Langmuir 1990,6, 1289-1294
1289
The Origin of the Swelling of Clays by Water Alfred Delville and Pierre Laszlo* Laboratoire de chimie fine, biomimttique et QUX interfaces, Ecole Polytechnique, 91128 Palaiseau C&dex,France Received August 22,1989. I n Final Form: February 16, 1990 The diffuse electric double-layer formalism allows calculation of the swelling pressure in good agreement with the observations made on dilute clay suspensions. We correct an error often made in the calculations that renders them incoherent. Analysis of the energies of the clay-water-cation system suggests the importance of solvation of the interlamellar cations.
Introduction Clay Swelling and Diffuse Double Layer Theoretical Approach. A characteristic structural Among clay minerals, smectites are defined by their feature of smectite clays is the occurrence of negative ability to swell. This property leads to many applications. charges on the sheets, due to isomorphous substitutions Quite a few studies have been devoted to it.l-lo More than in the tetrahedral and octaedral1ayers.l' Accordingly, the 50 years ago,Langmuir conjectured correctly that swelling goal of the diffuse-layer formalism is the calculation of the had to do with the presence of a diffuse electric double distribution of the counterions (necessaryto maintain eleclayer." Quantitative treatments were provided later.12J3 in the interlamellar space. Many such Overall, they agree with the m e a s ~ r e m e n t s . ~ * ~ * ~troneutrality) ~J~J~ calculations assume the "primitive model": only However, examination of the corresponding parameters electrostatic interactions between the parts of the system has convinced LOwgJs of the inconsistency of this (charged clay sheets and mobile ions) are considered, and appr0ach.~J6 Recently, an alternative treatment has the solvent is taken as a dielectric medium with a uniform appeared, with an empirical parametrization of the specific dielectric constant.20 Furthermore, ions are viewed as unclay-counterion interactions.16 penetrable hard spheres. All short-distance specific This article has two parts. First, we use the formalism interactions, such as ionic solvation or solvation of the of the diffuse layer to calculate the swelling pressure of charged sheets, are neglected. dilute clay suspensions. Readers are reminded that the Whenever the interaction of two particles stems from DLVO theory is applicable to suspensions in the limit of a potential-an electrostatic potentiel in this case-the infinite dilution.12J3 Its use for real systems at finite local concentration cij(r) of the ith particle at a radial concentrations introduces an error and renders the distance r from the centralj particle can be expressed from treatment incoherent, as observed by LOW.^^'^ We shall a mean force Wij potential as then proceed to show that the diffuse double-layer model gives a rather accurate description of the swelling of a dilute cij(r) = c: exp(-Wij(r)/kr) (1) clay suspension and does not require any parametrization. In the second part, we make a critical discussion of the where ci0 is the mean concentration of the ith particle, k double-layer formalism. We examine the dependence of is the Boltzmann constant, and T is the temperature. swelling on the nature of the cation and on the ionic Another equation, a closure relationship, is necessary in strength. order to solve for ci(r). The central approximation identifies the mean force potentiel Wij(r) with the (1) Schofield, R. K. Tram. Faraday SOC.B. 1946,42,219-228. electrostatic energy ej*(r). The latter is independent of (2) Bolt, G.H.;Miller, R. D. Soil Sci. SOC.Am. h o c . 1955., 19.. 285the chemical nature of the ion present at the distance r. 288. Such an approach neglects ion-ion correlation. It forms (3) Warkentin, B.P.; Bolt, G. H.; Miller, R. D. Soil Sci. SOC.Am. Proc. 1957.21.496-497. the basis of the Gouy-Chapman or of the Poisson(4Norrish, K.; Rausell-Colom, J. A. Clays Clay Miner., Proc. Conf. Boltzmann treatments, in which the Poisson relationship 1963,10,123-149. (5) Barclay. L. M.; Ottewill, R. H. Spec. Discuss. Faradav SOC.1970. 1,138-147. (6) Callaghan, I. C.; Ohwill, R. H. Faraday Discuss.Chem. SOC.1974, 57,110-118. (7) Low, P.F.; Margheim, J. F. Soil Sci. SOC.Am. J . 1979,43,473481. (8) Oliphant, J. L.; Low, P. F. J . Colloid Interface Sci. 1982,89,366-
-.
R7R
I.
(9) Viani, B. E.; Low, P. F.; Roth, C. B. J. Colloid Interface Sci. 1983, 96,229-244. (10) Lubetkin, S.D.;Middleton, S. R.; Ottewill, R. H. Philos. Trans. R. SOC.,London A 1984,311, 353-368. (11) Langmuir, I. J. Chem. Phys. 1938,6, 873-896. (12) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (13) Derjaguin, B.; Landau, L. D. Acta Physicochim. URSS 1941,14, 635. ...
(14) Iaraelachvili, J. N.; Adama, G. E. J. Chem. Soc., Faraday Tram.
I 1978, 74, 975-1007.
(15) Low, P. F.Langmuir 1987,3, 18-25. (16) Spitzer, J. J. Langmuir 1989,5, 199-205.
is coupled with the Boltzmann distribution law ci(r) = c: exp-(ei*(r)/kT) (3) where ci0 is the average concentration of ion i in the suspension of clay and where *(r) is the local electrostatic potential. (17) Van Olphen, H.An introduction to Clay Colloid Chemistry; Wiley New York, 1963. (18) Sparks, D. L. Soil Phvsical Chemistrv: - . CRC Press: Boca Raton, FL, i s m . (19) Attard, P.;Mitchell, D. J.; Ninham, B. W. J. Chem. Phys. 1988, 89,4358-4367. (20) Carley, D.D.J . Chem. Phys. 1967,46,3783-3788. (21) Marcus,R. A. J. Chem. Phys. 1955,23, 1057-1068.
0743-7463/90/2406-1289%02.50/0 .~ , 0 1990 American Chemical Society I
,
1290 Langmuir, Vol. 6, No. 7, 1990
Delville and Laszlo
t This potential is not completely determined; it may always be offset by translation to assume a specified value (0 for example) at a given point in space. This may be the midpoint between two clay sheets or the point where the local concentration ci(r) of the cations (or the anions) equals the mean concentration cia. T o avoid such constraints, let us write the conservation of the number of ions of each i type. This is done by normalizing the Boltzmann law21 ci(r) =
cf exp[-(eiWr)/ItT)](R
- a)
(4a)
LRexp[-(eiWz)/It771 dz where a is the ionic radius and where 2R is the interlamellar distance. After integration over the interlamellar half-volume, the relation XRci(r)dr = cio(R- a)
20
Lo z Figure 1 . Compared profiles for the counterion local concentrations from an analytical (*) and from a numerical (-) solution of the Poisson-Boltzmann equations.
(4b)
ensures conservation of the number of i ions. Such a conservation is mandatory for preservation of the electroneutrality for the clay suspension: ~ ( C + O
- c-O)(R - U ) + u = 0
(44
Use of a finite ionic radius allows introduction of an excluded volume effect between the condensed counterions and the clay surface. In the absence of added salt, the system of eqs 2 and 4 has been resolved analyticallyz2 (see also Appendix A). Otherwise, one has to resort to a numerical solution of eqs 2 and 4. For this purpose, a simple approach is to divide the interlamellar space into a series of layers parallel to the clay interface and to maintain a constant ionic concentration in each shell. To obtain a good digitalization of the ionic profile, the thickness of the layers must be small enough (less than 1 A). This digitalization greatly simplifies evaluation of the electrostatic potential (see appendix B). We then apply a self-consistent iterative procedure (Scheme I). Starting from a digitalized concentration profile, one calculates the electrostatic potential. Integration of the Boltzmann distribution law (eq 4a) for each layer yields a new profile and so on: the iteration is deemed to have converged when two successive concentration profiles become identical within a preset interval. In practice, convergence is achieved after 3040 iterations. As indicated in Figure 1,there is excellent agreement between such a numerical solution and the analytical solution of the Poisson-Boltzmann equations. The characteristic quantities in Figure 1 are an interlamellar distance of 88.7 A and a superficial charge density on t h e aluminosilicate sheet of 0.109 C/m2, very representative of montmorillonite clays. The cation radius was taken as 2.35 A in order to mimic an oxygen-tosodium van der Waals contack23 the sodium ion has an ionic radius of 0.95 A to which the oxygen van der Waals of 1.40A has to be added. The dielectric constant of water was taken as 78.5. The temperature was taken as 298 K. We had already performed a successful similar numerical (22) Enstrom, S.; Wennerstrom, H. J. Phys. Chen. 1978, 82, 27112714. (23) Cooker, H. J. Phys. Chem. 1976, 80, 2084-2091.
-2
t
-L *
fly-
- - - --
-
- -
-
- - - -
-
- -
- 6-
z ,
evaluation of the distribution of counterions around polyelectrolytes. ~ y l i n d r i c aor l ~ spherical25 ~ Let us turn now to a comparison of the concentration profiles (Figure 2) for sodium and for chloride ions. The charge density is the same as in Figure 1, the salt concentration is centimolar, and the interlamellar distance is 40 A. These are all conditions that are frequently encountered in swellingsituations2-l0for clays. While these two concentration profiles appear to mirror one another somewhat, their differences are much instructive. The local concentration of the (positive) counterions is greater than that of the (negative) co-ions a t all distance r. The s y s t e m a t i c large difference between t h e mean concentrations of both ionic types if also striking. A Mistaken Generalization. There is one alluring exception. A t an infinite distance between the charged sheets (2R m), an analytical solution is feasible (see Appendix C) because the local concentrations a t infinity
-
(24) Delville, A.; Gilboa, H.; Laszlo, P. J. Chem. Phys. 1982,77,204& 2050. (25) Delville, A,; Herwata, L.; Laszlo, P. Nouo. J. Chim. 1984,8,557562.
Langmuir, Vol. 6, NO. 7, 1990 1291
Swelling of Clays by Water
Table 1. Signs of the Electrostatic Potential 9 and of Its First and Second Derivatives. a
C' -
d2@/dz2
d@/dz 9
i
I
a
b)
Figure 3. (a) Variation of the local concentration corresponding to eq 5. (b) Electrostatic potential responsible for the distribution in Figure 3a. (c) Electrostaticpotential generated by this same
distribution. equal the mean salt concentration. Unfortunately, this result has been misleadingly generalized to finite interlamellar distances with a nonnormalized Boltzmann distribution (eq 3), with cia, the mean concentration of the i ions, taken as equal to co, the mean salt concentration. Such a mistaken treatment reduces eqs 2 and 4 to eq 5:
A* = 2c0 sinh (e*/k7')/(tot,) (5) where e is the proton charge. In like manner, the sum of the local concentrations is given by the relationship
c J r ) + c J r ) = 2c0 cosh (e\k/kT) (6) Equation 6 is frequently used in calculations of the swelling p~~~~~~~~2.4-7,9.10.12~16~17-1S
This formulation is wrong. Indeed, if it is always feasible to make the electric potential undergo a translation so that it will vanish at the point in space (denoted as y) where the local concentration of the co-ions is equal to the mean salt concentration, there is no reason why the local concentration of the counterions should become equal to the mean salt concentration at the self-same point y . Figure 2 provides a blatant counterexample. Let us go one step further and show that such an assumption is incoherent. We shall argue from the absurd. Assume that the above hypothesis is correct. One thus obtains ionic concentration profiles as in Figure 3a, generated by a potential such as shown in Figure 3b. Conservation of the total number of co-ions demands that the y point be included in the interval ( a , R ) , limits excluded. Hence, the local charge vanishes at y, is positive for smaller distances to the charged sheet, and is otherwise negative. The Poisson law dictates the sign of the second derivative (Table I): hence the first derivative of the potential d*/dz goes through a minimum at y and only at that point. Furthermore, we can obtain the values of this derivative d*/dz at each of the limits from application of Gauss's law: (d*/dz)z.a = -u/(totr) > 0 (74 (7b) (d*/dt),,R = 0 where u denotes the surface charge density on the charged
-
+ -
B
0 mas
Y
0 min 0
R
+ 0 -
At various points defined in the text.
aluminosilicate sheet. Thus, the derivative of the potential is negative at the point y, since it is an increasing function in the (y, R ) interval. There has to be a point, which we denote 0,in the (a, y) interval where the first derivative d\k/dz becomes zero. The potential * ( z ) thus is maximum at 0,is zero at y, and must be negative in the vicinity of the charged sheet so as to allow adsorption of the counterions. The potential \k(z) is positive in 0, and it becomes negative at the distance R since this function decreases in the interval (P, R). Such a potential, depicted in Figure 3c, is totally unable to produce concentration profiles of the type shown in Figure 3a from intervention of a Boltzmann law. Thus, one must conclude that the treatment embodied in eq 5 is inconsistent. This is why swelling calculations predicated on such an incorrect formalism require excessive values for the surface charge densitiesQ or p0tentia1s.I~
Results and Discussion Heeding the limitations inherent in the primitive model, which ignores any short-distance specific interaction of the water molecules and the cations with the clay sheet, we restrict consideration to dilute suspensions of clays with interlamellar separations of at least 25 A. We assume these specific interactions are invariant, since they concern only the first three solvation layers of the charged sheet.26 A t such interlamellar distances greater than 25 A, swelling has often been attributed to the electrostatic repulsion between the charged aluminosilicate sheets.17J8 Such an interpretation is flawed because it does not take the interlamellar counterions into account. Our Monte Carlo simulations show that the system consisting of the charged sheets plus the cations is net attractive from the electrostatic viewpoint.27 Such a behavior has been reported already both for univalent= and divalent29counterions. There is no fundamental difference between that electrostatic stabilization of the clay-cation system and the Madelung stabilization of ionic crystals. The long-range osmotic contribution to the swelling pressure II is obtainable from the van't Hoff lawl1J1
ll = RT(c+(R)+ c-(R)- 2 ~ ' ) (8) after calculation of the ionic distribution. Calculation of t h e swelling, however, is n o t straightforward. One needs to know the interlamellar distance to calculate the distribution of the ions. Many a ~ t h o r s ~ , ~evaluate , ~ ~ ~ this J ~ Jdistance ~ from the waterto-clay ratio characterizing the clay suspension:
R e - mw m,&
where m , and m, are the masses of water and of the clay, respectively, p is the mean water density (1g mL-I), and (26) Fripiat, J.; Cases, J.; Franpis, M.; Letellier, M. J.Colloid Interface
Sci. 1982,89,378-400.
(27) Delville, A.; Laszlo, P. Nouu. J. Chim. 1989, 13, 481-491. (28) Jonsson, B.; Wennerstrom, H.; Halle, B. J.Phys. Chem. 1980,84, 2179-2185. (29) Kjellander, R.; Marcelja, S.; Pashey, R. M.; Quirk, J. P. J. Phys. Chem. 1988,92,6489-6492.
Delville and Laszlo
1292 Langmuir, Vol. 6, No. 7, 1990
Table 11. Calculated Variation (Poisson-Boltzmann) of the Surface Potential as a Function of the Interlamellar
Distance I).
~~
25 50 75 100
-32.6 -50.6 -61.3 -68.9
19.6 33.2 42.2 48.9
a The clay has a surface charge density ionic strength is zero.
12.64 3.33 1.51 0.86 u = 0.109
C.m-2. The
Table 111. Influence of the Ionic Strength I on the Calculated Osmotic Swelling Pressure lI. n, atm D,%, I=OM I = 0.01 M I = 0.01 Mb ~~~
~~
50
D
100
Figure 4. Variation of the experimental (I)and the calculated (-, u = 0.093C-m-2; - - -,u = 0.187 Cam-2) swe pressures (atm) as a function of the interlamellar distance D ( ).
9
S is the specific surface of clay. This eq 9 assumes that all the water occupies the interlamellar space with a density equal to that of the pure solvent. Furthermore, the specific surface S is assimilated with that of isolated platelets: such an assumption ignores formation of t a c t o i d P and the relative distribution of the clay sheets.30 Tactoids are quasicrystalline aggregates of unswollen clay platelets, containing only structural water. In order to bypass such unlikely working hypotheses, we compare the calculated swelling pressure with simultaneous experimental determinations of the swelling pressure and of the interlamellar d i ~ t a n c e .If~tactoids are present, they do not interfere provided that the measured interlamellar distance is that between aggregates of platelets. Indeed, the time required for water exchange between the tactoids (internal intralamellar water) and the bulk is of the order of 1h.31 Thus, the system formed by clay aggregates and internal intralamellar water and cations must be viewed as a rigid entity whose individual components do not contribute to the osmotic pressure. Only interaggregate water molecules a n d cations a r e exchangeable elements contributing to the osmotic pressure on a relevant time scale. Our own measurements by deuterium and by oxygen-17 NMR show fast exchange of water on the outer surface of the tactoids with bulk water. We show (Figure 4) the variation of the swellingpressure 11as a function of the interlamellar distance (D= 2R) for eight montmorillonites with surface charge densities u between 5.774 and 11.67 me.A-2. The two solid curves shown are those calculated for these limiting charge densities from eqs 2, 4, and 8, with no parametrization whatsoever. The agreement is satisfactory. The diffuse double-layertreatment overestimates the swellingpressure at large interlamellar distances (beyond 65 A). Let us, for argument's sake, make the potential \k vanish when the local counterions concentration is equal to the mean concentration. The resulting surface potentials These (Table 11) agree with the values given by potentials gave negligible osmotic pressures, using the description of the double layer by eqs 5-6 and 8 (see Figure 3 of ref 15), while the correct formulation gives swelling pressure in agreement with experimental data. This strong difference points to the danger of such an erroneous treatment. ~~
~~~
(30)Fukushima, Y. Clays Clay Miner. 1984,32,320-326. (31) Cebula, D. J.; Thomas, R. K.; White, J. W. J . Chem. SOC.,Faraday Trans. 1 1980, 76, 314-321.
25 50 75 100
12.64 3.33 1.51 0.86
13.01 3.60 1.75 1.08
11.74 3.25 1.57 0.97
a The surface charge density is the same as in Table 11. The swelling pressure is calculated with the Wells32 activity correction (see text).
Table IV. Variation of the Calculated Swelling Pressure Il (in atm) with the Nature of the Interlamellar Cations.
D,A
Li+
Na+
K+
Rb+
cs+
25 50 75 100
11.88 3.23 1.48 0.85
12.64 3.33 1.51 0.86
13.55 3.45 1.55 0.88
13.92 3.49 1.57 0.89
14.38 3.55 1.58 0.89
a The surface charge density and ionic strength are the same as in Table 11.
Table V. Influence of the Nature of the Cations on the Size of the Tactoids cation Li+ Na+ K+ Rb+
cs+
ionic radius, A 1.94 2.35 2.79 2.96 3.16
nbr sheets per tactoid A?L,, kJ/mol 1.0 1.5 2.0 2.6 3.0
-530 -415 -320 -300 -277
Influence of t h e Experimental Conditions. We go now to the influence of the ionic strength on the osmotic pressure. As shown in Table 111, increase of the ionic M makes the osmotic pression II strength from 0 to rise by 0.2-0.3 atm. The greatest variation, Le., 25%,occurs at the greatest interlamellar distance. In complementary manner, we show in Table IV the influence of the chemical nature of the counterions on the osmotic pressure II. The dependence is introduced through the ionic radius: going from lithium to cesium ions the ionic radius increases by 1.20 A,Z3while the osmotic pressure increases by variable amounts, in the range 0.04-2.3 atm, depending on the interlamellar distance D. The greatest relative variation, 20 % ,occurs at the smallest interlamellar distance, as could be expected intuitively. These results (Tables I11 and IV) conflict with those of Lubetkin et al.,10 who observe a reduction of the swelling pressure with an increase of the ionic strength or with an increase of the ionic radius. One reason for the discrepancy is calculation of the osmotic pressure from the van't Hoff law (eq 8). Such an approach does not distinguish between solvation of the various alkali cations since it assumes an ideal solution. As shown by Table V, this hypothesis is far from reality: in going from lithium to cesium, the heat of solvation is reduced by almost 50%. Furthermore, an increase in ionic strength will amplify the deviation from ideality. In the theory of polyions with cylindrical symmetry, such
Langmuir, Vol. 6, No. 7, 1990 1293
Swelling of Clays by Water
as DNA, a correction has received widespread appliof the clay suspension such as swelling are determined by the number of free ions. They are insensitive to the cation.24%2-% It has been generalized for polyions with presence of site binding. Such a mutual compensation other symmetries, spherical26 or planar.37 It consists in between the two modes of attachment makes the multiplying the ionic concentration a t the midpoint determination of site binding tricky in the absence of direct between the polyions, c(R),by the activity coefficient y information on the dynamical properties of the countefor the added salt. For example, for sodium chloride and rion. an ionic strength of M, this factor y is 0.902.38 Values thus corrected are listed in Table 111. For the smallest inAnother approximation replaces the solvent by a terlamellar distances, this translates into a decrease of the continuousmedium characterized by a single bulk dielectric swelling pressure by up to 7%. For larger interlamellar constant. This approximation is valid only beyond the distances, the swelling pressure still increases with the ionic Stern layer.42 Image charges are one of the ways to take strength, and the maximum increase observed is 13%. into account the reduction of the dielectric constant near These results are at variance with the systematic reduction the clay surface. It induces a reduction of the local counof the swelling pressure upon an increase of the ionic terion concentration near the clay sheet. Such reduction strength reported by Lubetkin et al.1° whatever the is of the order of 8 % for a charge density of 0.08 C.m-2.43 magnitude of the interlamellar distance. It can amount to 50% for charge densities smaller by 1 However, these authorslo do not determine actual inorder of magnitude. However, the perturbation of the terlamellar distances; nor do they consider the formation counterion profile is significant only at distances to the of tactoids in their calculation of the interlamellar distance clay sheet smaller than 8 A.43 (eq 9). Yet, aggregation into tactoids is a well-known Finally, the major limitation of the formalism of the phenomenon. Furthermore, the size of these aggregates double diffuse layer is recourse to the primitive model that depends on the nature of the interlamellar ~ a t i o n sThe . ~ ~ ~ ~neglects all short-range interactions. The best way to avoid mean number of platelets per tactoid increases with the all these above approximations is to perform Monte Carlo ionic radius (see Table V). This makes the interlamellar simulations at the Born-Oppenheimer that is, by distance erroneously evaluated from eq 9 too small: in the simulation of the ion and water molecule distribution in case of cesium counterions, the real distance is larger by the vicinity of the clay surface. Such a treatment will be a factor of 3! Thus, we submit that the observed reduction able to reproduce the water organization by the clay surface of the swelling pressure when the ionic radius increases'O as detected by Israelachvili e t al.45 However, the might be an artifact due to an underestimation of the inconvergence of the Monte Carlo procedure requires a great terlamellar distance. In like manner, the size of the tacnumber (between 200 and 400) of counter ion^.^^ Thus, toids depends on t h e ionic strength.17 Thus, the a complete simulation of the clay-water-cation system at measurement of Lubetkin et al.1° cannot be interpreted large interlamellar distance would require a prohibitive in t h e absence of t h e required complementary number of particles. As an example, for an interlamelmeasurement of the interlamellar distances. lar distance of 100 A,one needs to simulate the distribution Limitations of the Formalism. The diffuse doubleof 95.000 water molecules and 400 ions for a surface charge layer formalism is a first approximation to modeling the density of the clay of 0.112 C.m-z. clay-water-cation system. Numerous approximations have Another possibility is offered by MO calculations. The had to be done. The first approximation assumes a role of short-range van der Waals attraction has long been uniform distribution of the charges on a smooth clay described with the Hamaker ~onstant.'~J~~'@ However, this surface, instead of taking explicit account of the real gives only a rough description of the short-range charges present on the atomic network of the clay. For interactions. It is unable to reproduce the fine structure cylindrical polyions, such an approximation does not affect of the clay solvation as measured by Israelachvili et al.45 the radial distribution of the counter ion^.^^.^^ To take into account the atomic structure of the clay sheet A second approximation is to ignore ionic correlation and the solvent, we have set about performing MO in the Gouy-Chapman or in the Poisson-Boltzmann calculations on clusters mimicking a clay surface. Results formalisms. However, no significant deviation is observed of these calculations will be reported in the near future. with the ionic concentration profiles obtained by Monte As noted above, the diffuse-layer formalism embodies Carlo simulationsz7 when the diffuse double-layer quite a few approximations about the clay-water interface. formalism incorporates excluded volume effect between Of course, other approaches are available. For instance, the condensed counterion and the clay surface. Spitzer's modeP is an alternative based on local binding Thirdly, specific site binding of the condensed counof the counterions at the clay surface. terions to the clay surface is not taken into account. Such We now return to the important question of the reason a specific interaction includes a partial desolvation of the why a smectite clay can differentiate between alkali metal hydrated counterion to let it coordinate to a specific site cations forming thicker tactoids (three times as many on the clay surface. However, in the framework of polyplatelets: Table V) with cesium than with lithium. What electrolyte theory, the total number of condensed and siteis the key factor in the dissociation of the tactoids? bound counterions remains constant. Colligative properties We shall propose a purely energetic analysis. Tactoids are unstable if the stabilization energy of the water (32) Wells, J. D. Biopolymer 1973, 12, 223-227. molecules entering the interaggregate space is greater than (33) Rinaudo, M.; Milas, M. Chem. Phys. Lett. 1976,41,456-459. the energy lost by the system upon swelling. The energy (34) Iwasa, K.; Mc.Quarrie, D. A,; Kwak, J. C. T. J. Phys. Chem. 1979, 82.1979-1985. - -, - - . - - - - -. gain stems from solvation of the charged aluminosilicate (35) Kwak, J. C. T.; Joshi, Y. M. Biophys. Chem. 1981,13,55-64. ~
~~
(36) Welock, D. J.; Diakun, G.P.; Edwards, H. E.;Phillips, G. 0.;Allen, J. C. Biochim. Biophys. Acta 1980,629,530-538. (37) Neal, C.; Cooper, D. M. Clays Clay Miner.1983,5,367-376. (38) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths Science Publishers: London, 1959. (39) Schra", L. L.; Kwak, J. C. T. Clays Clay Miner.1982,30,4C48. (40) Soumpasis, D. J. Chem. Phys. 1978,69,3190-3196. (41) Klein, B. J.; Pack, G. R. Biopolymer 1983,22, 2331-2352.
(42) Hunter, R. J. Zeta potential in Colloid Science; Principles and Applications;Ottewill, R. H., Rowell, R. L., E&.; Academic Press: London, 1981; Chapter 2. (43) Torrie, G. M.; Valleau, J. P.; Patey, G . N. J. Chem. Phys. 1982, 76,4615-4622. (44) Friedman, H. L. Faraday Discuss. Chem. SOC.1978,64, 7-15. (45) Israelachvili, J. N.; Pashley, R. M. Nature 1983, 306, 249-250. (46) Hamaker, H. C. Physica 1937,4, 1058-1072.
1294 Langmuir, Vol. 6, No. 7, 1990 sheet and from that of the interlamellar cations.47 On the other hand, the energy lost upon swelling stems from the decreased cation-charged sheet attraction and from the reduced water-water interactions in the bulk. The key factor in swelling appears to be an increase in the energy stabilization due to solvation of the interlamellar cations. There is a pronounced correlation between the size of the tactoids and the cation hydration energiesa AHwlv (Table V). Going from cesium to lithium, the solvation enthalpy is multiplied by a factor of 2, as the size of the tactoids reduces. However, this argument is incomplete; it considers only one water molecule in interaction separately with one clay sheet or with one cation. Monte Carlo simulations are required to calculate simultaneously the interaction between many water molecules and cations with the clay sheet. Grand canonical ensemble averaging will allow ~ . ~ the density of water simulation of an open s y ~ t e m , 4since near the clay surface may differ substantially from the bulk density. We work in that direction. These approaches will also serve to model the organization of water molecules near the clay surface.45
Conclusion In summary, a diffuse double-layer treatment provids an adequate model for the swelling of a clay, provided that the conservation of the number of ions is properly expressed, for dilute suspensions only. Interpretation of the swelling at shorter interlamellar distances, below 25 A, must await explicit inclusion of the solvent structure. Acknowledgment. We thank Dr. J. Baudracco, Universitk Paul Sabatier, Toulouse, for interesting discussions. We are grateful to the referees, who raised important points of presentation. Appendix A. Analytical Resolution of the Poisson-Boltzmann Equations In absence of added salt, the solution of eqs 2 and 4a is given byz2 = c(R)/cos' [ S ( Z - R ) / ( R- a ) ] (AI) where c(R),the local counterion concentration midway between the charged sheets, is given by
Delville and Laszlo s t a n s = -ue(R - a)/(2eof&T) (A3) This equation is resolved by a Newton-Gauss iterative procedure. u is the charge density of the clay sheet. This formalism gives a concentration profile in agreement with results from Monte Carlo simulations.27
Appendix B. Calculation of the Digitalized Potential The electrostatic potential generated by a sheet of surface charge u and by M sheets of volume charge p ( i ) in the interval Ix(i), x(i+l)l is
with V8k) = -z/(2tot,)
(B2)
V,(i,Z) = -z(x(i) - x(i+l))/(2eotr)
033)
if z < x ( i ) ,
V*(i,z)= -(2' - x(i+l)z - x ( i ) z + x(i)2)/(2cot,) if x ( i ) < z < x(i+l), and V,(i,Z)= -(zx(i+l) - x(i+1)2 - zx(i) + x(i)2)/(2tot,) if z
> x(i+l).
z is the distance to the clay sheet.
Appendix C. Infinite Dilution For infinite dilution, the local ionic concentrations may be written c+ = co exp(-@)
c- = co exp(4)
(C1)
where the reduced potential (4 = e \ k / ( k T ) ) is given by 4(z) = 2 In ( [ A + B exp(-~Z)]/[A- B exp (-Kz)])
(C2)
with
C(Z)
c ( R ) = 2 s2kTtocr/[(R- a)2e2(N,x lo3)] (A2) where e is the cationic charge. The value of s is the solution of the equation (47) Sposito, G.;Prost, R. Chem. Reo. 1982,82,553-573. (48) Rashin, A. A,; Honig, B.J. Phys. Chem. 1986,89, 5588-5593. (49) Adams, D.J. Mol. Phys. 1974,28,1241-1252. (50)Schoen,M.; Cushman, J. H.; Diestler, D. J.; Rhykerd, C. L. J . Chem. Fhys. 1988,88, 1394-1406.
K~
= 2coe2/(kTtoc,)
A = e ~ p ( 4 ~ /+ 2 )1
B = e ~ p ( 4 ~ /-2 1) The potential 40 is given by exp(4,/2) = [-F
+ (p+ 4)'/2]/(2F)
with
F = -~/(2tot,~~kT)'/'
((23)
