Langmuir 1994,10,2125-2130
2125
The Modified Gouy-Chapman Theory: Comparisons between Electrical Double Layer Models of Clay Swelling Jeffery A. Greathouse,*Scott E. Feller, and Donald A. McQuarrie Department of Chemistry and Institute of Theoretical Dynamics, University of California, Davis, California 95616 Received October 12,1993. I n Final Form: February 25, 1994@ The modified Gouy-Chapman (MGC)theory ofthe electrical double layer is used to calculatethe swelling pressure (n)between planar double layers using three possible boundary conditions: constant surface charge density (ao),constant surface potential (q,,),and constant potential at the plane of closest approach ofthe ions ( 1 ~ ~ )It. is shown that, at charge densitiestypical ofmontmorillonite clays, the swelling pressures ofthe three models are different only at plate separationsof a few ionic diameters. Beyond those separations, the three pressures are identical. The trend of increasing ll with increasing ionic size is shown for the two constant potential models, in agreement with previous work on the constant surface charge density model. Finally, the MGC theory is compared with experimental data from clay swelling experiments. The comparisons show that the MGC theory agrees qualitativelywith experiment,while the absence of a plane of closest approach results in poor agreement.
Introduction The study of the electrical double layer is a problem of great significance in colloid science. The classical theory of the single electrical double layer began with the work of Gouyl and Chapman,2where the ions in solution were modeled as point charges. An improvement of this theory, known as the modified Gouy-Chapman the0ry,3,~ included the finite size of the ions with respect to their interaction with the surface in the form of a plane of closest approach. While all of the ions remain in the diffise layer and interact with each other as point charges, they interact with the surface as hard ~ p h e r e s .While ~ the MGC theory has enjoyed great success in predicting many experimental observations, it has been criticized for its failure to describe systems containing divalent ion^^-^ due to the fact that short range correlations between the ions are neglected. Additionally, Monte Carlo computer simulations have shown that the MGC theory fails for concentrated solutions (> 1 M) of monovalent electrolytes? Recently, theories based on the hypernetted-chain approximation of liquidstate statistical mechanics have been shown to describe more accurately primitive model electrical double layers, including those containing divalent ions andor concentrated electrolyte solution^.^-^^ However, for electrical double layer conditions used in this study, namely those of interest in clay systems, the MGC theory and the hypernetted-chain theories produce nearly identical results. One application of the modified Gouy-Chapman theory has been to predict the swelling pressure in montmorilAbstract vublished in Advance ACS Abstracts, June 15,1994. (1) Gouy, G: J . Phys. 1910,9,457. (2)Chapman, D.L. Phil. Mag. 1913,25,475. (3)Henderson, D.;Blum, L. J . Chem. Phys. 1978,69,5441. (4)Torrie, G. M.;Valleau, J. P. Chem. Phys. Lett. 1979,65,343. (5) Torrie, G.M.; Valleau, J. P. J. Phys. Chem. 1982,86, 3251. (6)Valleau, J. P.; Ivkov, R.; Tonie, G. M. J. Chem. Phys. 1991,95, 520. (7)Lozada-Cassou,M.; Henderson, D.J. Phys. Chem. 1983,87,2821. (8)Torrie, G. M.; Valleau, J. P. J. Chem. Phys. 1980,73 (ll), 5807. (9)Kjellander, R.; Akesson, T.; Jonsson, B.; Marcelja, S. J . Chem. Phys. 1992,97(2),1424. (10)Lozada-Cassou, M.; Diaz-Herrera, E. J . Chem. Phys. 1990,92 (2) ~, 1194. , - - .. (11)Lozada-Cassou, M.; Saavdra-Barrera, R.; Henderson, D. J . Chem. Phys. 1982,77 (101,5150. (12)Feller, S . E.;McQuarrie, D. A. J. Colloid Interface Sci. 1994, 162,208. @
0743-7463/94/2410-2125$04.50/0
lonite, an expandable clay.13J4 The montmorillonite clay particles are thin plates with a diameter of about 2 pm and a thickness of about 10 A. Montmorillonite is composed of a lattice of A 1 2 0 3 and Si02 units, and when divalent metals such as magnesium substitute in the lattice for aluminum or silicon, a net negative charge develops on the clay surface. In the presence of an aqueous electrolyte solution, an electrical double layer is formed at the surface of the clay particle. The overlap of two double layers causes a repulsion between the particles which results in an osmotic swelling pressure if the clay is confined. Experimentally, the swelling pressure, ll, of montmorillonites can be measured as a function of the plate separation. The plates are initially far apart and are forced together as the external pressure is increased, and when equilibrium is established, the plate separation is measured by X-ray diffraction. Lubetkin, Middleton, and ll for several montmorillonites imO t t e ~ i l l measured '~ mersed in monovalent electrolyte solutions. Using the Gouy-Chapman (GC) theory with a Stern layer16 correction, good agreement was found with the Li+(aq) sample. Viani, Low, and Roth17 performed similar swelling pressure experiments and showed I-I to be an exponential function of plate separation. In order to compare double layer theory to the observed results, Viani et al. calculated theoretical swelling pressures using the GC theory with a constant surface charge density. After showing that an exponential relationship did not exist between their theoretical swelling pressure values and plate separation and noting no qualitative agreement between the swelling pressure versus plate separation curves for the GC theory and experiment, they concluded that double layer forces were not responsible for the observed swelling pressure. In the Gouy-Chapman model, either the surface charge density or the surface potential is typically held constant as the distance between two electrical double layers is varied, although another model based on surface com(13)Huerta, M.; McQuarrie, D. A. Electrochim. Acta 1991,36,(11/ 121,1751. (14)Huerta, M. M.; Curry, J. E.; McQuarrie, D. A. Clays Clay Miner. 1992,40 (5),491. (15)Lubetkin, S . D.; Middleton, S. R.; Ottewill, R. H. Phil. Trans. R. SOC.London, A. 1984,311,353. (16)Stern, 0.2.Elektrochem. 1924,30,508. (17) Viani, B. E.; Low,P. F.; Roth, C. B. J. Colloid Interface Sci. 1983,96,229.
0 1994 American Chemical Society
Greathouse et al.
2126 Langmuir, Vol. 10, No. 7, 1994 plexation was proposed in which neither the surface charge nor the surface potential remains constant.16 Israelachvili and Adamslg measured the force between mica sheets immersed in dilute monovalent electrolytes and compared their results using the GC theory with constant surface charge and constant surface potential. Their results were more closely fit by the constant surface potential curve, although the two theoretical curves were coincidentbeyond a separation of 100 A. Similar studies with mica sheets have been conducted by PashleyZ0and more recently by Shubin and KekicheffOz1 For montmorillonite clays, Miller and Lowz2indirectly measured the Stern potential16and concluded that this plane was characterized by constant potential and not constant charge. had previously calculated swelling pressures in which the Stern potential remained fured as the plate separation was varied. Because the MGC model does not include the feature of adsorbed ions, comparisons of the potential at the plane of closest approach (qs)with the Stern potential or the potential at the outer Helmholtz planex should be avoided. However, MGC calculations in which I), remains constant provide an alternate model in which neither the surface charge nor the surface potential remain fured as the plate separation is varied, similar to other double layer models. 16,18 To our knowledge, the calculations presented here are the first in which the potential at the plane of closest approach, as defined by the MGC theory, remains fxed as the plate separation is varied. It has been shown above that the theoretical quantity (if any) that should remain fured as two electrical double layers approach each other is not a unanimous choice. With regard to the physical reality of the three MGC models as they apply to montmorillonite swelling, the presence of permanent electrostatic charge on the clay surface due to isomorphic substitutions in the clay lattice suggests that the constant surface charge model is the most appr0~riate.I~ In terms of the MGC theory and its application to montmorillonite swelling in dilute electrolyte solutions, we intend to show that the three models produce nearly identical results in the range of experimentally measurable swelling pressures and plate separations. In addition, the effect of ionic size on the swelling pressure in the two constant potential models will be studied and compared. This trend has been studied using the MGC theory14 (constant surface charge) and with statistical mechanics25 (constant surface potential). Finally, a thorough comparison with experimental datal' will be made to show that the neglect of a plane of closest approach in creating theoretical pressure versus separation curves for the purposes of comparisons with experimental data can lead to incorrect assumptions concerning double layer theory.
Theory The model used in the calculations consists of two interactingelectrical doublelayers immersed in an aqueous solution of monovalent electrolytes. The double layers are modeled by charged plates of infinite extent, and the restricted primitive model is used to model the solution.
The ions are treated as point charges and are confined to the diffuse layer, which extends to a distance a12 from each plate, where a represents the ionic diameter of both the coion and counterion. It has been shown that the size of the coion is unimportant for surface charge densities typical of montmori110nites.l~The solvent (water)and all surfaces have a uniform dielectric constant. The swelling pressure, n, is calculated using
n = Pmw + PEL
(1)
where Il is the total swelling pressure, PELis the electrostatic or osmotic pressure, and PVDW is the van der Waals force given by
Pmw = -N6nh3
(2)
whereA is the Hamaker constant typically taken to be 2.2 x J and h = 2d,the distance between the two plates. due to the interThe electrostatic swelling pressure PEL acting double layers in the solution is calculated using
(3) where c is the bulk electrolyte concentration, kg is the Boltzmann constant, Tis the Kelvin temperature, e is the elementary charge, and I)d is the electrostatic potential at d, the midpoint between the plates. Here 1/, is a function of only one variable since the infinite plates reduce the problem to that of one dimension. Equation 3 is derived by subtracting the osmotic pressure at an infinite distance outside the plates from the osmotic pressure at the midpoint between the plates. In order to use eq 3 to find PEL, q d , the potential at the midpoint, must be found. We begin with Poisson's equation in one dimension: (4)
where @(x)is the charge density, 6 is the dielectric constant, go is the permittivity of free space, and x is the distance from one of the plates. This becomes Laplace's equation in the ion-free region (0 < x a/2) and the PoissonBoltzmann equation in the diffuse region [a/2 < x < (2d - a/2)1. For a 1:l electrolyte, the Poisson-Boltzmann equation is written
where q ( x ) = ev(x)/kBT is the reduced potential at the reduced distance 5 = KX and K is the Debye-Huckel screening parameter, defined by K = (2ce2/cc,,kBTP. Therefore, eq4 must be solved in two regions with solutions p'(0 4 5 < ~ a / 2 and ) ql' ( K U / ~< 5 < ~ d ) . There are three boundary conditions for the differential equations:
(18)Healy, T.W.; Chan, D.; White,L. R. PureAppl. Chem. 1980,52,
1207 --- . .
(19)Israelachvili,J.N.;Adams, G. E. J.Chem. Soc., Faraday Trans. 1 1978,74, 975. (20)Pashley, R.M.J. Colloid Interface Sci. 1981,83, 631. (21)Shubin,V.E.;Kekicheff, P. J. Colloid Interface Sci. 1993,156, 1 OR -"-.
(22)Miller, S. E.;Law, P. F. Langmuir 1990, 6, 572. (23)Low, P.F. Langmuir 1987,3, 18. (24) Grahame, D.C. Chem. Rev. 1947,41, 441. (25)Feller, S.E.;McQuarrie, D. A. J. Phys. Chem. 1993,97,12083.
where the derivative of the electrostatic potential at 5 = 0 must be equal to the negative of the electric field due to the charged plates; the second, (7)
Double Layer Models of Clay Swelling
Langmuir, Vol. 10,No. 7,1994 2127
where the electric field and therefore the derivatives of the potentials must be continuous across the plane of closest approach; and the third,
9
1
.
.
. . . . . . . tl 1.0 x 104M
I I
i
I
n
G
which is a result of the symmetry of the interacting double layers. The solution for q1is straightforward and is of the form P,
I
( E ) = -0:t + (Do
0
5 -Z
Ka/2
(9)
7
C
v
5
Y.
where boundary condition (6)was used and qo is the reduced surface potential. The fact that the potential is linear in this region4combined with the fact that the charge density at x = a12 (plane of closest approach)is zero allows the addition of the third equality in parentheses in eq 7. A n important relationship between qo, and qs,the reduced potential a t the plane of closest approach, is established by eq 9:
01,
*
K0,U
Po = Ps + 2
(10)
4
0
10
20
50
40
50
60
70
M)
90
100
hIA Figure 1. logbwelling pressure)vs separationcorresponding to u0= -0.10 C m-2 at infinite plate separation. The constant ljlg curve (solid line) has a constant value of vs= -264.0 mV, the constant surface potential curve (long dashed line) has a constant value of ljlo = -292.8 mV, and the constant surface charge density curve (short dashed line) has a constant value of a,, = -0.10 C m-2.
integrand in terms of qBand q0,giving
To solve for q", eq 5 is integrated once to obtain (11) where y = cosh[q'I(~)I,yd = cosh[p"(~d)l,and eq 8 is used as a boundary condition for the indefinite integral. After making the substitution Y = y - Y d and transforming dqlVdE to dYId6, eq 11becomes
where y s = cosh(qp,). Up to this point, the three double layer models have the same derivation. The difference between the three is the solution of eq 12 for Y d . For the constant surface charge density model, eqs 7 and 11 are combined, giving (13)
Equation 13 can then be used to change the upper limit in eq 12 to llz(oi)z. While keeping 0; constant, the resulting equation is solved for Y d as the value of d is changed. After a solution for yd is obtained, the potential at the plane of closest approach is calculated using eq 13, and the surface potential is calculated using eq 10. For the constant qSmodel, eq 12 is solved as it appears. The difference between this method and the constant surface charge density method is that the unknown variable yd appears both in the integrand and in the upper limit while y s remains constant as d is varied. Once yd is calculated, 00 is found using eq 13, and the surface potential is found using eq 10. Values of corresponding to a surface charge density at infinite plate separation are found using
For the constant surface potential model, eq 13is used again t o change the upper limit of integration in eq 12 to 1/2(u~)2, and eqs 10 and 13 are used to rewrite Y d in the
Once the value of y Bis found for a value of d, eq 10 is used to find ai,and eq 13 is used to find Y d .
Results The bulk electrolyte composition in each case consisted M solution ofa 1:l electrolyte. The dielectric of a 1.0 x constant for water, E , was assigned a value of 78.5 at a constant temperature of 25 "C. We have chosen to use an arbitrary set of ionic radii in our calculations rather than attempt to model any specific counterions. While the radii of unsolvated ions have been determined,26the precise size of an aqueous ion is not a well-known quantity. Table I1 of ref 14 shows that a wide range of literature values exists for monovalent cations, and in some cases the values from different authors do not even follow the same trends. To study ionic size effects on the swelling presure, the ionic radii ranged from 0.0 to 4.0 A, and a radius of 2.0 A was used to compare the three double layer models and to compare swelling pressures with experimental data. The difference between the left and right sides of eq 12 was minimized for each model using a univariate minimization algorithm found in the IMSL 1ibra1-y.~' The integral evaluations in eq 12were performed by a generalpurpose integrator from the IMSL library28that handles end point singularities. Also, all swelling pressures presented in the figures include the van der Waals force and have been calculated using eq 1. Model and Ion Size Effects on Swelling Pressure. To investigate the difference in swelling pressure between the three double layer models, the swelling pressures for two systems with parameters typical of montmorillonites were calculated. The bulk electrolyte solution was as described above, and the ionic radius ( d 2 )was 2.0 A.Figure 1shows the logarithm of the swelling pressure versus the separationh. The three curves represent the three models, (26) Marcus, Y. J. Solution Chem. 1983,12, 271. (27) Gill, P. E.; Murray, W. Minimization Subject to Simple Bounds on the Variables; NPL Report NAC72; National Physics Laboratory: London, England, 1976. (28) Piessens, R.; deDoncker-Kapenga, E.; Uberhuber, C. W.; Kahaner, D. K. QUADPACK; Springer-Verlag: New York, 1983.
Greathouse et al.
2128 Langmuir, Vol. 10, No. 7, 1994 7.0
9
:=*
t1
1.0 x I04M
0
6.5
%=LOA n
Q
6.0
t: v
8 5.5
cl
5.0
5
I
4
0
10
20
30
40
50
60
70
80
90
4
4.5
ID0
hldi
Figure 2. log(swel1ingpressure)vs separationcorresponding to a, = -0.20 C m-2 at infiniteplate separation. The meanings of the curves are the same as in Figure 1. The constant value of qsis -299.6 mV, the constant value of I ) ,is -351.2 mV, and the constant value of a, is -0.20 C m-2.
each corresponding to a clay with a, = -0.10 C m-2 at infinite plate separation. This corresponds to qo= -292.8 mV for the constant surface potential curve (long dashed line) and q8= -264.0 mV for the constant curve (solid line). Equation 14 was also used to verify the values of q5calculated from the constant surface charge density method, and the values of q, from this method a t h = 1000.0 A were always within 0.001 mV of qscalculated from eq 14. At plate separations on the order of a single ionic diameter, the pressures appear in the order a, > qo > qs,with both constant potential curves showing an attractive region due to the van der Waals force. At larger plate separations, the results converge rapidly, and beyond a plate separation of about 20 A,the three curves are identical. Figure 2 shows the three models with the same electrolyte composition except that a, = -0.20 C m-2 a t infinite plate separation, corresponding to qo= -357.2 mV for the constant surface potential curve (long dashed line) and qs= -299.6 mV for the constant q5curve (solid line). The pressures at very small plate separations follow the same trend as in Figure 1, and the plate separation at which the three curves become indistinguishable is smaller (-15 A). At this higher (negative) charge density, the two constant potential curves are less affected by the van der Waals force due to the fact that the repulsive swelling pressures are higher than the a,= -0.10 C m+ case. Thus for clays with a surface charge density ranging from -0.20 to -0.10 C m-2, a calculation of the swelling pressure is independent ofthe model for plate separations and electrolyte solutions of experimental interest. Using the constant surface charge density model, the trend of increasing swelling pressure with increasing ionic radius has previously been ~ h 0 w n . lBecause ~ the two constant potential models yield the same pressures as the constant surface charge density model for most plate separations and experimental conditions using montmorillonite clays (Figures 1and 2), it would be expected that the same pressure dependence on ionic radius found in the constant surface charge density model would occur with these two models as well. Figure 3 shows the dependence of swelling pressure on ionic radius for the fixed 3. case. The ionic radii W2) used were 0.0,1.0,2.0, 3.0, and 4.OA, and the constant potential was -273.4 mV. From eq 14,this corresponds to a, = -0.12 C m-2 at infinite plate separation. At small plate separations, the charge densities of the larger ion curves dropped considerably,
0
10
20
30
40
M
60
70
80
90
100
hiA Figure 3. log(swel1ingpressure) vs separation for the constant I## model. The ionic radii are (from bottom to top) 0.0, 1.0,2.0, 3.0, and 4.0 A.
. . . . . . . . .
7.5.
1: 1
7.0 n
6.5
-
1.0 x 1O4M '
G9 . 6.0
3 5.5
5.0
'
:* 0
10
20
- 30
40
- - - 50
60
70
80
90
100
hIA Figure 4. log(swel1ingpressure)vs separation for the constant surface potential model. The meanings of the curves are identical to those given in Figure 3. while a,,remained nearly constant for the point charge curve. For example, a t h = 10.0 A, a, ranges from -0.098 C m-2 for the point charge curve to -0.038 C m-2 for the 4.0-A-radius curve. The pressures for this model have already been shown to be the lowest of the three models, and even in the large ion case the van der Waals force prevents the pressure from increasing asymptotically. Figure 4 shows the same ion sizes as Figure 3, but here the surface potential is held constant a t qo= -300.0 mV as the plate separation is varied. The corresponding surface charge densities at infinite plate separation range from a,= -0.20 C m-2 for the point charge case to a, = -0.081 C m-2 for the 4.0-ibradius case. At this surface potential, only the point charge curve shows an attractive region, in contrast to Figure 3, where all but the largest ion curve shows an attractive region. The above results show that, for other than very small plate separations, the three electrical double layer models produce the same pressure versus plate separation results for typical montmorillonites immersed in a dilute 1:l electrolyte solution. Comparison to Experimental Data. The constant surface charge density model has previously been used to compare double layer theory for montmorillonites to experimental data using point charges" and also with the inclusion of a plane of closest approach.13J4 Because the swelling pressure of montmorillonites is typically measured at plate separations greater than 20 A,any of the three double layer models would serve to produce
Langmuir, Vol. 10,No. 7,1994 2129
Double Layer Models of Clay Swelling 2.5
a
i
h
.
. . . . . . .
.
7 ' 6 '
rr
se+5
E 4"
1.5
a
.
\
/
1.0
Y
c
I
0.5
0.0
0.010
0.015
0.020
t1
=
1 .
I 0.025
0.030
0.035
0.040
montmorillonite. The solid line represents the experimental results of ref 17. The diamonds are the calculated points, and the dashed line represents the best fit for the constant surface charge density model.
I
3.0
9
2 '
q/,=Z.oA so = 4 . 0 9 2 3 C m-'
h-' 1A-I
+
3
1.0 x 1O"M
Figure 5. ln(n 11,n in bars, vs llh,h in& forYellowWestern
1
2.5
0 0
10
20
30
40
50
60
70
80
90
100
hIA Figure 8. Range of swelling pressures for the montmorillonites used in ref 17. The solid lines represent the maximum and minimum experimental values for the best fit data from Table 2 of ref 17, and the dashed lines correspond t o the calculated pressures ofYellow Western (bottom curve) and Cameron (top curve).
+
Table 1. Line Fit Slopes (k)of Ln(ll 1) vs llh Data Where iI and h Have Units of Bars and 4 RespectivelyQ montmorillonite MGC exptl % diff Yellow Western
78.8 51.1 72.2
79.4 50.0 71.0
0.8 2.2 1.7
Cameron a The experimental slopes were taken from Table I1 of ref 17. The MGC slopes correspond to a 1.0 x M solution of ions with radii of 2.0 A.
U Y
1.0 I
0.5
I
0.0
I
0,010
0.015
0.020
0.025
0.030
0.035
0.040
h-' 1A-l
Figure 6. ln(n
+ l), JJ in bars, vs llh, h in A, for Cameron
montmorillonite. The meanings of the curves are identical to those given in Figure 5. 3.0
1
2.5
'
2.0
'
e3
h
7
'
1.0 x 1O"M
2.0
@
i
tl
1.5 '
C E l
5
1 1 . . . . . 1 1:l
I i ;
1.0 x 10%
0.5 0.0
0,010
a, = 4 . 0 9 2 3 C m-'
0.015
0.020
0.025
0.030
0.035
0.040
h-' 1A-l
Figure 7. ln(n+ 1),nin bars, vs llh, h in A, for Yellow Western montmorillonite. Decreasing dashed lines correspond to ionic
radii of 4.0, 2.0, and 0.0 A, and the solid line has the same meaning as that in Figure 5.
theoretical swelling pressure versus plate separation data for comparison. Figure 5-8 and Table 1 compare experimental and MGC swelling pressure data for several montmorillonites. The MGC pressures were calculated using the constant surface charge density model with a 1.0 x M bulk electrolyte solution containing mono-
valent coions and counterions having radii of 2.0 A, with the exception of Figure 7, which also contains calculations from solutions having ionic radii of 0.0 and 4.0A. All of the experimental curves were generated using data found in Table I1 of ref 17, where an exponential relationship was shown to exist between the swellingpressure II (bars) and the plate separation h (A)of the form ln(IT
+ 1) = Iz(l/h) + In b
(16)
where k and b are constants. In order to determine if the same relationship between ll and h exists in the MGC results, the MGC theoretical results were plotted in the form given by eq 16 for the two montmorillonites with the highest and lowest surface charge densities used in ref 17. Calculations were made over the experimental range of plate separations, and the constants k and b were determined for each charge density. The slopes (k)as defined by eq 16 are shown in Table 1 along with the experimentally determined slopes. These two samples represent the highest (Cameron) and the lowest (YellowWestern) surface charge densities, and the percent differencebetween the experimentally determined slope and the MGC slope is never greater than 2.2%.Two values of k were produced for the Yellow Western sample in accordance with ref 17, where two sets of constants were given for the four montmorillonites with the lowest surface charge densities: one set for small plate separations and one set for large plate separations. The slope k is the more important of the two constants in comparing theory and experiment because it is k that describes how the swelling pressure changes with changes in plate separation. From a comparison of the slopes in Table 1, then, it would appear that the MGC theory predicts this relationship quite well. Figure 5 shows the linear relationshi between ln(ll+ 1) and llh for the MGC model (a/2 = 2.0f f) and experimental
2130 Langmuir, Vol. 10, No. 7, 1994 data for Yellow Western montmorillonite (ao= -0.0923 C m-2) with 30.0 A < h < 90.0 A. The diamonds are the calculated theoretical values, and the two lines represent the best fits. The intercepts (ln[b]) for the two portions of the MGC line (dashed) are -0.168 and +0.455, while those for the experimental line (solid) are -0.472 and +0.273. Figure 6 shows the same linear relationship for Cameron montmorillonite(a,,= -0.187 C m-2). TheMGC (ai2 = 2.0 A)line is dashed, with diamonds representing the calculated points, and the experimental line is solid. Again the calculations were performed in the same range of plate separations, but here the values for the intercepts are +0.0127 for the MGC curve and -0.397 for the experimental curve. It should be noted that those constants apply only to pressure data in the range of plate separations listed above, and the value of the intercepts (ln[bl) refer to the swelling pressure at a plate separation out of the experimental range. The difference between the intercepts demonstrates the failure of the MGC theory to quantitatively account for the observed swelling pressures, however. The curious fact that the theoretical pressures are greater than the experimental pressures combined with the parallel nature of the lines suggests that the modification of even one parameter may cause the lines to become coincident. One such parameter is the clay plate thickness, which was assigned the traditional value of 9.3 A for all of the montmorillonitesexcept Cameron, which was assigned a thickness of 19.4 A.l' The dependence of the theoretical calculations on ion size is shown in Figure 7,which shows forYellowWestern montmorillonite the experimentalline (solid)and the MGC line with radius 2.0 A (short dashed) from Figure 5 along with the lines representing ionic radii of 0.0 A (long dashed) and 4.0 A (dot dashed). For the point charge, the slopes are 66.0 and 43.5 and the intercepts are -0.0520 and +0.496. The slopes for the 4.0-A-radius lines are 82.6 and 60.4 and the intercepts are -0.150 and +0.391. While the slopes of the 4.0-A lines are similar to those found in Figure 5, the slopes representing the point charge are clearly distinct from the others. In fact, one portion of this line crosses the experimental line, in agreement with Figure 7 of ref 17. This comparison clearly shows the inadequacy of the Gouy -Chapman theory in predicting montmorillonite swelling. Finally, a comparison of the range of swelling pressures (bars) is shown in Figure 8. The experimental curves l for each of the eight were generated by solving eq 16 for l montmorillonites using the constants provided in Table I1 of ref 17, finding the maximum or minimum value of ll for a value of h, and plotting the results. The MGC curves (dashed lines) are the cosntant surface charge density calculations for the montmorillonites with the highest surface charge density (Cameron, top curve) and the lowest surface charge density (YellowWestern, bottom curve). Both of the MGC curves represent ionic radii of 2.0 A. Again, the MGC curves are nearly parallel to the
Greathouse et al.
experimental curves and are displaced from the experimental curves by at most 15.0 A above I3 = 2.0 bars. This observation was also evident in the ln(n 1)versus llh results in which the best fit lines for the MGC theory and experiment had nearly the same slopes but different intercepts. It would appear Erom the preceding comparisons with experimental data that the use of a plane of closest approach in the MGC theory to predict swelling pressure trends agrees qualitatively but not quantitatively with experimental data, while the absence of a plane of closest approach results in poor agreement with experimental data.
+
Conclusion Despite the failure of the MGC theory for divalent ions and concentrated electrolyte solutions, the agreement between the MGC theory and the more rigorous hypernetted-chain theory and Monte Carlo computer simulations for dilute monovalent electrolytes and charge densities similar to montmorillonites is excellent. Therefore, the MGC theory should be able to produce very accurate electrical double layer pressure curves for these systems. Using charge densities typical of montmorillonites and a bulk electrolyte concentration of 1.0 x M, the nonlinear Poisson-Boltzmann equation was solved using the electrical double layer models of constant surface charge density, constant surface potential, and constant potential at the plane of closest approach, and the swelling pressures corresponding to the three models were compared. It was shown that, except at very small plate separtions, the swelling pressures of the three models were nearly identical. In this area of small plate separations, the order of swelling pressures was found to be u,, > qo=- qs. Also, the trend shown in the constant surface charge density model of increasing swelling pressure with increasing ionic sizeI3 was shown to exist for the two constant potential models. Therefore, for the values of plate separations used in most montmorillonite experiments, any of the three double layer models can be used t o produce swelling pressure data for comparison to experimental data. The constant surface charge density model was then used to compare the MGC swelling pressures to some experimental data for montmorillonites. It was shown that a qualitative agreement existed between experiment and theory in swelling pressure versus separation data. However, the use of point charges in the model solution did not produce the same relationship, and a comparison of experimental data to theoretical predictions using a model without a plane of closest approach could result in incorrect conclusions concerning double layer theory.
Acknowledgment. This work has been supported by the National Science Foundation under Grant NSF EAR 8910530.
