1026
Langmuir 1992,8, 1026-1029
On the Effect of Dielectric Saturation on the Swelling of Clays Joan E. Curry and Donald A. McQuarrie' Department of Chemistry and the Institute of Theoretical Dynamics, University of California, Davis, California 95616-5295 Received November .12,1991 The nonlinear Poisson-Boltzmann equation is solvedwith a field-dependentexpressionfor the dielectric constant to determinethe effect of dielectricsaturation on swellingpressure in clays. Calculationsincluding a scaled and an unscaled version of the approximate expression for the field-dependent dielectricconstant first derived by Booth are compared. It is found that the swellingpressure decreasesin all casesconsidered; however,the effect is significant only when the electricfield at the surface is high enoughto cause dielectric saturation. Nonlinear dielectric effects should be considered in models which predict surface electric fields in excess of lo7 V-m-'.
Introduction Swelling in clay systems is important in agriculture, petroleum engineering, and toxic waste disposal. Clays are typically modeled as interacting infinite charged, flat plates immersed in an electrolyte solution. Under appropriate conditions the clay particles interact repulsively and macroscopic swelling occurs. The process of swelling is not completely understood and much theoretical and experimental work is being conducted in this area. To this point, most theoretical attempts to predict experimental swelling pressures have been based on theories which consider the interacting plates to be immersed in a continuum solvent with a dielectric constant of that of bulk water within the entire region between the surfaces. Water is dipolar and is polarized in the presence of an electric field. At low field strengths, there is a linear relation between the polarization and the field strength and so the permittivity is constant. At higher field strengths, however, the dependence becomes nonlinear and at high enough fields, such as those surrounding ions in solution, all the solvent molecules near the surface are polarized in the direction of the field and the dielectric constant in that region is said to be saturated. The permittivity and therefore the dielectric constant in a saturated or partially saturated region are no longer constant and now depend on the electric field. Clay surfaces such as those in a montmorillonite clay produce electric fields that are often high enough to cause water to be saturated, although the degree and extent of dielectric saturation near a charged surface is not known exactly. In treating this effect in a theory of interacting clay particles, some investigators have proposed that an arbitrary region of low permittivity exists near the charged surface.' Although still within the continuum model, a more rigorous approach would include an expression for the dielectric constant in terms of the electric field. Debye2in his work on the theory of polar liquids derived an expression for the dielectric constant as a function of the electric field. This was later expanded upon by Onsager,3 K i r k ~ o o d and , ~ Frohlich5 but these theories are (1)Henderson, D.; Lozada Cassou, M. J . Colloid Interface Sci. 1986, 114 (I),180-183. (2)Debye, P. Polar Molecules; Dover Publications, Inc.: New York, 1929. (3)Onsager, L. J. Am. Chem. SOC.1936,58,1486. (4)Kirkwood, J. J . Chem. Phys. 1939,7,911. (5)Frohlich, H.Theory of Dielectrics; Oxford University Press: New York, 1949;Chapter V.
valid only at small field strengths. Booth6 adapted this work specifically to water and presented an approximate statistical mechanical expressionfor the dielectric constant of water which is valid at high field strengths. Booth's expression has recently been used to calculate the electrostatic potential profile in a reverse micelle.' In this paper we assume that the swelling pressure is due to the repulsive interaction between interacting electrical double layers formed at charged clay surfaces. We use a field-dependent expression for the dielectric constant, solve the Poisson-Boltzmann equation, and calculate the swelling pressure between two flat, charged plates in a 1:l electrolyte solution. We have used this simple and well-known approach in order to determine if a field-dependent dielectric constant should be considered in the more sophisticated theoretical treatments of the swelling pressure problem. Although the expression for c due to Booth is approximate, we nevertheless assume that its functional form is correct. Thus we also examine a scaled version of the Booth equation which causes a greater change in the dielectric constant for a given field strength.
Theory The theoretical model for this system consists of two infinite charged flat plates immersed in a 1:l electrolyte solution. The solution is composed of point ions immersed in a continuum solvent with a field-dependent dielectric constant. The general Poisson-Boltzmann equation for the electrostatic potential in the electrical double layer is d dv(x) c-zieni -ziecp(x) --(E*)-= -e)xp,( (1) 1 '0 dx dx where cp(x) is the electrostatic potential at some distance from the surface x, is the dielectric constant, E* is the electric field (dcptdx), zi is the valence, ni is the bulk concentration of the ith ionic species, e is the elementary charge, co is the permittivity of a vacuum, k is the Boltzmann constant, and Tis the temperature. Equation 1can be rewritten in the reduced form for a 1:l electrolyte as d d9 -(E)-= sinh 9 dE dF where the reduced potential 9 = ecp/kT, the reduced (6) Booth, F. J. Chem. Phys. 1961,19 (4),391-394. (7)Karpe, P. and Ruckenstein, E. J . Colloid Interface Sci. 1991,141
(2),534-552.
0743-746319212408-1026$03.00/0 0 1992 American Chemical Society
Langmuir, Vol. 8, No. 3, 1992 1027
Clay Swelling Pressure and Dielectric Saturation
75
p% I
i
\
\
I
I
\
15
I n "
0.1
E (v/m)
unmodified Booth dielectric constant (solid line), the Booth expression withy = 2 (longdashed line),and the Booth expression with y = 5 (short dashed line).
distance [ = ax, a2 = 2ne/&T, a constant similar to the Debye length K-', n is the bulk electrolyte concentration, and E = d\k/d[, the reduced electric field. The Booth expression in SI units for the dielectric constant in terms of the reduced electric field is
where L(x) = coth ( x ) - ( l / x ) is the Langevin function, N is the number density of water molecules, taken to be 3.34 X 1028 m-3, q is the refractive index of water (1.331, and p is the dipole moment of a water molecule, 2.025 D. Equation 3 can be rewritten in a simpler form as
in units of C.m-2. Three cases are the Booth expression for E with y = 1 (dots),with y = 2 (circles),and with y = 5 (squares).
menta at greater field strengths are difficult.8 Additionally, a molecular dynamics computer simulation of water under high field strengths predicts a dielectric constant of 55 f 7 for E = 2 X 108 which suggests y = 2.9 The Poisson-Boltzmann equation with e(E) given by eq 5 is
))
+ -$(coth
(BYE)- - " sinh \k B y E dg= which can be rearranged to give
(6)
"-)
AB + = sinh \k (7) dt2 sinh' ( B y E ) B ( T E ) ~ We consider the constant surface charge case, where a is the charge in the clay lattice resulting from isomorphic substitution. The boundary condition at the surface, then, is
(4)
whereA = 7N(q2+2)p/3(146nkTt,)1f2andB= 73n/18kTt,. A plot of e versus E as calculated from eq 3 is shown as the solid line in Figure 1. The dielectric constant begins to differ from the low field value of 78 at about lo8 V/m. The Booth equation predicts complete saturation at approximately 1O1O V-m-l. The exact dependence of t on E is not known, so we have also investigated the effect on the electrostatic potential profile of a scaled Booth expression for the dielectric constant, where y is a scaling factor of the electric field e = q2
0.4
(C/m-2)
Figure 2. Dielectric constantat the surface versus surface charge
3d( q 2
= q2 + A jj L ( B E )
0.3 --(I
Figure 1. Booth expression for the dielectric constant plotted versus the electric field in units of V-m-l. Three cases are the
t
0.2
A +.(BYE) YE
When y = 1 the original Booth expression is recovered. In Figure 1, in addition to the unscaled or y = 1 Booth expression, the dielectric constant as a function of field strength is also plotted for y equals 2 (long dashed line) and 5 (short dashed line). A value of y greater than 1 causes the dependence of the dielectric constant on the field strength to become nonlinear at a lower value of the field strength. It seems reasonable to consider a scaled Booth equation since the actual numerical dependence of t on E is not known exactly. Experiments suggest that the dielectric constant begins to behave nonlinearly at field strengths greater than lo7 V-m-', although measure-
"'
(8)
[=0 Equation 8 can be solved independently of the PoissonBoltzmann equation to determine the electric field and therefore the dielectric constant at the charged surface. Figure 2 shows a plot of the dielectric constant at the surface as a function of the surface charge for y equal to 1 , 2 , and 5 . In addition to the boundary condition at the surface, due to the symmetry of the system, the boundary condition at the midpoint is
E = ad The repulsive electrostatic force between two similar charged flat plates can be calculated using
P = 2nkT(cosh 9, - 1) (10) where \ k is~the reduced potential at the midpoint between ~~
~
(8) Kolodziej, H.A.; Jones, G. P.; Davies, M. J. Chem. Soc., Faraday Trans. 2 1975, 71 (2), 269-274. (9)Nagy, G.; Heinzinger,K. J. Electroanal. Chem. 1990,296,549-558.
Curry and McQuarrie
1028 Langmuir, Vol. 8,No. 3, 1992
2o
r---7
80 70
60 50 u
40 30
2o
i
tl/
I 20
40
60
0
80
2
4
6
Distance (A)
60
I
80
Distance (A)
Figure 4. Electrostatic pressure in atmospheres as a function of the total distance (A) between the two surfaces. Electrolyte concentration is M and the surface charge density is -0.20 C-m-*. The four cases are t = 78.8 (dots)and the Booth expression for c with y = 1 (circles),with y = 2 (squares),and with y = 5 (triangles). the surfaces.1° Equation 7 subject to the boundary conditions in eqs 8 and 9 is solved numerically using a finite difference algorithm. The calculation time was negligible. The electrostatic potential a t the midpoint is used to calculate the electrostatic contribution to the hwelling pressure using eq 10. We have not included other contributions to the swelling pressure such as van der Waals forces since their inclusion only introduces a constant.
Results For conditions typical of clay systems, surface charge density equal to -0.125 C-m-2and electrolyte concentration equal to M, the repulsive pressure versus plate separation is plotted in Figure 3 for e = 78.8 (dots), the Booth expression for t with y = 1 (circles), with y = 2 (squares), and with y = 5 (triangles). Figure 4 shows the same comparison a t a surface charge densityof -0.20 C-m-2. (10) Verwey, E. J. W.; Overbeek, J. Th. G. The Theory ofthe Stability New York, 1948; Chapter V.
of Lyophobic Colloids; Elaevier:
14 18 18 20
(Y
Table 1. Swelling Pressure Values for Increasing Surface Charge and Surface Potential for a Plate Separation of 30 A u, C-m-2 v P,atm cluli -0.25 -0.30 -0.35 -0.40
40
12
Figure 5. Dielectric constant as a function of r for aolutioaeto eq 7 (solid l i e ) and eq 11 (solid line with dots) for a surface charge density of -0.20 C.m-2, a concentration of 10-3 M,and a plate separation of 40 A. For the upper curves y = 1, a = 0.512, and 6 = 1 and for the lower curves y = 5, = 0.971,and 6 = 0.35.
1
0' 20
10 X(A)
Figure 3. Electrostatic ressure in atmospheres as a function of the total distance (A) getween the two surfaces. Electrolyte concentration is M and the surface charge is -0.125 C.m-a. The four cases are c = 78.8 (dots)and the Booth expression for t with y = 1 (circles), with y = 2 (squares), and with y = 5 (triangles).
2o
8
Cp9
0.339 0.350 0.359 0.367
4.76 4.76 4.75 4.75
2.17 2.09 2.04 2.00
In all cases, the pressure values for the Booth expression for c are less than those obtained with a constant value of t = 78.8. This is consistent with the results of Oosawa et al.," who modeled the dielectric constant as a function of the electric displacement field. If e at the surface is near the saturation value of 1.77,then there is a significant reduction in the pressure. The amount the pressure is reduced due to the field-dependent t depends on the reduction of c at the surface. For example, in Figure 3, for a surface charge density of -0.125 C.m-2, there is little reduction in pressure using a fixed value of t = 78.8 compared to using the Booth expression for c with y = 1. There is about a 6 % difference for y = 2 and about a 30% difference for y = 5. By referring to Figure 1 it can be seen that the dielectric constant at the surface drops to only 65 with y = 1, while it drops to 3 with y = 5. For a = -0.20 C-m-2,Figure 4, there is a larger effect and this too can be related to the dielectric constant at the surface. When y = 1 in Figure 4 the dielectric constant at the surface drops to 36, while for y = 5 it is 2.5. Interestingly, if the surface electric field is high enough to cause a drop in the dielectric constant to near its saturation value, the repulsive pressure becomes independent of the surface potential and surface charge. Note that the y = 5 curves in Figures 3 and 4 nearly coincide. Table I contains pressure values for higher surface charges at a 30-A plate separation, which show there is no variation in repulsive pressure for any value of surface charge and corresponding potential. It is common in calculations of the electrical double layer to account for nonlinear dielectric effects at the charged surface by expressing the dielectric constant as an explicit function of x . We have found that one explicit function of x which approximately reproduces the Booth results is of the form (11) Maeda, H.; Oosawa, F. Biophys. Chem. 1980,12, 215-222.
Clay Swelling Pressure and Dielectric Saturation e = e*[1 - a exp(-j3x)l
(11) where a and j3 are parameters and e* is the low field value of the dielectric constant. The parameter a fixes the value oft a t the surface and the parameter j3 controls the slope of the 4 x 1 function. Figure 5 shows a plot of 4 x 1 obtained by solution to eq 7 (solid line) contrasted with the solution to eq 11(solid line with dots) for a surface charge density of -0.2 electrolyte concentration of M and a plate separation of 40 A. The parameters for the upper curves are y = 1, a = 0.542,and j3 = 1 and for the lower curves are 7 = 5, a = 0.971, and j3 = 0.35.
Conclusion The Poisson-Boltzmann equation is solved with an electric field dependent dielectric constant to account for polarization of the solvent near the charged surface. In all cases the nonlinear dielectric effect causes a decrease in electrostatic repulsion between the surfaces. The
Langmuir, Vol. 8, No. 3, 1992 1029
repulsion is significantly different from that calculated using a fixed dielectric constant only if the electric field at the surface is strong enough to cause the dielectric constant a t the surface to drop to near its saturation value. For a model which includes an unscaled Booth expression, surface charges much higher than those in swelling clays are required before there is an influence on the repulsive pressure. If, however, a scaled version is used, then there could be a significant effect. In general, dielectric saturation shouldbe included in more sophisticated treatments if the electric field at the surface is greater than 108Vsm-'. Data on the behavior of the dielectric constant of water at field strengths greater than lo7V-m-l would be helpful in this area.
Acknovledgment. This work has been supported by the National Science Foundation under Grant NSF EAR 8910530.
