distance from the center of the inter oral space, in A, for interlamellar distances: (a) 8 A; (b) 9 $ (c) 10 A.
function yields the hydration state of the ions as a function of the water content of the pore. (iii) The local concentration of Na+ counterions in the interface, useful to check the validity of the DLVO theory which assumes a monotonic decrease of the local ionic concentration as a function of the separation from the clay surface.3638 Figure 4 shows the organization of water molecules for an interlamellar distance of 8 A. The pore contains only one shell of water, confined in a layer of thickness less than 2 A. The maximum local concentration is high (greater than 200 M) compared to bulk water density, but this layer only forms a 2D film hydrating the clay surface. This may be quantified by a determination of the order parameter S = (3 cose2- l),g3 where 8 is the Euler angle between the z director and each oxygen-oxygen pair of atoms. Indeed, this function gives information on the structure of the confined fluid: for 3D isotropic fluid, no direction is privileged and the mean value of the order parameter is zero while the order parameter of a pure 2D fluid is -1. The mean value calculated for the confined water at an interlamellar distance of 8 A is -0.985, clearly showing its 2D organization. Despite the high local concentration (200M), the mean water-water connectivity is lower than in bulk water, because of the planar organization of the water molecules. It is deduced from the integration of the first peak of the oxygen-oxygenradial distribution function; it is equal to 10 f 1for bulk water and 7.5 f 0.5 for the confined fluid. The Na+-oxygen radial distribution function is given in Figure 5a. A first solvation peak clearly appears with aminimum located at 3 A. This position is useful to define the water content of the first hydration shell of the ions, by integration of the radial distribution function between 0 and 3 A. This upper limit corresponds to an inflection point of the variation of the mean number of water molecules included in a sphere centered on Na+ counterion (Figure 5a). As shown in Figure 5a, only four water molecules are in the first hydration shell of the Na+ ions. This coordination number is smaller than the mean solvation number (namely 6) of Na+ in bulk water owing to the small available space in the pore forcing all species (water and ions) to fit in a layer less than 2 A thick. As seen in Figure 6a, the Na+ ions are located near the center of the pore, as are the water molecules. Under these conditions, four water molecules coordinate in a planar configuration around each Na+ ion. From this partial de(63)Saupe,A. 2.Naturforsch. 1964, 19a, 161.
(A)
Figure 5. Radial distribution functiong(r)and ita integral (n(r)), describing the mean organization of the water molecules around the cations, for interlamellar distances (a) 8 A, (b) 9 A, and (c) 10 A.
I
0 1 -5
'
'
'
'
'
0
2-D/2
'
'
' 5
(A)
Figure 6. Local Na+ concentration CN&)
in the pore versus distance (in A) from the center of the pore for interlamellar distances (a) 8 A, (b) 9 A, and (c) 10 A. solvation of the Na+ ions, one can predict a strong repulsive regime because of the high affinitp2 of the Na+ ion for water molecules. Increasing the pore thickness by 1 A induces great changes in the water distribution (see Figure 4b): the center peak splits, with shoulders located near the clay surfaces. The maximum local concentration is drastically reduced to 100 M: the interlamellar water molecules, confined between 3 and 7 A, start to behave as a 3D liquid. The mean coordination number of Na+ increases to 5 molecules per ion (see Figure 5b), while the Na+ counterions remain immobilized a t the center of the pore. A further increase of 1 8, in pore width induces more changes in tlfe organization of the water molecules (Figure 4c): the available space in now sufficient to allow two well-separated hydration shells, with a maximum local concentration 190f 10M. This high value is characteristic of two separate 2D films. At the center of the pore, where the Na+ ions remain localized with a mean solvation number of 6 (Figure 5c), the local concentration of water is 20 f 5 M. The interlamellar space is now large enough to accommodate an octahedral coordination of the water molecules around the cations. Two hydration layers are observed for an interlamellar distance of 11A (Figure 7a) with a smaller maximum for the local concentration (170 f 10 M). Both peaks are asymmetrical, with a shoulder extending toward the inner part of the pore. The increased available space is not large enough to accommodate a third layer of water, but the interlamellar space does not remain empty: the two hydration shells extend in the inner region to occupy the
Langmuir, Vol. 8,No. 7,1992 1801
Structure of Liquids at a Solid Interface 600
I
'
-8
I
I
-6
-4
-2
0 2-D/2
2
4
6
8
(A)
Figure 7. Same as Figure 4,for interlamellar distances (a) 11 A, (b) 12 A, and (c) 13 A.
6oo
Id)
A
t:
w
500
1
200 100
0 -12
-8
-4
0 r-D/2
4
8
I
12
(A)
Figure 8. Same a8 Figure 4,for interlamellar distances (a) 14 A, (b) 14.5 A, (c) 15 A, and (d) 17 A.
free space. The local concentration at the center of the pore is 20 f 5 M. The Na+ counterions remain localized at the center of the pore, with a coordination number 6.7 f 0.2. This value is slightly higher than the solvation number of Na+ in bulk water, perhaps as a consequence of the higher density of the adsorbed water compared to bulk water. An increase of the pore interlamellar distance to 12 A induces as asymmetrical broadening of the two hydration shells which now totally overalp (Figure 7b). The maximum local concentration of water is reduced to 140 f 10 M and the local concentration at the center of the pore increases to 50 5 M,a result characteristic of a 3D layer of water confined in the pore. The two hydration shells are now strongly connected. The Na+ counterions remain localized at the center of the pore, without any change of their mean solvation number. An increase of 1 A of the pore thickness permits the formation of a third hydration shell (Figure 7c) at the center of the pore. Its maximum water concentration is 105 f 7 M. The two first hydration layers sharpen, with an increase of their maximum density (160 f 10 MI. The minimum local density of water is about 50 M. The interlamellar Na+ ions remain at the center of the pore. The distribution of the water molecules remains nearly unchanged when the thickness of the pore increases to 14 A (Figure 8a), the only change being a broadening of the central hydration layer. The water distribution for an interlamellar distance of 14.5 A (Figure 8b) is rather complex. Shoulders appear at the inner part of the first hydration layers and the inner peak begins to split. The minimum local concentration of water is about 40 M, and
the maximum local concentrations are 150 f 10 M,for the first hydration shells, and 97 f 7 M for the inner peaks. The Na+ cations spread through the available space, without any change of their mean solvation number. For an interlamellar space of 15A, the water distribution seems unchanged (Figure 8c) and the central peaks do not show larger separation. The major change concerns the distribution of Na+ ions which splits into two wellseparated layers (Figure 6b), located 6 A from the closest clay surface. The comparison of Figures 8c and 6b shows an alternation of layers of water molecules and Na+ ions. Thus, the layering of the water molecules is a balance of many interactions, including not only the clay-water and water-water interactions but also clay-ion, ion-ion, and ion-water interactions. It seems a priori impossible to isolate a single interaction which, alone, could be responsible for the observed results. Finally, for an interlamellar distance of 17 A, the distribution of water molecules shows four well-separated layers (Figure 8d). The maximum local concentration of water is around 150 f 10 M for the outer hydration layers and 90 f 10 M for the inner layers. The interlamellar counterions are organized in two well-separated peaks, alternating with the hydration shells. There are no ions at the center of the pore. To summarize, the distribution of water molecules and Na+ ions shows clear layering, with coupling between the two phenomena. However, these results are somewhat different from the layering obtained with simple hard or soft spheres confined by smooth ~ a l l s . 1 2 ~ 1 5 ~ 1As ~ ~ an example, layering of the water molecules in the presence of molecular clay particles is the result of interactions among many sites which can result in local interferences. Even more, the final water organization results from a balance of many interactions: clay-water, water-water, ion-clay, ion-water, and ion-ion. It leads to complex patterns (see Figure 8b,c as an example),and the transition from one to two, three, or four hydration layers is not as marked as for simple fluids composed of hard or soft spheres, confined by smooth wa11s.112J5J8 The reason for the discrepancy of the DLVO theory at small interlamellar distances is now well illustrated. The basic approximation of that approach is the neglect of the structure of the solvent which is replaced by a continuous medium in which the ions are distributed without any constraint. The present results however clearly show coupling between the water organization and the distribution of the interlamellar Na+ counterions (see Figures 6 and 8). In some cases, the local concentration of counterions at the center of the pore is zero (see Figure 6b,c), leading to a swelling pressure equal to zero in the DLVO approach.36-38 That approach will therefore be valid only when the width of the pore greatly exceeds the diameter of the solvent molecule (more than 20 A). (B) Water Content of the Pore. Figure 9 shows the variation of the number of water molecules confined in the pore as a function of the interlamellar distance. The result is surprising; despite the layering of the water molecules, the variation displayed in Figure 9 is nearly linear. Hard spheres confined by smooth walls would exhibit a step variation;'*2 the number of spheres confined in the pore increases only when the pore width is a multiple of the sphere diameter. The results shown here are in complete opposition with this prediction, for a simple reason: water molecules at room temperature do not behave has hard spheres! This is well illustrated in Figure 7a-c. In these exam les, the interlamellar distance increases from 11to 13 At 13A, the pore is large enough
1.
~ 0 ~ ~ ~ ~ ~
1802 Langmuir, Vol. 8, No. 7, 1992
Delville
Nw 400
300
200
100 6
8
10
12
14
16
18
D (A) Figure 9. Variation of the water content of the pore as a function of interlamellar distance.
to accommodate three hydration shells. For 11and 12 A, the available space allows only the filling of two hydration shells, but these are asymmetric and progressively overlap in order to fill the increasing available space. This induces a continuous increase of the water content of the pore, overcoming the expected step variation. A step variation would probably exist at a temperature below the freezing point of the liquid phase, but we restrict our study to temperatures at which the water is liquid. Another factor masking the step variation is the heterogeneity of the surface energy of the clay particle. The water organization results from the interference among effects due to many different sites, leading to a reduction of solvent layering induced by the pore surfaces. (C)Swelling Energy. Table IV gives the variation of the total energy and ita various components as a function of the interlamellar distance. Two classes of terms contribute to the energy of the clay-water interface: First, interactions between the two components of any pair of ‘solute’ species (ions and clay surfaces). The different contributions are clay-clay (E-), clay-ion (E&, and ion-ion (Ecc)interactions, plus the energy of the ions resulting from the external field31(Exc)which compensate for the truncation of the Coulomb potential. Second, interactions involving the solvent molecules: water-water (Em), clay-water ( E m w ) , and the ion-water (E,) interactions, plus the energy resulting from the orientation of the water dipole in the external field (Exw). The clay-clay interaction (E-) is the sum of an electrostatic and a van der Waals interaction. The clayclay electrostatic energy is given by
E,, = --a2SD
2tO where a is the charge density of the clay, S is the total surface of the interface, and D the interlamellar distance. The van der Waals interaction is given by
where A is the Hamaker constant (2 X Ja for the clay-water interface) and e is the thickness of the clay sheet (=7 A). This van der Waals energy is negligible in the whole range reported here (less than 5 kJ/mol). We now have all the information necessary to understand the enthalpic contribution to the swelling of the clay at low interlamellar distances. As illustrated in Figure 10, the reference states used in the calculations correspond (61)Israelachvili,J. N. Intermolecular and Surfaces Forces;Academic
Press: London, 1985; Chapter 11.
to totally dissociated species. Thus the solvent molecules are in a gaseous phase, without interactions between each other. However the reference state in a swelling experiment is liquid water. One must therefore add to the waterwater interaction the energy of cohesion of liquid water, which is approximated by the vaporization enthalpy (40 kJ/mol). The resulting swelling energy is shown in Figure 11. The swelling process between 8 and 15A is exothermic, the energy of the clay-water interface decreases by 8.5 MJ/mol. The slope in Figure 11 decreases rapidly and the energy of the interface becomes steady for a pore width of 15 A. This result is characteristic of clay hydration? the enthalpy of immersion of the clay decreases rapidly as a function of the initial water content of the clay paste3 and becomes steady when the clay interface contains three or four hydration shell^.^ The total enthalpy variation of 8.5 MJ/mol corresponds to a heat release of -0.6 J/m2 when the water content of the clay interface increases from one to three hydration layers. This result is compatible with data reported for laponite3 (-0.5 J/m2) and hectorite3 (-0.4J/m2) in the same range of swelling. All the ‘solute-solute” (clay-clay,clay-ion, ion-ion, and external field-ion) interactions are collected together in Figure 12. These interactions are mainly electrostatic except at very small interlamellar distance (8A) and were already treated in a McMillan-Mayer approach of the claywater interface.*12 The trend was the same: the electrostatic contribution to clay swelling is attractive and is a linear function of the width of the pore.12 Thus, swelling of the clay is not due to an electrostatic repulsion between the clay surfaces bearing the same negative charges and any treatment neglecting the neutralizing charges of the interlamellar counterions would be incomplete. The contribution of the solvent molecules to the enthalpy of swelling contains three terms: the water-water, clay-water, and ion-water interactions. Figure 13a displays the water-water contribution to which the enthalpy of vaporization of the water molecules confined in the pore has been added. The final contribution is attractive. Figure 13b displays the clay-water energy which decreases monotonically with an exothermiccontribution to the clay swelling of -10 MJ/mol. Figure 13c shows the ion-water contribution, also decreasing monotonically. The exothermic contribution to clay swelling due to cation hydration is also -10 MJ/mol. Clay and cation hydration are the two driving forces of the swellingof montmorillonite. Together, they contribute 50% of the energy released. This result depends on the chemical nature of the clay particle. Replacement of the Na+ counterions by Li+ or Ca2+ions will increase the ionwater contribution, since these cations coordinate strongly to water molecules,52 but replacing Na+ counterions by K+, Rb+, or Cs+ ions will reduce the ion-water contributi0n.5~The chemical composition of the clay network is also important. An increase of the charge density of the clay will increase the clay-water and clay-ion interact i o n ~ Similarly, .~~ a higher degree of substitution of the Si(1V)surface atoms of the clay byAl(II1) or Fe(II1) atoms will introduce a larger number of strong interaction sites3l between the clay particle and the water molecules or the interlamellar ions. Thus, the swelling energy of other kinds of clay may be changed, but the qualitative variations of the different contributions will probably be the same. The enthalpic contribution to clay swelling vanishes when the interlamellar distances become larger than 16 A. At this distance, the interlamellar counterions are fully hydrated and exchangeable, introducing an osmotic con-
Langmuir, Vol. 8, No. 7, 1992 1803
Structure of Liquids at a Solid Interface
Table IV. Variation of the Water Content and of the Total Energy (4)as a Function of Pore Width (A). D 8 9 10 11 12 13 14 14.5 15 17
Nw 131 167 231 264 300 349 391 410 433 495
Et -53.5 -57.3 -62.3 -64.8 -67.1 -69.5 -71.8 -73.0 -74.1 -76.4
E-4.0 -4.5 -5.0 -5.5 -6.0 -6.5
Eme -53.2 -53.0 -51.9 -50.6 -49.2 -48.0 -46.7 -46.1 -45.6 -43.5
Ecc 14.9 14.6 14.6 14.6 14.5 14.1 14.0 13.9 13.8 13.3
Exc 2.7 3.0 3.4 3.8 4.2 4.6 4.9 5.1 5.3 6.0
E, -1.8 -2.6 -4.1 -5.1 -5.8 -7.4 -8.6 -9.2 -9.9 -12.1
Em, -4.0 -5.0
E, -8.0 -10.0 -12.3 -13.5 -14.7 -15.3 -16.1 -16. -16.5
-7.0 -8.4 -10.0 -10.8 -11.9 -12.5 -13.3 -13.9
Exw -0.003 -0.02 -0.04 -0.08 -0.13 -0.19 -0.31 -0.35 -0.35 -0.51
-7.0 -7.3 -7.5 -8.5 -17.1 0 The contributiona to the energy are montmorillonite-montrillonite (Emm), montmorillonite-cation (Emc),cation-cation (Ecc),external field-cation(Exc),water-water (Ew), montmorillonite-water (Emw),cation-water (Em),and external field-water (Exw).All energies are given in MJ/mol. -30
clay
-1
t
1
1
I
I
I
10
12
14
16
Eel
clay
-35
steam swelling energy -40
6
Figure 10. Difference between the swelling energy of the claywater interfaceand the total energy (Et)calculatedby the Monte Carlo procedure (see Table IV). -45
E -50
-55
- 60 6
8
10
12
14
16
18
D (fi) Figure 11. Variation of the total swelling energy as a function of pore width.
tribution to clay swelling. The calculation of the swelling pressure in such a transition regime needs a complete evaluation of the solvation forces. This is the purpose of the next section. (D)Solvation Forces. Table V displays the solvation force of the clay-water interface and its numerous contributions. The contribution of the clay-clay interaction is dominated by the electrostatic term, given by
ne,= fJ2/2to (14) This term is constant and equal to 7.36 katm. The van der Waals contribution is given by and its absolute value is smaller than 0.02 katm, and therefore negligible.
8
18
D (A) Figure 12. Variation of the electrostatic energy of the claywater interface resulting from the sum of all “solute-aolute” contributions (see text).
The internal pressure of bulk water is evaluated by the same Monte Carlo procedure with a sample containing 380 water molecules simulating a pure liquid. This pressure is subtracted from the water-water contribution to the solvation force because the swelling measurements are not performed in vacuum but by immersing the clay particles in liquid water. Figure 14shows the variation of the total solvation force as a function of the interlamellar distance. The general trend is a decrease of the pressure when the width of the pore increases. The initial slope gradually decreases, as expected from the previous discussion on the hydration state of the interlamellar counterions. Indeed, at small interlamellar distances, the Na+ counterions are partially desolvated. This process requires a strong compression of the interface because of the high solvation e n e r d 2 of the Na+ cation. Small oscillations appear for interlamellar distances of 10and 14A, but, here too, more calculations and better accuracy would be necessary to ascertain the existence of these oscillations. Despite the strong layering shown by the water organization at the clay interface, the solvation pressure displays only small oscillations. Once again, it is possible to separate all the contributions to the pressure in two classes: First, the contribution resulting from ‘solute-solute” interactions, with a dominant clay-ion term (see Table V and Figure i5a). This term decreases rapidly for interlamellar distances between 8 and 10 A, shows a broad minimum between 11 and 13 A, and starts to increase slowly above 13 A. Second, the contribution of the solvent molecules, summed in Figure 15b and displaying a marked maximum around 10 A, with a small oscillation at 14 A. The calculated pressure is the s u m of many contributions of the same order of magnitude, but sometimes with
Deluille
1804 Langmuir, Vol. 8, No. 7, 1992 a EWW
6
I
I
I
I
1
I
8
10
12
14
16
18
D
(A)
b
Emw -5
-10
-15 6
8
12
10
D C
14
16
18
14
16
18
(A)
-5
Ecw -10
-15
6
8
10
12
D
(A)
Figure 13. Variation of the solvent contributions to the swelling energy: (a) water-water; (b)montmorillonite-water; (c) cation-
water interactions.
opposite variations. Under these conditions, the variation of the total pressure (Figure 14) may not be monotonic. Furthermore, the small oscillations are not artifacts but are due to the solvent contributions (see Figure 15b). However, it is impossible to explain the observed pattern on the basis of a single interaction. The reported solvation force is 100 times larger than the pressure calculated on the basis of the DLVO theory by a Poisson-Boltzmann treatment of the interlamellar counter ion^^^^^^ distributed between two smooth surfaces bearing the charge density of the clay particles. The same ratio was already reported2I6between the measured solvation force and the prediction of the diffuse layer theory. This discrepancy clearly illustrates the impossibility of the DLVO theory to explain the swelling property of colloids at small interparticle distances in the presence of a structured liquid. (E) Influence of the Parameters. The simulations presented here describe a particular system: Na-montmorillonite, with a charge density of 7.17 X 10-3 e/A2 in the presence of pure water ( p l k t = -18.378),55 without
added salt and at room temperature. It is interesting to try to predict, at least qualitatively, the influence of the conditions adopted for the simulations on the results. A first parameter characterizing the clay particle is its charge density. An increase of charge density of a smooth wall induces a higher organization of the interlamellar water molecules, with a stronger layering.25 This effect will appear if montmorillonite is replaced by another clay bearing a higher charge density, such as mica or vermiclite.^^ The effect however is not direct, because the charge distributed on the network of the clay particle is neutralized by the interlamellar counterions. Thus, increasing the charge on the clay will concomitantly increases the number of interlamellar counterions which disturb the layering induced by the clay surfaces. Thus, evaluation of the global influence of the charge density of the clay is not simple. Another parameter characteristic of the clay particle is the ratio of tetrahedral substitution. The ratio of tetrahedral to octahedral substitution of montmorillonite is rather 10w,39940 but some clays (like beidellit@ have mainly tetrahedral substitution. The surface sites interact strongly with the water molecules31even if the total charge of the clay is not changed. Under these conditions, increasing the number of tetrahedral substitutions will induce stronger layering. The Na+interlamellar counterions are exchangeable and may be replaced by Li+ or Ca2+ cations, which strongly coordinate water mo1ecules151interfering with the structure of water induced by the clay surface. Conversely,replacing the Na+ cations by K+, Rb+ or Cs+ ions will reduce ion solvation51and increase layering. An interesting case is that of the K+ cation in the presence of mica layers; because the desolvated K+ has exactly the size required to fit in the hexagonal cavity of the clay, it behaves as an unexchangeable ion, resulting in a nonswelling clay. On a cleaved external surface of the clay, the K+ cation may stay at the center of the hexagonal cavity, with water molecules coordinated above it. The clay surface is thus neutral but highly polar and wettable, but the ions are totally fixed and will not interfere with the water organization induced by the clay surface. The total energy of this system thus contains less contributions and will probably display stronger layering. Adding an electrolyte to the suspension reduces the electrostatic contribution of the swelling pressure35-3' if the interlamellar distance is larger than K - ~ where , ~ ~ K , the Debye screening constant, is given by (16) where I is the ionic strength of the suspension. This will have no effect in the range of interlamellar distances studied here, except for a decrease in the chemical potential of the water molecules, with a reduction of the amount of water molecules confined in the pore and a decrease of the layering. Finally, reduction of the temperature is a simple way of mimicking the behavior of hard spheres and increasing layering of the solvent. However the range of temperatures accessible with liquid water is too limited. (F) Limitations. These simulations are but a first approach to a molecular description of the clay-water interface. Many hypotheses are implied pairwise additivity of the potential; rigidity of the water molecules and clay particles. Even more, empirical potentials are used to describe the water-water and clay-water interactions.
Langmuir, Vol. 8, No. 7, 1992 1805
Structure of Liquids at a Solid Interface
Table V. Variation of the Total Swelling Pressure (I&) and Its Different Contributions as a Function of Pore Width (A).
D
ut
8 9 10 11 12 13 14 14.5 15 17
34.8 27.3 23.8 13.6 9.7 6.9 6.7 3.6 4.3 2.1
nec
&C
0.21 0.14 0.15 0.12 0.16 0.21 0.70 0.79 0.98 1.8
-3.3 -13.4 -17.6 -19.4 -19.4 -19.2 -18.5 -18.2 -17.4 -15.9
&e
nww
ma
ncw
nsw
1.8 1.7 1.8 1.8 1.8 1.7 1.7 1.8 1.7 1.7
11.5 21.9 26.5 25.7 27.4 27.4 26.8 25.1 25.1 22.1
17.6 11.9 11.0 6.9 4.5 3.4 2.1 0.97 0.80 -0.48
-0.31 -1.7 -4.6 -7.1 -9.6 -11.0 -10.6 -11.1 -10.3 -9.6
-0.25 -0.50 -0.85 -1.8 -2.4 -2.9 -2.8 -3.1 -3.6 -3.2
a The pressure contributions are montmorillonite-cation (IImc),cation-cation (II& external field-cation (nSc), water-water (rim), montmorillonite-water (IImw),cation-water (TIcw), and external field-water (IIxw). All pressures are given in katm. I
I
I
I
I
1
a
0
1
I
I
I
I
nmc
2o
t 6
8
10
12
14
D (A) Figure 14. Variation of the swelling pressure function of the pore width.
16
6
18
8
12
10
nt (katm) as a
b
nw
35
I
I
14
16
I
1
18
(A)
D
One may wonder about the reliabilityof results obtained by using empirical potentials suited to the description of bulk water whose concentration is 56 M, and yielding rather drastic conditions corresponding to highly compressed 2D films with local concentration 150-200 M. We have no definite answer to these questions, and assuredly many improvements may be brought to this model, but the high local density of water molecules in contact with the adsorbing surface is a direct consequence of the latter’s attractive potential. Many similar results have been r e p ~ r t e d . ~The ~-~ MIND0 ~ energy of interaction between the surface of montmorilloniteand a water molecule shows minima at -47.2 kJ/molfor a perfect clay31(no tetrahedral substitution) and -63.2 kJ/mol near a substituted tetrahedral site.31 The resulting potential well is deep enough to justify the observed local concentration. Finally, the clay-water interface contains only 16 Na+ counterions; this number is too small to give a good statistical precision. The results shown in Figure 6 are rather noisy and are used only in a qualitative discussion of the layering of the Na+ ions in the pore. Conclusion We used a molecular description of the clay-water interface to model the organization of the water molecules and the Na+ counterions as a function of the pore width. Strong laverine of both distributions amears. with an alternation of t i e different layers. There&& are however 1
-
1
\
I
30 25 20 15
10 5
Figure 15. Main contributions to the swelling pressure: (a) montmorillonite-cation; (b) sum of the solvent contributions.
in contradiction with predictions based on a hard sphere model. Despite the strong layering of the solvent, the variation of the water content, the swelling energy and the pressure do not show important singularities (steps, oscillations). The reason for this behavior can be traced to interferencesamong many contributions to the observed results. A Drecise determination of the influence of each parameter iequires new simulations.
Acknowledgment. We cordially thank Drs. H. Van Damme, Setton, and s, Sokolowski for helpful discussions.
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Registry No. HzO, 7732-18-5.
