Langmuir 1989,5, 199-205
199
Electrostatic Calculations on Swelling Pressures of Clay-Water Dispersions J. J. Spitzer* Department of Chemistry, University of Lethbridge, Lethbridge, Alberta, Canada N7T 7M2 Received May 20, 1988. I n Final Form: August 10, 1988 An electrostatic model of counterion adsorption between two planar highly charged surfaces is found to quantitatively describe available data on short-range repulsions in aqueous clay systems over two magnitudes of repulsive pressures and ionic strengths. These repulsions arise from adsorption of counterions, which is in agreement with a similar conclusion based on measurements of repulsions between crossed mica cylinders (Pashley,R. M. J. Colloid Interface Sci. 1981,83,531). Forces arising from surface-modified solvent structure need not be considered explicitly in these particular experiments. Such structural forces probably contribute indirectly to the ionic adsorption equilibrium in the double layer. In addition, the model predicts only a weak dependence of the Stern potential (or potential) on ionic strength for the case of single planar double layers, in agreement with experimental observations. Unlike the Stern theory, the model also predicts charge-potential discontinuities that may be investigated experimentally. In particular, the double-layer differential capacity vs potential is not a continuous function near the point of zero charge, the discontinuity being largest at low ionic strengths.
Introduction Experimental results on swelling pressures of gels that are composed of flat charged particles of naturally occurring clays provide one of the most direct routes for quantitative tests of theories of colloidal close-range repulsive forces.'s2 Such close-range forces are often rationalized as hydration forces,1*3-7 but other experimental data indicate that double-layer electrostatic forces are also i n v o l ~ e d . ~ ~In * *particular, ~ Low1 shows by elegant experimentation that less than 2% of total charge is diffuse; using the classical double-layer theory with a low Stern potential, he calculates negligible repulsive forces between the clay platelets, in disagreement with experiment. Similarly, Lubetkin et al.? by assuming a high Stern potential, show that the classical double-layer theorylO can account for some of their experimental results, but the theory does not explain why the repulsive forces depend on the kind of counterions. In principle, some combination of Stern adsorption theory and the classical double-layer theory should provide a better interpretation of the experimental data, but no such theory appears available in a simple and usable form. Indeed, attempts to combine these two concepts quickly lead to mathematical complexities because of the nonlinear nature of the underlying equations. In order to avoid such mathematical difficulties, a relatively simple (linear) model of the double-layer counterion adsorption has been derived and applied to some experimental data.l1-l3 The purpose of this paper is to give a complete derivation of this approximate theory with detailed applications to the data of both LOW^^ and Lubetkin et al.2 Such interpretation of the experimental data gives reasonable values of the theoretical parameters that describe the specificity of counterion adsorption, and also the amount of diffuse charge is predicted to be below 2%, in agreement with Low's estimate.l In related experiments with crossed cylinders of mica sheets in electrolyte solutions, I~raelachvili~ and PashleyG identified and quantified the short-range repulsive forces in terms of deviations from the exact solutions of the nonlinear Poisson-Boltzmann equation, which was incorporated into the model of surface dissociation phenomena and charge regulati~n.'~Neither of these concepts could predict the large magnitude of the measured re-
* Current address: BASF Canada Inc., Dispersions R&D, 453 Christina Street South, Sarnia, Ontario, Canada N7T 721.
pulsive forces, though the charge regulation model could quantify the forces more precisely.6 Pashley's analysis led to the conclusion that the strong repulsions become operative only when there is an appreciable counterion site adsorption. Such conclusion is supported by Low's estimate2 of low amount of diffuse double-layer charge and by the observation of Healy et alS6 that high surface charge density polymer latices remain stable in highly concentrated electrolyte solutions when a layer of hydrated counterions prevents surface-to-surface contact. The theory developed below describes approximately the conditions under which such a hydrated layer of counterions is formed and attributes the large close-range repulsions to "atmospheric" type counterion binding. However, the theory is derived for two flat plates, and as yet cannot be used directly for the more complicated geometries of two approaching spheres or cylinders.
Derivation of Theory Setting Up the Model. The essential features of the model are shown schematically in Figure 1. The approach to solve this problem is similar to the Manning theory of ion condensation16 in that the ion distributions are assumed to be linear (Debye-Huckel approximation); however, in a more realistic fashion, we assume that the counterions approach the primary charge, expressed as a positive surface charge density ao(+), to a distance of (1) Low. P. F. Langmuir 1987,3, 18. (2) Lubetkin. S.D.: Middleton. S. R.: Ottewill. R. H. Philos. Trans. R . Soc. London'1984,A311, 133. ' ( 3 ) Viani, B. E.; Low, P. F.; Roth, C. B. J.Colloid Interface Sci. 1983, 96,229. (4) Israelachvili, J. N.; Adams, G. E. J. Chem. SOC.,Faraday Trans. 1 1978, 74, 975. (5) Healy, T . W.; Homola, A.; James, R. 0. Faraday Discuss. Chem. SOC.1978, 156. (6) Pashley, R. M. J. Colloid Interfuce Sci. 1981,83, 531. (7) Le Neveu, D. M.; Rand, R. P.; Parsegim, V. A. Nature 1976;259, 601. (8) Norrish, K. Faraday Discuss. Chem. SOC.1954, 18, 120. (9) Friend, J. P.; Hunter, R. J. Clays Clay Miner. 1970, 18, 275. (10) Vervey, E . J. W.; Overbeek, J. Th. G. Theory of Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. ( 1 1 ) Spitzer, J. J. Nature 1984, 310, 396. ( 1 2 ) Spitzer, J. J. Colloids Surf. 1984, 12, 189. (13) Spitzer, J. J. In Particulate and Multiphase Processes; Ariman, T . , Veziroglu, T. N., Eds.; Washington: Hemisphere Publishing Corp., 1987; Vol. 3, pp 11-21. (14) Chan, D. Y. C.; Pashley, R. M.; White, L. R. J. Colloid Interface Sci. 1980, 77, 283. (15) Manning, G. Acc. Chem. Res. 1979, 12, 443.
0 1989 American Chemical Society
Spitzer
200 Langmuir, Vol. 5, No. 1, 1989
Electrostatic Solution o f the Model. All the equations are derived in SI units. a. Region 1, 0 I x Ia . Solution of the Laplace equation gives
i
, ut'*' I
(4)
i
,
where x is a distance variable, Pois the surface potential, t1 is the local dielectric constant, and eo is the permittivity of a vacuum. b. Region 2, a Ix 5 b . The Poisson equation with the Debye-Huckel distribution of counterions p 2 ( x ) = z-en4(l - z - e \ k 2 ( x ) / k T ) (5)
I ,
'
I
2
3
I i
!
o
a
b
d
gives a linear Poisson-Boltzmann equation
Figure 1. Adsorption model of electrical double layer at large
surface separation (and high surface charge densities). Distance a is Stern layer, b is co-ion exclusion distance, and d is midpoint separation distance. closest approach a. At this distance the counterions are assumed to adsorb to give rise to a negative surface charge density uA(-), the Stern layer. At this point it is useful to utilize the general results of the classical double-layer theory,1° which predicts an extremely low concentrations of co-ions near a highly charged surface. We therefore make an approximation that at distances shorter than b the local concentration of co-ions is actually zero, and their concentration begins to increase at distances greater than b according to the Debye-Huckel-Manning linear distribution laws (c.f. Figure 1). Therefore, the potential at the distance b must be Pt(+)
Pt(+) = kT/z+e
(1)
which may be termed the thermal electrostatic potential of co-ions (where k T is thermal energy and z+e is the charge carried by counterions). In Figure 1, d is the midpoint separation distance between two charged plates and el, t2, and t3 are a dielectric profile in the double layer (which turns out to be unimportant). The volume charge densities p2 and p3 are calculated as in the Debye-Huckel theory. The total adsorption of the counterions is described by the "site" adsorption charge density uA(-), and by "atmospheric" adsorption gab(-) given by (c.f. Figure 1 )
A reasonable interpretation of this kind of adsorption is that the counterions that are "site adsorbed" are partially dehydrated, which leads to specific ion-surface binding forces in addition to electrostatic attraction; the counterions that are "atmospherically adsorbed" are fully hydrated and adsorbed by electrostatic forces alone. Such hydration/dehydration distribution of adsorbed counterions may be described by a constant K A 'Jab(-)
= KAgA(-)
V2PZ = P ( \ k , ( X )
(6)
with (7)
where n4 is the bulk concentration of counterions with
Pt(-) = kT/z-e
(8)
which can be compared with eq 1. The solution of eq 6 is P2(x) = A2 exp(-kc)
+ B2 exp()uc)+ Pt(-)
(9)
Equation 6 also arises in a problem of low-charge double layer where counterions and co-ions approach the surface charge to different distances of closest approach,16with no counterion adsorption. c. Region 3, b 5 x Id . Here the usual Debye-Huckel solution applies to give P3(x) = A3 exp(-Kx) + B3 exp(Kx) (10) where K is the well-known reciprocal of Debye's length. In order to determine the integration constants A2,B2, AB,and B3,the boundary value conditions are as follows. continuity of potentials: *,(a) = *,(a)
(11)
Pdb) = *3(b) (12) presence and absence of surface charge densities at x = a and x = b:
symmetry condition at x = d :
(z)d=o
(3)
in keeping with the simple linear nature of the model. Variations of this constant with ionic strength and with the kind of counterions will be discussed in the light of actual values obtained for the clay-water systems. At this stage of the development of the ionic adsorption model, eq 3 has no theoretical basis, and perhaps there exist other more justifiable relations that might work equally well (e.g., simple proportionality of the Stern charge uA(-) and the surface charge go(+)). Indeed, other phenomenological models may rationalize the experimental data on a basis of different arguments.
- P,"))
total co-ion exclusion condition at x = b, c.f. eq 1: P3(b) = P,(+)
(16)
The ionic hydration-dehydration condition is given by eq 3.
From the above conditions, one obtains the transcendental equation F ( b ) for the free boundary b as (16)Spitzer, J. J. J. Colloid Interface Sci. 1983, 92, 198.
Electrostatics of Clay- Water Swelling Pressures
1 e3K 2 ez*
- --\Et(+)
+ exp[-X(b - a)] X
tanh [ ~ ( -d b)]
(;[-\Ett)
Langmuir, Vol. 5, No. 1, 1989 201
1 t3K 2 fZ#
- -\Et(-)I - - -*$+)
tanh [ ~ ( -d b)]
The first derivative of this equation is F'(b), given by 1 t3K K F'(b) = ---\Et(+) - X exp[X(b K A t2# cosh2 [ ~ ( -d b)] 1 E3K a)] -[-\Et(+) - -\E (-)I - --\Et(+) tanh [ ~ ( -d
[
+2
t2*
(' ::i
+ exp[X(b - a ) ] 2 --\Et(+)
1 E3K 2 ea#
- --\E$+)
cosh2 [:(d - b)]
l
o
a
d
Figure 2. Adsorption model of electrical double layer at close surface separation, showing only the Stern layer a and midpoint separation distance d. The concentration of co-ions between the surfaces is assumed to be zero.
and the repulsive pressure calculated by the electrostatic formula
tanh [ ~ ( -d b)]
is
P(d) = '/zfot3K2-\E32(d)
(29)
If we now investigate how the repulsive pressure depends on the separation distance, we find that when and the distance b can be obtained by the Newton method as bl = bo - F(bo)/F'(bo) (19) to a required degree of accuracy, where bo is the initial guess of the distance b and bl is the first approximation. The integration constants are then given by Az exp(-Ab) = 1 1 t3K -[-\Et(+) - -\E,(-)] + - --\Et(+) tanh [ ~ ( -d b)] (20) 2 2 t2X Bz exp(Xb) = 1 1 E3K -[-\E,(+) - -\Et(-)] - 5 --\Et(+) tanh [ ~ ( -d b ) ] (21) 2 E2# exp(-Kb) A3 = -\E$+) (22) exp(-2rtb) + exp(-2Kd) eXP(Kb) B3 = -\E$+) exp(2Kb) + exp(2Kd) The charge distributions are given by
(23)
-\E,@)
Az[exp(-Xa) - exp(-Xb)]
---
+ B2[exp(hb) - exp(Xa)] (25)
-
= -\Et(+)
(30)
which corresponds to d = b, all the co-ions are expelled from between the plates. In other words, the DH region b-d disappears. We shall call the midpoint separation distance d at which eq 30 becomes valid as the critical midpoint separation distance, denoted d*. This distance can be calculated from eq 18 by setting b = d* to give (cf. ref 11)
At distances closer than d < d*, the electrostatic model is shown in Figure 2 (cf. Figure 1). The solution is the same as given by eq 4,6, and 9. To avoid confusion we can write eq 9 with different integration constants as -\E2(x) = C2 exp(-Ax) B2 exp(Xx) + -\Et(-) (9a)
+
By use of the symmetry condition of eq 15 with \E3(x) replaced by -\Ez(x), eq 9 and 13, and the hydrationldehydration condition of eq 3, the constants C2 and D2 are given by (go(+) / eoe2X) ( K A/ ( 1 + K A ) )eXp (-XU) c2 = (32) exp(-2Xa) - exp(-2Xd)
gob(-) --=
eoe2#
= *k,(b) = -\E&)
D2 =
(go(+)/totzN(K~/(1+ KA)) exp(Xa) exp(2Xd) - exp(2ha)
The charge density
uA(-)
(33)
can be calculated from
EoezX
As[eXp(-~b)- eXp(-~d)]+ Bg[eXp(~d)- eXp(~b)](26) and it can be shown that the electroneutrality condition go(+) + U A ( - ) + a,*(-) 4- abd(-) = 0 is satisfied independently. The value of the midpoint potential -\E3(d)is then -\E3(d)= P,(+)(cosh [ ~ ( -d b)])-' (271
(cf. eq 24 for the case of d 3 d*). The charge density ad(-) can be calculated from (cf. eq 26)
202 Langmuir, Vol. 5, No. 1, 1989
Spitzer
Table I. Comparison between Calculated and Experimental Double-Layer Layer Pressure" d, A
15 17.5 20 25 30 35 40
MPa 0.67 0.48 0.34 0.20 0.14 0.10 0.070
Pexptlt
MPa 0.69 0.49 0.36 0.22 0.15 0.11 0.082
Pcpld,
W), mV 1910 1600 1370 1070 880 740 640
" Experimental values are estimates from Figure 10 of ref 3 for a variety of clays at 0.0001 M NaCl. Independent parameter, uo(+) = 14.0 &/em2; model parameters, KA = 0.0068, a = 2.0 A; calculated parameters, d* = 422 A, 1 / = ~ 304 A. since C2exp(-Ad) = Dzexp(Xd) according to the symmetry condition of eq 15. From eq 35 and 34 the electroneutrality condition is again satisfied independently. The midpoint potential is now given from eq 9 as
The repulsive pressure is calculated from
to give P(d) =
+
1 / 2 ~ 0 ( ~- 3~~2 X ~~)(\k,(+))~
t&12\kt(-)[ \kt(+)- \k,(d)]
+ f/2~otzX~\k2~(d) (38)
It is the last term in eq 38 that leads to large repulsive pressures at close separations. The first term is negligible, and the second term contributes by about only 10% to the total pressure.
Discussion Application of the Theory to a Clay-Water System. The above theory can be applied directly to the available data2i3on repulsive pressures in clay gels. All the calculations were done by trial and error, and the derived parameters are not the statistically -bestn ones; therefore, in some cases better agreement with experiment could be obtained. However, it was felt that the calculated values are generally within the experimental error of the measured values. In Table I the data of Viani et al.3 (their Figure 10) are compared with calculated repulsive pressures by using eq 38 since the experimental range of separation midpoint distances is smaller (15-40 A) than the critical separation distance of 422 8,,when the gel is in equilibrium with lo4 M NaC1. The attractive van der Waals pressure has been
shown previously to be relatively small;' nevertheless, it was used in all the calculations using eq 10 of ref 3. There is little doubt that the theory can account satisfactorily for the data with a minimum of parameters. The theory is quite sensitive to the constant KA = 0.0068 and somewhat less sensitive to the distance of closest approach a = 2.0 A. The dielectric profile el, tz, and t3 can be used as a constant, with t1 and tz = t3 = 78.5 equivalent to the dielectric constant of water (the results are independent of el). In agreement with Low's estimate,' the diffuse charge is only 0.68% of the total counterion charge. From the electrostatic model one can conclude that the origin of these repulsive pressures is electrostatic in nature but determined by hydration f dehydration adsorption equilibrium near the charged surface. The constants KA and a are a little different from those used in the preliminary calculations1' in order to fit better with the parameters derived from the data of Lubetkin et a1.2 for other alkali metal counterions that are discussed below. In Table I1 experimental data for lithium Wyoming bentonite2 are compared with calculated values by usin eq 38; however, eq 29 was used for the point at d = 75 in M LiCl (critical separation distance d* = 52.6 A). Again we obtain a satisfactory agreement between the theory and experiment for 2 orders of magnitude of pressures and 2 orders of magnitude of ionic strengths. We also observe that the interactions take place at constant charge (increasing midpoint potential) and that the product of the Debye length 1/ K and the "degree of dissociation" KA of the site adsorbed charge uA(-) (cf. eq 3) is constant to within 20% over 2 orders of magnitude of ionic strength. With this empirical observation, the ad hoc condition, eq 3, could be expressed as
!.
cab(-) = PKQA(-)
(39)
KA = PK
(40)
with Thus the whole set of data can be described by on1 two parameters, the distance of closest approach a = 4.0 and the constant p = 3.8 A in the one-parameter empirical eq 39. This empirical equation describes the adsorptive counterion equilibrium between specifically (site) adsorbed ions and electrostatically (atmospherically) bound ions. Similar conclusions can be drawn from the data in Table I11 for potassium Wyoming bentonite. For this system the distance of closest approach a = 1.5 8, and p = 2.0 A, both of which indicate that the potassium ion is more easily dehydrated than lithium ion. In Tables IV-VI the remaining comparisons between the theory and experimental data2are shown. Except for the
Table 11. Comparison of Theory and Experiment for Lithium Wyoming Bentonite" lo4 M LiCl M LiCl lo-* M LiCl pelput MPa P~ded,MPa *(d),mV Pexpd, MPa Pealed, MPa Wd),mV Paxpd, d, A MPa Pcpld,MPa 12.5 7.1 5.4 5340 3.9 3.3 1300 1.6 2.3 25 0.87 0.88 2150 0.51 0.54 510 0.34 0.34 37.5 0.34 0.35 1340 0.21 0.21 310 0.13 0.10 50 0.18 0.18 ' 970 0.11 0.11 220 0.057 0.033 I5 0.078 0.077 610 0.043 0.041 130 0.015b 0.010 100 0.039 0.042 450 0.021 0.020 80 125 0.022 0.026 350 0.011 0.010 60 150 0.015 0.017 280 1/K, A 304 96 30 d*, 8, 637 175 52.6 a, A 4.0 4.0 4.0 KA 0.015 0.038 0.11 KAIK, 8, 4.6 3.6 3.3 "Experimental values are graphically estimated from Figure 3 of ref 2 with Q(+) = 11.6 FC/cmZ. b d 3 d*.
K
Wd),mV 330 110 60
30 17
Langmuir, Vol. 5, No. 1, 1989 203
Electrostatics of Clay- Water Swelling Pressures
Table 111. Comparison of Theory and Experiment for Potassium Wyoming Bentonite' lo-' M KC1 M KCl 10-2 M KC1
Puptl,MPa
d, 8, 6.25 12.5 25 37.5 50
Pdd,MPa
1/K, A d*, A a, A
KA KA/K,
W d ) ,mV
P-u, MPa
Pdd,MPa
W d ) ,mV
P-u, MPa
Pdd, MPa
W d ) ,mV
4940 2120 980 630 460
3.6 0.66 0.19 0.078 0.040
3.6 0.71 0.16 0.065 0.034 96 121 1.5 0.023 2.2
1370 580 260 160 120
1.5 0.34 0.077 0.028b
1.7 0.34 0.055 0.023
310 120 40 25
4.2 0.83 0.19 0.080 0.044 304 401 1.5 0.0077 2.3
3.6 0.80 0.24 0.11 0.056
A
30.4 30.6 1.5 0.055 1.7
"Experimental data from ref 2, Figure 5, uo(+) = 11.6 pC/cm2. bThis point calculated from equations relating to d 3 d* (37.5 which gives the co-ion exclusion distance b = 28.8 A.
Table VI. Comparison of Experiment and Theory for Lithium Beidellite' lo-' M LiCl
Table IV. Comparisons of Calculated and Theoretical Repulsive Pressures for Sodium and Cesium Wyoming Bentonites at lo-' M NaCl or CsCl" d, A Pexptl,MPa Pdd,MPa W),mV Sodium Wyoming Bentonitello-' M NaCl 4170 10 2.2 3.2 0.52 1840 20 0.65 1170 0.25 0.27 30 0.16 860 40 0.15 0.11 670 50 0.091 0.054 430 75 0.039 310 0.028 100 0.021 0.018 240 125 0.013 0.012 200 150 0.0091 d* = 521 A a = 2.0 A KA = 0.011 Cesium Wyoming Bentonite/lO-' M CsCl 5 2.2 0.58 2030 7.5 0.28 0.26 1090 10 0.077 0.16 740 20 0.0095 0.043 320 a = 1.7 8, d* = 120 A KA = 0.003 Experimental data from ref 2, Figure 6.
Table V. Comparison of Experimental and Calculated Results for Cypern Montmorillonite" lo-' M LiCl M LiCl
d,A 12.5 25 50 75 100
A d*, 8, KA a, A
KA/K,
A
Po, tl,
Pdd,
Mha 3.2 0.58 0.14 0.054 0.025
MPa 3.4 0.56 0.12 0.048 0.026 304 550 0.0134 4.0 4.1
W),Po mV 4240 1700 760 480 350
tl,
Pcdcd,
Mya 1.1 0.28 0.049 O.O1lb
MPa 2.2 0.31 0.028 0.0095
W), mV 320 110 27 16
30.4 51.2 0.12 4.0 3.6
"Experimental data from Figure 7 in ref 2, uo(+) = 10.3 pC/cm2. *This point calculated from equations relating to d 3 d* (75 > 51.2), with co-ion exclusion distance b = 42.8 A.
case of cesium Wyoming bentonite, the theory can be fitted satisfactorily to the experimental data; in the cesium case the gel probably contained some face-to-face associated clay particles in equilibrium with individual platelets.2 In Table VI1 the parameters for different counterions and different clays are collected. The distances of closest approach for different counterions indicate that Li+ adsorbs in the most hydrated form, and Na+, K+, and Cs+ are partially dehydrated when adsorbed in the "Stern layer" x = a . The amount of diffuse charge decreases uniformly from Li to Cs in accord with expectations; the amount of diffuse charge also appears to decrease with increasing surface charge density uo(+) for different con-
> 30.61,
d , 8, 12.5 25 50 75 100 125
Perptl, MPa
Pdd,MPa
2.1 0.44 0.12 0.062 0.037 0.023
2.8 0.46 0.095 0.039 0.021 0.013
Q(d), mV 3830 1530 690 430 310 240
'Data from Figure 8 of ref 2, uo(+) = 13.5 pC/cm2. KA = 0.0092, A, and d* = 514 A.
a = 4.0
Table VII. Summary of Model Parameters for Different Counterions and Different Clays
a,A KA lo-' M Chloride Solutions counterion Li Na Nan K cs
4.0 2.0 2.0 1.5 1.7
lo-' M LiCl clays all claysb 2.0 Wyoming bentonite 4.0 Cypern montmorillonite 4.0 beidellite 4.0
ud+),~C/cm~
0.015 0.011 0.0068 0.0077 0.0030
11.6 11.6 14.0 11.6 11.6
0.0068 0.015 0.0134 0.0092
14.0 11.6 10.3 13.5
'From data in ref 3, all other parameters from data in ref 2. bFrom data for all clays in ref 3 in lo-' M NaCl, all other parameters from data in ref 2.
stant charge materials, though the variations in surface charge density are not large enough to make a firm conclusion. It can then be concluded that the above model of counterion adsorption can satisfactorily account for double-layer repulsion forces in well-dispersed clay-water systems. In this model the hydration effects are described by the Stern adsorption distance a and by the site-atmospheric adsorption constant KA, both of which are incorporated into a linear version of the double-layer theory. The distance a is an exclusion distance for counterions; such a parameter is necessary for any realistic model of an ionic double layer. The constant KA describes how easily the counterions adsorb into the Stern layer; if KA 0, the Stern adsorption is largest, and if KA m , the Stern adsorption diminishes to zero. Accordingly, the repulsive forces become larger as KA increases since the concentration of counterions in the diffuse double layer also increases. These repulsive forces may be termed ionic hydration forces,G and their existence does not necessarily rule out the concept of structural hydration forces, Le.,
-
-
204 Langmuir, Vol. 5, No. 1, 1989
Spitzer
Table VIII. Dependence of Double-Layer Charge Distributions, Co-Ion Exclusion Distance b , the "Stern"Potential * ( a ), and the Surface Potential qoon the Surface Charge Density for Single Nonoverlapping Double Layersa on(+). r C b, A * ( a ) , mV u A ( - ) , r C cm-2 u.h(-), rC cm-2 Ob-, r C cm-2 \kn,b mV \kn/on, V m2/C lo4 M 1:l Electrolyte 107 14.79 0.15 0.06 15 472 950 6.4 78 9.84 0.10 0.06 10 360 640 6.4 211 0.05 50 4.89 0.06 5 6.6 330 37 2.5 116 2.42 0.02 0.06 180 7.1 30 1 49 0.06 0.93 0.01 8.6 86 0.00 11 0.44 0.5 28 25 0.06 56 5.1 0.04 0.00 25.9 0.1 32 0.06 32 25.7 0.05868c 3.0 0.00 0.00 49 0.06 29
M 1:l Electrolyte 15 10 5 2.5 1 0.5868c
46 35 21 11 4.9 3.0
a p = 3.0 8, and a = 3.0 A.
* tl = 6.0.
97 71 45 34 27.4 25.7
13.1 8.56 4.01 1.74 0.38 0.00
0.59 0.59 0.59 0.59 0.59 0.59
940 630 330 170 84 59
6.3 6.4 6.6 7.0 8.4 10.0
-
Critical surface charge density below which the Debye-Huckel theory applies.
those that arise from modified water structure near a surface. Such structural hydration forces probably indirectly contribute to the values of the constants KA and a because the extent of the ionic dehydration on adsorption into the Stern layer depends also on the original hydration of the surface itself. Therefore, the structural hydration forces should be preferably looked for at surfaces near a point of zero charge where the interference of ionic adsorption should be the least. General Remarks about the Site and Atmospheric Adsorption Model. The above examples show that the linear approximation that is based on the general predictions of the classical theory,1° together with the site and atmospheric phenomena of adsorption that are described self-consistently within the electrostatic model, is relatively successful for highly charged systems of 10-15 pC/cm2. It is therefore worthwhile to find out whether the model predicts new phenomena in systems with variable surface charge density. The equations presented so far can be easily modified for systems with single nonoverlapping double layers by letting the midpoint separation distance go to infinity. Therefore, no new derivations are needed for single double layers. The general dependence of charge distributions and the co-ion exclusion distance b on decreasing surface charge densit no(+) is shown in Table VI11 for p = 3.0 A and a = 3.0 , in 0.0001 and 0.01 M 1:l electrolyte (with d m to give results for single plates). As the surface charge density is decreased, the co-ion exclusion distance b decreases. (Incidentally, the distance b depends also on the curvature of the surface.) The Stern potential * ( a ) also decreases, and we notice that the values of this potential are quite reasonable. If we assume that the Stem potential is equal to the {potential, then the model predicts a very weak ionic strength dependence of the { potential. This prediction is in excellent agreement with recent measurements of electrophoretic mobilities of montmorillonites." The classical Gouy-Chapman theory and the Gouy-Chapman-Stern theory incorrectly predict a very strong ionic strength dependence of the Stern potential. Thus the model is successful in rationalizing experimental data for both single double layers and two overlapping flat double layers. For the particular assumed value of p almost all counterions are "site" adsorbed at high surface charge densities, but as the charge density decreases most of the charge is being shifted into the diffuse DH between
K
1.31 0.86 0.40 0.17 0.04 0.00
-
(17)Horikawa, Y.; Murray, R. s.;Quirk, J. P. Colloids Surf. 1988,32, 181.
b and d m , until the critical surface charge density no(+)* (denoted by an asterisk) is reached. At this level of surface charge, all specific adsorption ceases and all counterions are now in the DH diffuse region. These observations also appear reasonable, though whether and how the constant p depends on no(+) needs further investigation. The critical surface charge densities were calculated before;12however, a more general equation is given below for two flat plates separated by a distance 2d no(+)* = t o t 3 ~ \ k , ( +tanh )
[K(d - b ) ]
(41)
In Table VIII, the surface potential is also calculated by using eq 4 with the boundary condition of eq 11. The "inner" double-layer dielectric constant c1 was taken as equal to 6, a value generally gssumed appropriate on the basis of electrochemical studies.'* The ratio of surface potential to surface charge is also given. These data show that there are two linear regions in which surface charge and surface potential are related linearly. The first region is around the point of zero charge below the critical surface charge density (asgiven by the Debye-Huckel theory) and need not be discussed further. The second region is quasi-linear and begins to appear at charge densities of about 5 pC/cm2; the linearity is seen to increase with increasing surface charge. consequently, the differential double-layer capacity is constant in a narrow region near the point of zero charge and nearly constant in the high surface charge region, in qualitative agreement with experiment and the classical Gouy-Chapman-Stern theory (GCS).1° The main difference between the GCS theory and the site-atmospheric adsorption model is the predictions for the capacity/charge relationships. The GCS theory predicts a smooth (continuous) function everywhere; the site-atmosphericadsorption model predicts a discontinuity in the capacityfcharge function unless a specific relationship between the constant KA and dielectric constants c2 and c3 is assumed.13 Such discontinuity is predicted to be most noticeable in dilute solutions near the point of zero charge, where there are few detailed experimental data available. However, at least one set of recent raw datalg shows reasonable indications of such discontinuities. Though these discontinuities need to be investigated further, they appear to represent disorderforder surface phase transitions between the diffuse three-dimensional adsorption (classical double-layer theory) and the begin(18) Bockris, J. O'M. private communication, 1983. (19) Cooper, I. L.; Harrison, J. A.; Holloway, J. Electrochim. Acta 1981, 26, 1265.
Langmuir 1989,5, 205-210 ning of a two-dimensional site adsorption (ion “condensation”). One incidental outcome of the above calculations is that nonlinear chargepotential functions can be derived on the basis of linear differential equations (using linear boundary value conditions) by appropriate definitions of boundaries and related boundary values. It therefore appears that experimental observations of nonlinear charge-potential behavior do not ipso facto support either the nonlinear Poisson-Boltzmann equation or its linear version.
Conclusions An ionic adsorption model of an electrical double layer has been derived and applied to calculations of repulsive pressures in clay-water systems. The model can rationalize nearly all the available data to probably within the experimental error over 2 orders of magnitude of pressures and ionic strengths. Unlike the classical double-layer theory, which neglects ionic adsorption phenomena, the model predicts very large repulsions at close double-layer separations. These repulsions are attributed to counterion hydration/dehydration equilibrium close to a charged surface. Therefore the concept of hydration forces that may originate from modified solvent structure in the proximity of a (charged) surface need not be invoked for
205
these particular experiments. Nevertheless, the present model does not rule out the possibility of such structural forces, the existence of which would be best demonstrated at surfaces near ‘the point of zero charge. The ionic adsorption model also predicts the Stern potential (t potential) to be nearly independent of ionic strength, and this prediction appears to be confirmed experimentally. The Gouy-Chapman-Stern theory predicts too strong ionic strength dependence of the ( potential. Qualitative calculations with the new model suggest that, unlike the Gouy-Chapman-Stern theory, the differential capacity of an electrical double layer may not be differentiable in respect to surface charge at predicted critical surface charge densities. Such surface phase transitions could be most easily observed at low ionic strengths near the point of zero charge.
Acknowledgment. I thank Prof. D. Dolman, S. F. O’Shea, and Loren Hepler for encouragement and comments and the University of Lethbridge and the Natural Sciences and Engineering Research Council for financial support. Prof. B. A. Pethica, J. N. Israelachvili, J. O’M. Bockris, J. Lyklema, G. S. Manning, C. Tanford, P. F. Low, V. A. Parsegian, R. H. Wood, and H. P. Bennett0 are thanked for critical comments.
Aqueous Sodium Halide Solutions of Nonionic and Cationic Surfactants with a Consolute Phase Boundary. Light Scattering Behavior T.Imae Department of Chemistry, Faculty of Science, Nagoya University, Chikusa, Nagoya 464, Japan Received May 25, 1988. In Final Form: September 13, 1988 The static and dynamic light scattering of hepta(oxyethy1ene)tetradecyl ether (Cl4E7) in 1 M NaC1, tetradecyldimethylammonium chloride (CI4DAC)in 2.6 M NaC1, and tetradecyldimethylammoniumbromide (C14DAB)in 4.3 M NaBr has been measured at various temperatures, and the micelle size and the intermicellar interaction have been evaluated at a finite micelle concentration in the dilute regime. Rodlike micelles grow with a rise in temperature, and an increase in the aggregation number produces a large value of the frictional virial coefficient. Therefore, the negative, large value of the hydrodynamic virial coefficient is raised, while the second virial coefficient of rodlike micelles is small, independent of the temperature. The solution behavior of rodlike micelles in the semidilute regime obeys the scaling laws with the characteristic exponents, v, of 0.58-0.75 for C14& and 0.57-0.63 for C14DACand C14DAB. The angular dependence of the effective diffusion coefficient in the semidilute regime is classified by two types, depending on whether the initial slope is positive or negative. The relation between the threshold micelle concentration of overlap, (c - co)*, and the consolute phase boundary is discussed.
Introduction Aqueous solutions of oligo(oxyethy1ene) alkyl ethers (c,~,)separate into two liquid phases when the ternperature rises above the lower consolute phase boundary or the cloud point.1-Q The lower consolute temperature
varies, if electrolytes are added to aqueous C,E, solutions.’b14 We have recently reported that aqueous Sohtions of alkyldimethylammonium chlorides and bromides (CnDAC, CnDm) and dodecylammonium chloride induce the liquid-liquid phase separation when sodium halides
(1) Balmbra, R. R.; Clunie, J. S.; Corkill, J. M.; Goodman, J. F. Trans. Faraday SOC.1962,58, 1661; 1964,60,979. (2) Clunie, J. S.; Corkill, J. M.; Goodman, J. F.; Symons, P. C.; T a b , J. R. Trans. Faraday SOC.1967,63, 2839. (3) Clunie, J. S.; Goodman, J. F.; Symons, P. C. Tram. Faraday SOC.
(7) Zulauf, M.; Rosenbusch, J. P. J.Phys. Chem. 1983,87, 856. (8) Corti, M.; Minero, C.; Degiorgio, V. J.Phys. Chem. 1984,88,309. (9) Corti, M.; Minero, C.; Cantii, L.; Piazza, R. Colloids Surf. 1984,12, 341. (10) Maclay, W. N. J. Colloid Sci. 1956, 1 1 , 272. (11) Schick, M. J. J. Colloid Sci. 1962, 17, 801. (12) Tokiwa, F.; Matsumoto, T. Bull. Chem. SOC. Jpn. 1975,48,1645. (13) Deguchi, K.; Meguro, K. J. Colloid Interface Sci. 1975,50, 223. (14) Zulauf, M. Physics of Amphiphiles, Micelles, Vesicles,and Mi-
1969,65, 287. (4) Ekwall, P. Aduances in Liquid Crystals; Brown, G. H., Ed., Academic: New York, San Francisco, London, 1975; Vol. I, pp 1-142. (5) Lang, L. C.; Morgan, R. D. J. Chem. Phys. 1980, 73, 6849. (6) Mitchell, D. J.; Tiddy, D. 3.T.; Waring, L.; Bostock, T.; McDonald, M. P. J. Chem. SOC.,Faraday Trans I 1983, 79, 975.
0743-7463/89/2405-0205$01.50/0
croemulsions;Degiorgio, V., Corti, M., Eds.; North-Holland Amsterdam, 1985; p 663.
0 1989 American Chemical Society
