Application of Double-Layer Theories to the Extensive Crystalline

We have used the Poisson-Boltzmann (PB) theory and DLVO double-layer theory together with the hypernetted chain formalism based on the primitive model...
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Langmuir 1997, 13, 6241-6248

6241

Application of Double-Layer Theories to the Extensive Crystalline Swelling of Li-Montmorillonite J. P. Quirk* Department of Soil Science and Plant Nutrition, Faculty of Agriculture, University of Western Australia, Nedlands, W.A. 6907, Australia

S. Marcˇelja Department of Applied Mathematics, Research School of Physical Science & Engineering, Australian National University, Canberra ACT 0200, Australia Received May 9, 1997. In Final Form: August 21, 1997X We have used the Poisson-Boltzmann (PB) theory and DLVO double-layer theory together with the hypernetted chain formalism based on the primitive model for electrolyte solutions to examine recently published results for extensive crystalline swelling of Li-montmorillonite as revealed by d(001) spacings over the pressure range 0.05-0.9 MPa and the concentration range 1-10-4 M LiCl. The analysis based on Langmuir’s use of elliptical integrals reveals that the theory satisfactorily predicts surface separation over the range 18-120 Å and that the DLVO theory, with a 5.5 Å Stern layer, indicates Gouy plane potentials ranging from -58 to -224 mV over the concentration range 0.1-10-4 M LiCl and a constant Gouy plane charge of 0.038 C m-2, which is 30% of the crystal lattice charge. The anisotropic hypernetted chain (HNC) analysis, also based on lattice charge, satisfactorily predicts pressure-distance relationships over the range of concentrations. There is no evidence for the operation of the “secondary hydration force” at a surface separation of 18 Å.

Introduction The swelling of the mineral montmorillonite in its lithium or sodium form has been studied by a number of workers with the intention of defining the character and properties of the electric double layer and also because it is perceived as a model for interpreting the behavior of sodium-affected soils.1-3 This mineral has the ideal unitcell formula of (Si8)IV(Al3.34 Mg0.66)VIO20(OH)4-Na0.66, in which the sodium ions balance the charge of the crystal lattice deriving from the isomorphous replacement of Al3+ by Mg2+ in the octahedral layer of the clay lattice. The crystals consist of parallel-aligned elementary aluminosilicate lamellae which are approximately 10 Å thick and 1000-2000 Å in lateral extent. Water molecules can enter between the lamellae and thus cause swelling of the crystal, and as a consequence of this swelling the sodium ions can be exchanged for other ions. The initial stepwise hydration of the crystal results in an increase in the c-axis of the crystal to give d(001) spacings for Li-montmorillonite of 12.5, 15.5, 19, and 22.5 Å, corresponding to one, two, three, and four “layers” of water molecules within the structure. This process is referred to as limited crystalline swelling. Norrish and Quirk4 reported that at a concentration of 0.3 M NaCl the d(001) spacing for Na-montmorillonite increased sharply from 19 to 43 Å, and Norrish5 reported that with further dilution the basal spacing increased in proportion to C-0.5, where C is the molar concentration; similar results were found for Li-montmorillonite. This behavior is described as extensive crystalline swelling and spacings well in excess of 100 Å can be achieved. This behavior contrasts with that of Ca-montmorillonite, which exhibits only limited crystalline swelling, as the d(001) X

Abstract published in Advance ACS Abstracts, October 1, 1997.

(1) Quirk, J. P.; Schofield, R. K. J. Soil Sci. 1955, 6, 163. (2) Quirk, J. P. Philos. Trans. R. Soc. London 1986, 316A, 197. (3) Quirk, J. P. Adv. Agron. 1994, 53, 121. (4) Norrish, K.; Quirk, J. P. Nature 1954, 173, 255. (5) Norrish, K. Discuss. Faraday Soc. 1954, 18, 120.

S0743-7463(97)00484-8 CCC: $14.00

value does not exceed 19 Å at any concentration even in distilled water.4 There have been two distinctly different explanations for the extensive crystalline swelling of Li- and Namontmorillonite. On the one hand various authors5-8 have attributed this type of swelling to the interaction of the diffuse double layers of contiguous clay surfaces in terms of DLVO theory. On the other hand Low9,10 and Viani et al.11 have discounted the influence of diffuse double-layer interactions; they11 measured the swelling pressure of Namontmorillonite in equilibrium with 10-4 M NaCl over a range of separations between 30 and 100 Å and concluded that the structural perturbation of the interfacial water under the influence of the alumino-silicate surfaces, that is the structural component of the swelling pressure, is primarily responsible for the extensive crystalline swelling of montmorillonites. In the mid-1930’s Derjaguin12 distinguished two components of the disjoining (swelling) pressure between interacting particle surfaces; the electrostatic component and the structural component. The electrostatic component results from the diffuse distribution of counterions in the vicinity of a particle surface in aqueous solutions, and this together with the attractive London-Van der Waals force constitutes the basis for the DLVO theory of colloid stability.13,14 The structural component of the disjoining pressure results from the overlapping of two structurally modified boundary layers of water; this force (6) Bolt, G. H. Geotechnique 1956, 6, 86. (7) Aylmore, L. A. G.; Quirk, J. P. Clays Clay Miner. 1962, 9, 104. (8) Lubetkin, S. D.; Middleton, S. R.; Ottewill, R. H. Philos. Trans. R. Soc. London 1984, 311A, 353. (9) Low, P. F. Langmuir 1987, 3, 18. (10) Low, P. F. In Clay-Water Interface and Its Rheological Implications; Gu¨ven, N., Pollastro, R. M., Eds.; The Clay Minerals Society: Boulder, CO, 1992. (11) Viani, B. E., Low, P. F.; Roth, C. B. J. Colloid Interface Sci. 1983, 96, 229. (12) Derjaguin, B. V. Langmuir 1987, 3, 601. (13) Derjaguin, B. V.; Landau, L. D. Acta Physicochem. 1941, 14, 633. (14) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948.

© 1997 American Chemical Society

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hypernetted chain closure; thus, ion size and ion-ion correlations are not neglected as in the DLVO treatment.

is also referred to as “hydration force” or “secondary hydration force”. Pashley,15 Pashley and Israelachvili,16 and McGuiggan and Pashley,17 using the surface force apparatus, have shown that, for separations of molecularly smooth muscovite mica (a layer lattice alumino-silicate) surfaces of less than 25 Å, an additional repulsive force, above that expected from the DLVO theory, is encountered; this force was described as a “secondary hydration force”. It is especially noteworthy that when two K-muscovite surfaces interact, seven oscillations in the force-distance curves were found16 in 1 and 10-3 M KCl solutions with a periodicity of 2.5 ( 0.3 Å, which is close to the diameter of a water molecule (2.76 Å). In this paper we seek to apply double-layer theory to the extensive crystalline swelling of montmorillonite as affected by applied pressure (suction) and electrolyte concentration. In a recently published paper, Zhang et al.18 have provided the first comprehensive set of experimental observations relating d(001) values for Li-montmorillonite to applied pressure and LiCl concentration; the pressure range used extended from 0.05 to 1 MPa, and the concentrations were 1, 0.5, 0.3, 0.1, 10-2, 10-3, and 10-4 M LiCl; the surface separations measured ranged from 18 to 80 Å. LiCl rather than NaCl was used because the extensive crystalline swelling of Li-montmorillonite takes place over a wider concentration range; Norrish5 found that the transition to extensive crystalline swelling for dried oriented flakes took place at 0.66 M LiCl whereas the transition in NaCl took place at 0.25 M. Zhang et al.18 commented that DLVO theory predicts that the separation of interacting surfaces decreases with increasing electrolyte concentration and that fulfillment of this prediction has been taken, by others, as evidence in favor of the theory. They18 further indicate that the ions of added electrolyte will tend to disrupt any arrangement of water molecules imposed by the particle surfaces and, consequently, will reduce the repulsive force due to the hydration of these surfaces. They conclude that the decrease in particle separation with increasing concentration of electrolyte is “not necessarily attributable to the repression of electric-double layers by added electrolyte”. We will show that the extensive swelling behavior of Li-montmorillonite accords closely with double-layer theory. We have used classical double-layer theory, that is nonlinear Poisson-Boltzmann (PB) and DerjaguinLandau-Verwey-Overbeek (DLVO) theories, to analyze the above authors’ results. We have also used the double layer theory developed by Kjellander and Marcˇelja,19 who evaluated the properties of the primitive model of electrolyte solutions in the inhomogeneous fluid between charged plates. In the primitive model of aqueous electrolyte solutions the ions are treated as hard spheres with an assigned size, and the aqueous solvent is treated as a dielectric continuum. The interactions comprise electrostatic and hard core terms. The analyses use the charge density at the surface, that is the crystallographic charge density arising from isomorphous replacement in the crystal lattice. The correlation functions for the inhomogeneous ionic fluid between the surfaces are obtained from the solution of integral equations using the

in which 1/κ is the Debye length, x is the half-distance of the separation of Gouy planes for interacting particles, YG is the reduced Gouy plane electric potential, U is the reduced electric potential at the midplane, and F1 denotes an elliptical integral of the first kind. The Stern layer thickness was taken as 5.5 Å, which relates to the fact that in the extensive crystalline swelling range (d(001) values > 35 Å) for Li-montmorillonite5 and Livermiculite23 a linear relationship is found between the d(001) values and C-0.5 and when this relationship is projected to the ordinate, an intercept of 21 Å is found;5,23 this is considered to be made up of an elementary lamella thickness of montmorillonite of 10 Å and two Stern layers of 5.5 Å on each of the interacting surfaces. The surface to surface distance is obtained from the X-ray diffraction results and is denoted by 2X so that x is X 5.5 Å. Equation 3 has been used to obtain Gouy plane reduced potentials. For each concentration (activity) of LiCl solutions a value of 1/κ is combined with the half-distance of surface separation minus 5.5 Å for each applied pressure to give values of κx. The values of the reduced midplane potential, U, have been calculated by substituting into eq 2 the activity and the total pressure, that is the experimentally applied pressure plus the Van der Waals contribution (eq 1) calculated using the siloxane to siloxane distance. When each value of κx and U obtained from the experimental information is substituted into eq 3, values for F1(exp((-U),arcsin exp(-YG - U)/2))] are obtained, from which values for YG for each pressure at the various LiCl concentrations are obtained. The values of YG so obtained can be substituted together with the values of U in the following equation14 to obtain the surface density of charge at the Gouy plane (σG),

(15) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531. (16) Pashley, R. M.; Israelachvili, J. N. J. Colloid Interface Sci. 1983, 101, 511. (17) McGuiggan, P. M.; Pashley, R. M. J. Phys. Chem. 1988, 92, 1235. (18) Zhang, F.; Low, P. F.; Roth, C. B. J. Colloid Interface Sci. 1995, 173, 34. (19) Kjellander, R.; Marcˇelja, S. J. Chem. Phys. 1985, 82, 2122.

(20) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1959. (21) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1978, 74, 975. (22) Langmuir, I. J. Chem. Phys. 1938, 6, 873. (23) Norrish, K.; Rausell-Colom, J. A. Clays Clay Miner. 1963, 10, 123.

Application of Theories al.18

Zhang et in their Figure 5 provide surface separations, obtained by X-ray diffraction, plotted against pressure for Limontmorillonite in equilibrium with LiCl solutions ranging in concentration from 0.5 to 10-3 M LiCl; these curves are reproduced in our Figures 4-8. In our calculations we have used activities rather than concentrations.20 To their experimentally imposed pressures we have added the Van der Waals attractive contribution (PV) given by

Pv )

[

]

A 1 1 2 + 6π D3 (D + 2t)3 (D + t)3

(1)

in which D is the surface to surface distance, t is the lamellae or plate thickness, taken as 1 nm, and A is the Hamaker constant, which has the value 2.2 × 10-20 J.21 When the Van der Waals pressure is added to the pressure applied to the montmorillonite gel in the experiment, we obtain the total pressure (PT) opposing the operation of repulsive double-layer forces. This pressure is then substituted into the following equation22

PT ) 2RTa(cosh U -1)

(2)

to obtain the reduced midplane potential U for each pressure and activity a. The values of U are used in several ways. Langmuir22 has provided the following equation,

[ (

κx ) 2 exp(-U/2) F1 exp(-U),

π 2

)

]

F1(exp(-U),arcsin exp(-(YG - U)/2))

(3)

Crystalline Swelling of Li-Montmorillonite σG )

(

)

10-3aRT (2 cosh YG - 2 cosh U)1/2 2π

Langmuir, Vol. 13, No. 23, 1997 6243 (4)

In Figure 1 the Gouy plane reduced potential YG is plotted against the total applied pressure for all concentrations and pressures. Table 1 sets out the information on the surface density of charge at the Gouy plane, σG. If we ignore the change of double-layer structure in the Stern layer, eq 3 can also be used to obtain values of X. The charge at the actual surface, referred to as the crystal lattice charge, is substituted into eq 4 to obtain values of Ys, which is the reduced surface potential. The values of Ys increase to a small extent with increased pressure. Substitution of Ys and U into eq 3 together with 1/κ enables values of Xcalc to be obtained. In Figure 2 these values are plotted against the values of X obtained by X-ray diffraction.18 Similar results23 for the swelling of a single crystal of Li-vermiculite in relation to pressure in equilibrium with a 0.03 M LiCl solution are used to arrive at Figure 3, in which the calculated values of X are compared with those obtained experimentally. The ideal unit cell for Wyoming montmorillonite has a charge of 0.66 e, and the unit-cell parameters are a ) 5.17 Å and b ) 8.95 Å, which indicates a unit-cell area of 92.5 Å2 or one unit of charge per 139 Å2. We have used the value of one unit charge per 135 Å2, corresponding to 0.119 C m-2. From the above information and the ideal unit-cell composition, the surface area of the clay available to water can be calculated as 760 m2 g-1. Similar considerations using the information given by Mathieson and Walker24 lead to one unit of charge per 75 Å2 for Kenya vermiculite, and this corresponds to 0.214 C m-2. The area available for water adsorption is 740 m2 g-1. The anisotropic hypernetted chain analysis (AHNC), as developed by Kjellander and Marcˇelja19 uses the crystallographic charge density and an adsorption potential for counterions in contact with the surface. The Li ion is assigned a radius of 2.7 Å. The adsorption potential is not known and can be used as a single free parameter. In the Kjellander and Marcˇelja19 analysis the net pressure between the charged plates is given by

Pnet ) Pionic + PV - Pbulk

Figure 1. Reduced Gouy plane potentials for Li-montmorillonite in relation to pressure and LiCl concentration. The range of κx is shown for each concentration. The lines are derived by using eq 2 with a surface density of charge of 0.038 C m-2 (Table 1).

(5)

in which Pbulk refers to the bulk or external solution, PV is the Van der Waals (Hamaker) contribution, and Pionic is made up of several components

Pionic ) kTΣni + Pel + Pcore

(6)

where ni is the number concentration of ions of species i at the midplane and provides the kinetic component of the pressure, Pel is the electrostatic force across the midplane arising from correlations between ions, and Pcore is the pressure component arising from core-core interaction across the midplane. The accuracy of the method has been confirmed by the Monte Carlo simulation.25 Kjellander and Marcˇelja26 have remarked that despite its wellknown deficiencies the nonlinear Poisson-Boltzmann equation is still the most practical and widely used method for describing the electric double layers in aqueous ionic solutions. They showed that the effects of the electrostatically induced ion-ion correlations and of finite ion size, which are both neglected in the PB and DLVO theory, tend to compensate each other in calculations of double-layer interactions.

Discussion Surface Potentials. It can be seen from Figure 1 that the values of YG vary with concentration in the manner expected from DLVO theory. It can also be seen from the experimental points that at each concentration YG increases with increasing pressure. We have used YG values corresponding to 0.4 MPa, as shown in Figure 1, to calculate the surface charge density at the Gouy plane (24) Mathieson, A. McL.; Walker, G. F. Am. Mineral 1954, 39, 231. (25) Kjellander, R.; Akesson, T.; Jo¨nsson, B.; Marcˇelja, S. J. Chem. Phys. 1992, 97, 1424. (26) Kjellander, R.; Marcˇelja, S. J. Phys. Chem. 1986, 90, 1230.

Figure 2. Half-distance for surface to surface separation for Li-montmorillonite (Xcalc), obtained using the experimental information of applied pressure, concentration, and reduced surface potential, compared with the experimental values (Xmeas) obtained from X-ray diffraction measurements. The values for the reduced surface potential were obtained using eq 4 together with the crystal lattice charge 0.119 C m-2.

using eq 4. It can be seen from Table 1 that σG is constant over the range 0.1 to 10-4 M LiCl and has the value 0.038 C m-2 and that the surface potentials vary from -58 to -224 mV. For comparison with the YG values given in Table 1, the respective values of the reduced surface potential (Ys) based on a crystal lattice charge of 0.119 C m-2 are 2.54, 2.99, 3.96, 6.11, 8.33, and 10.6. It would naturally be expected that the charge at the siloxane surface of the clay lamellae would be constant, since this charge arises from isomorphous substitution in

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Figure 3. Half-distance of surface-to-surface separation for Li-vermiculite in 0.03 M LiCl (Xcalc), obtained using the experimental information of applied pressure, concentration, and reduced surface potential, compared with the experimental values (Xmeas) obtained from X-ray diffraction measurements. The value for the reduced surface potential is obtained using eq 4 together with the crystal lattice charge 0.214 C m-2. The open symbols indicate that a condensed separation of about 5 Å is also present. Table 1. Gouy plane Reduced Potentials (DLVO) for Li-Montmorillonite at Approximately 0.4 MPa Pressure and Various Concentrations Together with Gouy Plane Charge activity/ mol L-1

Gouy plane reduced potential (YG)

Gouy plane chargea (σG)/C m-2

percentage charge in diffuse layer

0.3695 0.224 0.0791 0.0089 0.00096 0.0001

0.94 1.39 2.26 4.26 6.38 8.71

0.023 0.033 0.038 0.038 0.037 0.038

19.3 27.7 32.0 32.0 31.1 32.0

a For comparison, the crystallographic surface charge density is 0.119 C m-2.

the clay crystal lattice and in the case of montmorillonite (Wyoming) the principal substitution is Mg2+ for Al3+ in the octahedral layer of this 2:1 mineral; this substitution is about 4.5 Å from the siloxane surface. The magnitude of this charge is 0.119 C m-2, so that about 30% of the total surface charge resides in the diffuse part of the electric double layer in the concentration range 10-410-1 M LiCl. The Gouy plane charge is 0.033 C m-2 at the concentration of 0.3 M LiCl and decreases to 0.023 C m-2 at 0.5 M LiCl. These lower values are probably occasioned by the fact that the concentration is close to 0.66 M LiCl, at which concentration Norrish5 reported a transition from a d(001) value of 22.5 Å to 35 Å and then a subsequent increase in d(001) values in proportion to C-0.5. When the d(001) spacing is 22.5 Å, the counterions are shared by the two surfaces,27 but when expansion takes place, each surface has its own set of counterions forming a diffuse double layer. The lines in Figure 1 were obtained by using eq 4 with the constant surface charge (0.038 C m-2), as shown in Table 1. The close correspondence between the experi(27) Slade, P. J.; Stone, P. A.; Radoshlovich, E. D. Clays Clay Miner. 1985, 33, 51.

mental points and the curves in Figure 1 for pressures above and below 0.4 MPa indicates the term [2 cosh YG - 2 cosh U]1/2 in eq 4 is operative and that the change in YG with increasing pressure at each concentration is satisfactorily accounted for by the DLVO theory. The values of YG which are below the lines in Figure 1 at pressures of about 0.1 MPa or less may be accounted for by some loss of parallel alignment especially at lower concentrations. The range of κx values at each concentration is indicated in Figure 1. There is obviously no need for charge regulation, as with muscovite mica, for which the proton has a significant exchange advantage over alkali metal cations.15 Pashley and Quirk28 have explained this advantage as arising from the nature of the exchange sites which result from the substitution of Al3+ for Si4+ in the tetrahedral sheet of muscovite, and as a result the negative charge is associated with a triad of oxygens immediately above the site of isomorphous replacement. The competitive advantage of the proton is because it is able to combine with a water molecule at the exchange site to form an hydronium ion at the surface whereas alkali metal cations are restrained from approaching the surface because of their hydration shell. Farmer and Russell29 from infrared and other evidence have concluded that for montmorillonite the charge, which arises from isomorphous replacement in the octahedral sheet of the lattice, is distributed over many oxygens and so can be regarded as smeared out. Pashley15 has shown that the pKm values for hydronium and Li ions indicate a marked preference for the hydronium ions. Lubetkin et al.,8 using a technique similar to that of Zhang et al.,18 obtained pressure-distance curves for Limontmorillonite (Wyoming) in equilibrium with 10-2, 10-3, and 10-4 M LiCl. To interpret their results, they used the following equation for weak interaction

P ) 64n0kT tanh2(eΨs/kT) exp(-κh)

(7)

applicable for h > 2/κ, where h represents the surface to surface distance obtained by dividing twice the water content of the clay by the surface area (760 m2 g-1); ψs is the surface potential, and the other symbols have their usual physicochemical meaning. A plot of ln P against h should, according to eq 7, be linear, the negative slope should yield values of κ, and the intercept at h ) 0 provides a value for the surface potential ψs. However these plots for Li-montmorillonite for 10-2 and 10-3 M LiCl were linear only for pressures less than 0.1 and 0.02 MPa. The values for h overestimate the particle to particle separation, since the water content of the clay would involve water in pores in addition to that in the interlamellar spaces. For instance at 0.1 MPa pressures in equilibrium with 10-2 M LiCl the value for 2X from Zhang et al.’s results is 61 Å and the corresponding Lubetkin et al. value for h is 85 Å. Because of such differences it is not possible to use eq 3 to interpret the Lubetkin et al.8 results, which encompass a range of pressures similar to that of Zhang et al.18 Electrokinetic Measurement of Surface Potentials. Callaghan and Ottewill30 reported ζ potential values for Na-montmorillonite (Wyoming) between -50 and -44 mV over the concentration range 10-1-10-4 M NaCl. Low,31 also using Na-montmorillonite, reported a value of -69 mV in 10-4 M NaCl, and subsequently Miller and (28) Pashley, R. M.; Quirk, J. P. Soil Sci. Soc. Am. J. 1989, 53, 1660. (29) Farmer, V. C.; Russell, J. D. Trans. Faraday Soc. 1971, 67, 2737. (30) Callaghan, I. C.; Ottewill, R. H. Discuss. Faraday Soc. 1974, 57, 110. (31) Low, P. F. Soil Sci. Soc. Am. J. 1981, 45, 1074.

Crystalline Swelling of Li-Montmorillonite

Langmuir, Vol. 13, No. 23, 1997 6245

Low32 reported a constant potential of -71 mV for Limontmorillonite over the concentration range 10-1-10-4 M LiCl. Horikawa et al.,33 working with Li-montmorillonite from the same source, obtained ζ potential values ranging from -41 to -69 mV over the concentration range 10-2-10-4 M LiCl. Reference to Figure 1 shows that if we take the values of the reduced Gouy plane potential at the ordinate (no interaction), the surface potentials are seen to vary from -45 to -208 mV over the concentration range 10-1-10-4 M LiCl, the crucial issue being that over this concentration range, as shown in Table 1, the charge at the Gouy plane is constant at 0.038 C m-2. Thus ζ potential measurements are not appropriate for predicting the swelling behavior of Li-montmorillonite. The difference in the surface potentials obtained by the two methods may arise from the fact that the 10 Å lamellae when in suspension do not correspond to rigid plates, an outward shift in the shear plane in more dilute solutions, or surface conduction. Film Thickness. In Figure 2 the measured halfdistance of surface to surface separation for Li-montmorillonite, X, is compared with that calculated using eq 3. The calculated values are based on the values of U obtained from the total pressure, the activity of the solution, and the surface potential Ys, which is obtained by substituting 0.119 C m-2 into eq 4. Substitution of these values into eq 3 together with the corresponding values of U for each pressure should yield reliable values of X, since the values of U are derived from experimental pressures which can be measured accurately and the surface density of charge should also be accurate. In eq 3 the principal term in the evaluation of X, 2 exp(-U/2)[F1(exp(-U),π/2)], is, in general, much larger than the second term, 2 exp(-U/2)[F1(exp(-U),arcsin exp((Ys - U)/2))]; the value of the second term varies between 2.8 and 3.0 Å over the whole range of experimental information provided by Zhang et al.18 Thus only quite small errors would be introduced in the calculation of X values if the second term was considered as being constant with a value of 3 Å. Equation 4 indicates that as σs increases the value of (2 cosh Ys - 2 cosh U)1/2 also increases and that hence, if U remains constant, Ys must also increase. Furthermore, the values of [F1(exp(-U),arcsin exp(-(Ys - U)/2))] will decrease. We have indicated that for montmorillonite the product of this term with 2/κ exp(-U/2) is approximately constant at 3 Å; thus as σs increases, this value must decrease. For instance it can be shown that, with the surface density of charge of muscovite mica (0.340 C m-2), this value becomes virtually constant at 1 Å. Thus X is not very sensitive to changes in surface density of charge. For a group of montmorillonites, with charge density ranging from 0.08 to 0.18 C m-2, Low10 has reported surface to surface separations (2X) measured by X-ray diffraction,11 for applied pressures of 0.1, 0.4, and 0.69 MPa in equilibrium with 10-4 M NaCl. This information demonstrates that the values obtained for 2X are almost constant at each pressure over the above range of surface density of charge. This is in accordance with the above considerations of diffuse double-layer theory, and it cannot therefore be argued that the relative insensitivity of surface separation to surface density of charge is evidence that “double layer repulsion does not contribute significantly to swelling”.10 It can be seen from Figure 2 that there is good agreement between Xcalc and Xmeas for measured values between 9

and 21 Å involving 22 experimental observations. Above Xmeas values of 21 Å or a surface separation of 42 Å there are clear deviations from the measured values (six points) in that the calculated values are appreciably larger than the measured values of X. These points are for pressures of 0.1 and 0.2 MPa for concentrations of 10-1-10-4 M LiCl, with the discrepancy becoming quite large at 0.1 MPa and 10-4 M LiCl. At small applied pressures and especially at small concentrations, X-ray diffraction peaks are very broad and it seems that the modal values are an underestimate, as they do not relate to a well aligned system of clay lamellae. It is relevant that the opportunity for disorder increases under the conditions stipulated above, since the montmorillonite exists in a form of tactoids which are quasicrystals,34 exhibiting extensive crystalline swelling. The notable characteristic of a quasicrystal is that it lacks a-b order in the stacking of the lamellae. Furthermore, Li-montmorillonite produces stable suspensions at 10-2 M LiCl, indicating a broad and shallow secondary potential minimum. Thus the development of disorder seems to be the most likely explanation of the deviations shown in Figure 2. Norrish and Rausell-Colom23 carried out a similar experiment to those of Zhang et al.18 using a single-crystal Li-vermiculite immersed in 0.03 M LiCl (a ) 0.0265), to which loads corresponding to pressures ranging from 0.7 to 0.004 MPa were applied. The plot of Xcalc versus Xmeas derived from this information is presented in Figure 3. From this figure it may be noted that there is good agreement between the measured and calculated values of X up to 60 Å, that is a surface separation of 120 Å. The two open points in the figure denote mixtures of the expanded and condensed spacing at 0.7 and 0.53 MPa pressure, with the proportion of the condensed d(001) spacing of 15 Å being greater at 0.7 MPa. As the Livermiculite is a single crystal, there would be a-b order and hence the lamellae would be well aligned especially as the crystal would have been in the laboratory environment of about 70% relative humidity; application of the Kelvin equation for the vapor pressure above a curved meniscus and the Young-Laplace equation for the pressure difference across a curved meniscus indicates the equivalent of a pressure of about 50 MPa; the distance of surface separation under these circumstances is approximately 5.5 Å, and as a result the Van der Waals force would be large. The maximum pressure applied to the montmorillonite used by Zhang et al., was about 0.9 MPa, so that the development of some disorder at small concentrations and pressures would not be unexpected. Thus it can be concluded that the DLVO theory is well obeyed over a range of separations from 18 to 120 Å. We believe that the above information is the first comprehensive treatment of swelling pressures in terms of DLVO theory which, in the main, has been applied in a semiquantitative way to colloid stability in suspensions. The agreement between Xmeas and Xcalc for surface separations of about 18 Å indicates that at this separation there is no additional pressure contributed by a “secondary hydration force”. This is consistent with recent theoretical findings of Marcˇelja,35 who found that, for Na-montmorillonite for surface separations between 10 and 30 Å, the “secondary hydration forces” were small. However for muscovite mica with a surface charge density (0.340 C m-2) about three times that of montmorillonite, the additional force in excess of that expected in DLVO theory was considerable. These analyses introduced the potential

(32) Miller, S. E.; Low, P. F. Langmuir 1990, 6, 572. (33) Horikawa, Y.; Murray, R. S.; Quirk, J. P. Colloids Surf. 1988, 32, 181.

(34) Quirk, J. P.; Aylmore, L. A. G. Soil Sci. Soc. Am. Proc. 1971, 35, 652. (35) Marcˇelja, S. Nature (London) 1997, 385, 689.

6246 Langmuir, Vol. 13, No. 23, 1997

Figure 4. Experimental points ([) for the pressure-surface separation relationship for 10-3M LiCl shown together with the predicted values using the nonlinear PB equation (s) and the anisotropic hypernetted chain formalism with zero (‚‚‚) and 2.5kT (- - -) adhesion energy for the Li ion.

of mean force for water-separated Na-Na ion pairs into the anisotropic HNC formalism. Pressure-Distance Relationship Using a Primitive Model. Figures 4-8 show a plot of pressure versus separation results corresponding to the experiments of Zhang et al. and calculated using the anisotropic HNC formalism. The inputs to these calculations are the crystallographic surface charge density 0.119 C m-2, an assigned radius of 2.7 Å for the Li ions, the surface separation of the siloxane oxygen surface of interacting montmorillonite lamellae, and the selected value for the counterion-surface adhesion free energy. From the above information, curves are derived for the expected pressure which are compared with the experimental values. Two cases are shown in the figures: one in which adhesion energy is zero and another for which an adhesion energy of 2.5kT is assigned to the counterions in contact with the surface. The value of 2.5kT is a free parameter, which is however kept fixed throughout the range of electrolyte concentrations. The pressure-distance relationships generated in the above manner can be compared with the experimental results for pressure versus distance of separation, as shown in Zhang et al.’s Figure 5, which is reproduced individually in Figures 4-8. Also included in the figures is a plot based on the PB equation, for which the only input other than the electrolyte concentration was the surface density of charge. From Figures 4, 5, and 6, for which the LiCl concentrations are 10-3, 10-2, and 10-1 M, it can be noted that there is good agreement between the AHNC curves and the experimental points when an adsorption energy of 2.5kT is used. There is also reasonably good agreement with the PB equation. The anisotropic HNC formalism has not been applied to the 10-4 M LiCl swelling results because of accuracy problems, associated with ion pairing.36 In Figure 7, for 0.3 M LiCl, it can be seen that the experimental points lie between the two AHNC curves. The 2.5kT curve shows the Van der Waals jump into “contact” at a separation of about 10 Å. Figure 8, for the concentration 0.5 M LiCl, corresponds to an experimental system where equilibrium is established between states with different separations. The effect of discrete solvent structure is not reproduced in primitive model calculations. (36) Kalyuzhnyi, Yu. V.; Holovko, M. F.; Haymet, A. D. J. J. Chem. Phys. 1991, 95, 9151.

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Figure 5. The experimental points ([) for the pressuresurface separation relationship for 10-2 M LiCl shown together with the predicted values using the nonlinear PB equation (s) and the anisotropic hypernetted chain formalism with zero (‚‚‚) and 2.5kT (- - -) adhesion energy for the Li ion.

Figure 6. Experimental points ([) for the pressure-surface separation relationship for 10-1 M LiCl shown together with the predicted values using the nonlinear PB equation (s) and the anisotropic hypernetted chain formalism with zero (‚‚‚) and 2.5kT (- - -) adhesion energy for the Li ion.

Figure 7. Experimental points ([) for the pressure-surface separation relationship for 0.3 M LiCl shown together with the predicted values using the nonlinear PB equation (s) and the anisotropic hypernetted chain formalism with zero (‚‚‚) and 2.5kT (- - -) adhesion energy for the Li ion. The theory predicts the Van der Waals jump.

For an electrolyte concentration of 1.0 M the calculated results are attractive throughout the separation range. The collapse at high electrolyte concentrations is thus seen as arising from the correlation attraction between

Crystalline Swelling of Li-Montmorillonite

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Figure 8. Experimental points for the pressure-surface separation relationship for 0.5 M LiCl are shown together with the predicted values using the nonlinear PB equation (s) and the anisotropic hypernetted chain formalism with zero (‚‚‚) and 2.5kT (- - -) adhesion energy for the Li ion. The theory predicts the Van der Waals jump. Table 2. Effect of Pressure and Concentration on the Transition of Li-Montmorillonite from Extensive Crystalline Swelling to the Condensed State18 pressure/ MPa

concn/M

surface separation/Å

condition

0.9 0.5 0.4 0.05

0.3 0.5 0.5 1.0

17.3 17.0 and 13.3 17.6 10.0

expanded partially condensed expanded condensed

the adsorbed ions, free ions and the surface charge, which is sufficient to overcome weak double-layer repulsion and lead to the condensed state. Conditions for Lamellae Condensation. The lamellae of Li-montmorillonite can exist in an expanded state for which the surface separation is greater than 18 Å or in a condensed state when the surface separation is about 10 or 13.3 Å, as shown in Table 2. The difference between these two states is that in one case diffuse layer forces are operative and each surface has its own set of cations whereas in the condensed state the counterions are shared by the two surfaces.27 It can be seen from Table 2 that at a concentration of 0.5 M LiCl condensation is partially attained between pressures of 0.4 and 0.5 MPa. At 0.3 M LiCl no condensation occurs when a pressure of 0.9 MPa is applied, and at 1.0 M LiCl a condensed state corresponding to a 10 Å separation is present at all pressures including the smallest of 0.05 MPa. For Li-Kenya vermiculite with a surface charge density of about twice that for the montmorillonite (0.214 C m-2), Norrish and Rausell-Colom23 reported partially condensed structures at 0.7 and 0.5 MPa in the presence of 0.03 M LiCl. At 0.5 MPa pressure the modal separation is 47 Å, and this exists together with a surface separation of 5.3 Å (d(001) ) 14.8 Å). The authors give no details of the distribution of spacings for 5 MPa applied pressure, but such information is provided for 0.2 MPa, at which pressure no condensation is evident, and for 0.7 MPa. On the basis that the smallest separation in a distribution is most likely to condense, we have estimated that these values are respectively 35 and 30 Å for 0.2 and 0.7 MPa. Thus it may be inferred that for separations of about 30 Å vermiculite lamellae would collapse when the applied pressure is 0.7 MPa. The pressures at which condensation or partial condensation take place are similar, but the concentrations are very different, being 0.5 M for Li-montmorillonite and 0.03 M for Li-vermiculite. The difference between the

structural origin of the charge for the two surfaces may be a factor involved in promoting the more ready condensation of the vermiculite lamellae. For each eight silicon atoms in the tetrahedral sheet of vermiculite, 2.29 are replaced by aluminum and there is an excess of positive charge due to replacement in the octahedral sheet of the crystal lattice so that the resulting surface charge of 0.214 C m-2 is less than the 0.340 C m-2 for muscovite, which has two of each eight silicon atoms in the tetrahedral sheet replaced by aluminum and a neutral octahedral sheet.37 By contrast, virtually all the negative charge of the montmorillonite crystal lattice arises from isomorphous replacement in the octahedral sheet. Pashley15 has reported that muscovite surfaces jump into “contact” at separations of 20 Å in an 0.01 M LiCl solution, and when “contact” probably involves the separation of surfaces by two layers of water molecules,38 then the actual separation at which the Van der Waals jump takes place would be about 25 Å. He also found that, at a concentration of 0.06 M LiCl, the jump into contact, as expected from DLVO theory, did not occur and that the energy of interaction increased considerably with decreasing distance; this is the experimental basis for the “secondary hydration force”. Pashley15 obtained similar results for other alkali metal cations, the difference between the alkali metal cations being that the larger the unhydrated radius the smaller the concentration at which the transition to a secondary hydration force occurs.15 This has been explained in terms of a competition between protons and the alkali metal ions. The Van der Waal’s jump seems to occur when the occupancy of the surface sites by alkali metal ions has been reduced to one-third,15,39 that is to a value similar to that for montmorillonite if the hydronium ions at the immediate surface are so strongly retained as to, in effect, reduce the surface charge. Significance for Future Work. For soil and clay science the most significant feature of the Kjellander and Marcˇelja theory is that, for close distances of surface approach (about 10 Å or less) and when polyvalent ions (radius < 3 Å) balance the surface charge, an attractive pressure is predicted. Kjellander et al.40 have successfully predicted the magnitude of these attractive pressures for divalent montmorillonite and vermiculite and the surface separation at which the particles exist within the primary potential minimum in each case. It is our intention to combine the findings of this work with those of the present analysis to obtain a quantitative prediction of mixed cation systems with different proportions of monovalent and divalent ions balancing the clay particle charge. Such a study is of fundamental importance for the understanding of the behavior of sodic soils.2,3,41 We emphasize that unlike primitive model calculations PB and DLVO treatments do not provide electrostatic attractive forces for close distances of approach when particle charge is balanced by divalent ions. Conclusions In general terms the analyses reported here show that the DLVO theory and the anisotropic HNC formalism satisfactorily describe the interaction of Li-montmorillonite lamellae over the concentration range 10-1-10-4 M LiCl and for surface separations ranging from 18 to 60 Å. For a single crystal of Li-vermiculite with a-b order and hence well-aligned lamellae, the DLVO theory is (37) Norrish, K. Proc. Int. Clay Conf. 1972, 417 (38) Quirk, J. P.; Pashley, R. M. J. Phys. Chem. 1991, 95, 1660. (39) Pashley, R. M. Chem. Scr. 1985, 25, 22. (40) Kjellander, R.; Marcˇelja, S.; Quirk, J. P. J. Colloid Interface Sci. 1988, 126, 194. (41) Shainberg, I.; Lety, J. Hilgardia 1984, 52, 1.

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applicable up to separations of 120 Å, the limit of experimental measurement in 0.03 M LiCl. The Gouy plane potentials obtained by using a Stern layer thickness of 5.5 Å, justified on the basis of diffraction studies, show the theoretically expected variations with respect to concentration and applied pressure. The variation with respect to concentration is markedly different from the restricted range of values obtained by electrokinetic methods. The range of Gouy plane potentials was from -15 to -225 mV over the concentration range 0.5-10-4 M Li Cl. This analysis also indicated that the surface charge at the Gouy plane is constant at 0.038 C m-2, being about 30% of the crystal lattice charge. This constancy is not surprising in view of the fact that the surface charge arises from isomorphous replacements in the crystal lattice whereby a lattice cation is replaced by another with a lower valency. Unlike the muscovite

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surface, no regulation is required because protons do not have an exchange advantage because the negative lattice charge is distributed over many siloxane oxygen atoms. At surface separations of 18 Å for Li-montmorillonite there is no evidence for a “secondary hydration force” or the structural component of disjoining pressure. This is different from the results reported for muscovite mica, for which the three time greater surface density of charge seems to be the significant factor. Acknowledgment. We are pleased to acknowledge helpful discussions with Prof. R. M. Pashley of the Australian National University and Dr. K. Norrish of the CSIRO, Division of Soils. LA970484L