2884
Langmuir 1994,10,2884-2891
Electrical Theory of Clay Swelling? M. V. Smalley Polymer Phasing Project, ERATO, JRDC, Keihanna Plaza, 1-7 Hikari-dai, Seika-cho, Kyoto 619-02, Japan Received September 14, 1993. In Final Form: May 2, 1994@ The Coulombic attraction theory of colloid stability, proposed by Sogami and Ise, is adapted to the properties of the two-phase (gel) region of colloid stability which is observed in clay swelling. A review of the recent debate about the nature of the thermodynamic electrostatic interaction potential between colloidal particles in electrolyte solutions is given, and the essential experimental facts about the n-butylammonium vermiculite system are recalled. In the two-phase region, it is shown that Sogami theory combined with the Dirichlet boundary condition (constant surface potential) yields the prediction that the ratio (s) of the salt concentration in the supernatant fluid t o the average salt concentrationin the gel phase be constant. For a surface potential of 70 mV, s is equal t o 2.8, in excellent agreement with the experimentalresults on the n-butylammoniumvermiculitegels. This result is used to calculatethe interlayer spacings in the gel phase as a function of the electrolyte concentration (c) and initial volume fraction ( r ) of the clay in the solvent medium, again in excellent agreement with recent studies of n-butylammonium vermiculite swelling. This in turn yields a prediction for the position of the r-c phase boundary between the one-phase and two-phase regions of colloid stability, which is also expressed in terms of the position of the minimum in the pair potential between parallel clay plates. This is a central prediction of the Sogami theory, in which there is a weak attractive tail in the electricalinteraction potential. The standard theory of colloid stability, the DLVO theory, is shown to be a limiting case of the Coulombic attraction theory, in the one-phase region.
1. Introduction The interaction between the charged particles in ionic colloidal solutions is usually treated in terms of the DLVO theory, which was developed independently in the 1940s by Derjaguin and Landau1 and Venvey and Overbeek.2 Accordingto this theory, the thermodynamic pair potential that describes the Coulombic interaction between the charged particles is a pure repulsion. The stability of lyophobic colloids is then attributed to the van der Waals force. The DLVO theory has long been one of the foundations of colloid chemistry, but the basic intuitive concept on which it is based, namely the repulsive nature of the Coulombic interaction between the like-charged macroions, has been questioned by Ise and co-workers (see ref 3, for a recent review). They have argued that the ordering observed in highly charged macroionic solutions is due to Coulombic attraction between the like-charged particles through the intermediate counterions. This new concept was first given theoretical expression by Sogami in 19834 and generalized by Sogami and Ise in 1984.5 The Coulombic attraction theory first received attention in the West when Overbeek disputed it in 1987.6 It has already been shown that Overbeek‘s criticism of the new theory is fundamentally flawed and that the Coulombic attraction theory provides a logical and self-consistent basis for describing the interparticle interactions in colloidal system^.^ The Coulombic attraction theory has since been placed on a rigorous basis by calculations of the Helmholtz free energy for the case of the interaction This paper is dedicated to my father, Gordon Smalley, who died April 24, 1994. Abstract published inAdvance ACSAbstracts, August 15,1994. (1)Derjaguin, B.V.; Landau, L. Acta Physiochem. 1941,14,633. (2) Verwey, E.J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyobhobic Colloids; Elsevier: Amsterdam, 1948. (3) Dosho, S.;Ise, N.; Ito, K.; Iwai, S.; Kitano, H.; Matsuoka, H.; Nakamura, H.; Okumura, H.; Ono, T.; Sogami,I. S.;Ueno, Y.;Yoshida, H.; Yoshiyama, T. Langmuir 1993,9, 394. (4) Sogami, I. Phys. Lett A 1983,96, 199. (5) Sogami, I.; Ise, N. J . Chem. Phys. 1984,81,6320. (6)Overbeek, J. Th. G. J . Chem. Phys. 1987,87,4406. ( 7 )Smalley, M. V. Mol. Phys. 1990,71,1251.
of two flat double layer^.^,^ In ref 8 the one-dimensional colloid problem was solved a t the mean field theory level subject to the hypothesis of “counterion dominance”, and in ref 9 the exact mean field theory solution to the problem was given. The latter paper has proved rigorously the existence of effectivelong-range attraction between highly charged plates in an electrolyte solution and provides a new basis for the analysis of a variety of phenomena in macroionic solutions. The main phenomena to be discussed here are the interlayer spacings between clay plates immersed in an electrolyte solution and the distribution of electrolyte in the two-phase region of colloid stability observed in clay swelling. First, however, we note that the work in refs 7-9 on the interaction of highly charged plates in a n electrolyte has been criticized by Levine and Hall,lo by Ettelaie,ll and by Overbeek.12 We have replied to these criticisms13J4and begin with a brief overview ofthe debate, which serves to give the theoretical background to the work presented here.
2. Review of the Debate Both the Coulombic attraction theory of colloid stability and the DLVO theory are based on the calculation of the electrical contribution to the free energy of the region bounded by the macroions. In both theories it is assumed that the motion of the macroions is adiabatically cut off from that of the simple ions and that the distribution of simple ions is determined by the Boltzmann distribution. In both theories the total electrostatic energy of the solution is obtained by solving the Poisson-Boltzmann equation using the primitive model, in which the solvent (8) Sogami, I. S.;Shinohara,T.; Smalley, M. V. Mol. Phys. 1991,74, 599. (9)Sogami, I. S.; Shinohara,T.; Smalley, M. V. Mol. Phys. 1992,76, 1. (10) Levine, S.;Hall, D. G. Langmuir 1992,8, 1090. (11) Ettelaie, R.Langmuir 1993,9, 1888. (12)Overbeek, J. Th.G. Mol. Phys. 1993,80,685. (13) Smalley, M.V. Langmuir, submitted for publication. (14) Smalley,M.V.;Sogami,I. S.Mol. Phys., submittedforpublication.
0743-7463/94/2410-2884$04.50/0 0 1994 American Chemical Society
Electrical Theory of Clay Swelling
Langmuir, Vol. 10,No. 9, 1994 2885
Figure 1. Geometry of the problem: (a)schematic illustration of the equipotential lines around two charged plates immersed in an electrolyte; (b) representation of the one-dimensional idealization of this situation; (c) variation of the electrostatic ) the coordinate x, whose origin is taken at potential ~ ( xwith the left-hand plate. d denotes the midplane between the two plates and xmlnthe equilibrium separation of the plates (from
Sogami et alJs
is treated as a dielectric continuum. In both theories the Helmholtz free energy is obtained from the electrical energy by the Debye-Huckel charging procedure. The model used in the SSS formulation of the Coulombic attraction theory is illustrated in Figure 1. The essential feature of the results in the SSS theory is that there are three terms which contribute to the adiabatic (Helmholtz) pair potential between the plates, which is expressed in the following form
V(x)= v"'(x)
+ V?(x)+ V,"(x)
(1)
where x is the plate separation. The purely repulsive part V:(x) represents the osmotic contribution of the ions which are trapped in the inner region Ri, and the purely attractive part Vo0(x) originates in the osmotic pressure of the small ions in R, exerted against the surfaces of the plates from outside. The other component Vel(x)results from the electric interaction in the equilibrium system of the small ions and the plate charges. The DLVO potential, given by eq 2, where K is the inverse Debye screening
VDLvo(x)= exp(-m)
(2)
length, corresponds, approximately but essentially, to the sum of the osmotic components Vio(x) V,,"(x)in our adiabatic pair potential. The other component Vel(x)of the pair potential leads to a long-range attraction between the plates. It results from the electric field energy in the inner region Ri through the charging-up procedure. A delicate balance of two diametrically opposed effects, the electrical and the osmotic, leads to the pair potential having a minimum. The DLVO potential ignored the component Vel(x) and, as a result, predicted a pure repulsion. Ironically, Overbeek12verifies the three terms in eq 1 and thereby verifies the first 40 of the 43 equations which we use in ref 9. The crux of his argument is that he then
+
claims that in eq 1 (eq 41 in ref 9)we have missed two nonelectrical, nonthermal, nonmechanical terms, which lead back to DLVO theory. The reason we did not include these terms is that they do not exist. In the limited space available here, we cannot prove our c o n c l ~ s i o nthat ~~ inclusion ofeither or both ofthese terms leads to the system coming out of thermal equilibrium, but the reader may be able to feel intuitively that it is inherently unlikely that adding simple terms of the type k T In ni (a chemical potential term) and e@*(an electric potential energy term) to a function which is expressed in terms of elliptic integrals will lead to a n exact cancellation ofthe attractive branch in eq 1, as claimed by Overbeek in ref 12. Likewise, space forbids us from proving here that Levine and Halllo make a mathematical blunder in taking the x-derivative of the field energy term and that the term which Ettelaiel' tacks onto our thee$ also leads to the system coming out of thermal equilibrium. These are demonstrated in ref 13. We wish to emphasize that we have answered the criticisms of the SSS formulation of the Coulombic attraction theory and that ref 9 is the exact solution to the problem. The SSS formalism gives a rigorous description of the interaction, but it is difficult to extend it because of the mathematical complications. In the present paper, we therefore develop the Gibbs free energy formalism of the interaction between charged plate^.^ This is a direct development of the original Sogami-Ise interaction potential for spheres5and lends itself readily to comparison with experiment because of its mathematical simplicity. Of course, if a Gibbs description is adopted, it has to be used consistently. Overbeek in ref 12 glosses over our rebuttal' ofhis previous criticism6of the Sogami potential based on the Gibbs free energy. As we use this formalism in the present paper, we recall the argument here. Overbeek6 suggested that the Gibbs free energy expression employed by Sogami and Ise5 was incomplete in that the electrical contribution of the solvent was ommited. The equation which he used to justify this was
(3) which states an independent relation between the chemical potentials of the solvent molecules and the small ions in a region that also contains the macroions. Ise et al.15 first pointed out in their footnote 32 that Overbeek violated the Gibbs-Duhem equation by omitting the contribution of the macroions in eq 3, the correct Gibbs-Duhem equation being
Niui + NsolvPsolv + N p m + ~ p a r= t 0
(4)
and we amplified this rebuttal, showing that eq 3 leads to the inescapable and implausible conclusion that "there is no free energy associated with the electrical double layers". As pointed out by Schmitz16in an independent review of the debate, "Since this conclusion is clearly wrong, one must seek a different argument from inclusion of the solvent to invalidate the theory proposed by Sogami and Ise". It remains the only argument forwarded by Overbeek to date. In ref 7, we went on to show that eq 4 leads to
AG = AF -k AE = AF
+ PAV
(5)
and sought a conceptually simpler way of obtaining this (15) Ise, N.; Matsuoka, H.; Ito, K.; Yoshida, H.;Yamanaka, J. Langmuir 1990, 6, 296. (16) Schmitz,K.S.Macroions i n Solution and Colloidal Suspension; VCH: New York, 1993.
2886 Langmuir, Vol. 10,No. 9, 1994
Smalley
relation, using the principle of virtual work. Overbeek objects to this derivation in ref 12, where he states that it should be replaced by
AG=O
(6)
Since AG is the free energy change a t constant T and P, the work done in compressing the macroionic phase, PAV, results in no change in the free energy of the system according to this equation: any mechanical work done on the system simply disappears. This contravenes the first law of thermodynamics. Although this is a serious blunder, our opponents have been making a n even more serious one in refs 6 and 10- 12. They have been ignoring a scientific concept so basic that it appears on the very first page of the Feynman Lectures on Physics.17 “The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific truth.” (Feynman’s italics). 3. n-Butylammonium Vermiculite Swelling In ref 7 we treated the theoretical ideal of a stack of perfectly parallel charged plate macroions. We are fortunate to have a model system which approximates quite well to this ideal, namely the n-butylammonium vermiculite system which was first reported by Walker.ls As long ago as 1962, Garrett and Walkerlg suggested that the homogeneous-lookinggreen- yellow gels formed when n-butylammonium vermiculite crystals are soaked in water or dilute aqueous solutions of n-butylammonium salts could be used to provide a sensitive test of theoretical models of colloid stability, and in 1963, reporting a preliminary X-ray study of the gels, Norrish and RauselColomzOnoted that the cohesive force in the system a t interplate separations of the order of 200 A was far too great to be explained by van der Waals forces. In recent years the system has been studied as a function of five variables: the volume fraction of the crystals in the condensed matter system (r),the electrolyte concentration of the soaking solution (c),the temperature (T),hydrostatic pressure (P),and uniaxial stress along the swelling axis ( p ) . Smalley et al.zl studied the P,T behavior a t r < 0.01, c = 0.1 M,p = 0, Braganzaet aLz2studied thec,Tbehavior a t r < 0.01, P = 1 atm, p = 0, Crawford et al.z3studied thep,c behavior a t r < 0.01, T = 10 “C, P = 1 atm, and most recently, Williams et al.24studied the r,c,Tbehavior a t P = 1 atm, p = 0. The essential facts are as follows. (1)The gels show no tendency to disperse into the solvent medium when the latter is held in gross excess (r < 0.01) with respect to the m a c r o i o n ~ . l ~Although -~~ eq 4 tells us that there must be a finite number of macroionic plates in the salt solution which exists in equilibrium with the gel phase, the experimental results tell us that this number is negligibly small. (2) There is a reversible phase transition, with respect to both t e m p e r a t ~ r eand ~ ~ ,hydrostatic ~~ pressurez1 be~~
~
(17) Feynman, R. P. The Feynmann Lectures on Physics; AddisonWesley: Reading, MA, 1963. (18)Walker, G. F.Nature 1960,187,312. (19)Garrett, W. G.; Walker, G. F. Clays Clay Miner. 1962,9,557. (20) Norrish, K.; Rausel-Colom, J. A. Clays Clay Miner. 1963,10, 123. (21)Smalley, M. V.; Thomas, R. K.; Braganza, L. F.;Matsuo, T. Clays Clav Miner. 1989.37. 474. (i2) Braganza,L. F.;Crawford, R. J.; Smalley, M. V.; Thomas, R. K. Clays Clay Miner. 1990,38,90. (23) Crawford, R. J.; Smalley, M. V.; Thomas, R. K. Adu. Colloid Interface Sci. 1992,34,537. (24) Williams, G. D.; Moody, K. R.; Smalley, M. V.; King, S. M. Clays CZay Mzner., to be published.
tween the crystalline (primary minimum) and gel (secondary minimum) states of the system. (3)Throughout the concentration range 0.001 M < c < 0.1 M, the n-butylammonium chloride concentration in the supernatant fluid is 2.6 & 0.4 times the average concentration in the gel phase.z4 (4)Throughout the concentration range 0.001 M < c < 0.1 M, the d-spacing between the parallel plates varies approximately as the inverse square root of the concentration of the soaking solution. In other words, the plate separation is equal to a constant number of Debye screening (5) Throughout the concentration range 0.001 M < c < 0.1 M, the d-spacing between the parallel plates decreases linearly with the logarithm of the applied uniaxial stress along the swelling axis, the quantitative relation showing that the surface potential is equal to 70 mV and is constant with respect to the salt c o n c e n t r a t i ~ n . ~ ~ Fact 1 shows that there is a long-range (10-100 nm) attractive force between the charged plates. Fact 2 shows that the colloidal (gel)state is a true thermodynamic phase of the system. This already invalidates DLVO theory, which is based on the assumption that lyophobic sols are thermodynamically unstable. Facts 3 and 4 will be addressed later in the paper, being objects of its study. Fact 5 is particularly important for the calculations presented in section 5. We first note that the relationship between the (net) surface charge, 20and the (effective) surface potential, ~ 0 is , given by9
where AB is the Bjenum length. It is very difficult to measure the net (as opposed to the analytical) surface charge on a colloidal particle. The net charge ofmacroions and latex particles can sometimes be determined by transference experiments, first carried out by Wall et aLZ5 and subsequently by Ito et a1.,z6but this method cannot be applied to clay particles. obtained a similar result to fact 5 for the related montmorillonite clays, finding y j o = 60 mV to be constant with respect to the electrolyte concentration from thermodynamic measurements. Fact 5 itself was obtained from the effect of uniaxial stress on the n-butylammonium vermiculite gels, which were studied over a wide range of applied stresses and salt concentration^.^^ This study provided unusually sharp information about the net surface charge on the particles, which increased with the square root ofthe concentration of the salt solution. This proved experimentally that the surface potential on the plates is constant with respect to the electrolyte concentration. The parameter K , which is proportional to the square root of the salt concentration, was also measured from the effect of uniaxial stress on the gels, and the quantitative analysis gave the result vo= 70 mV, irrespective of the value of the electrolyte concentration, c, in the range 0.001 M < c < 0.1 M. We wish to emphasize that this is a thermodynamically measured quantity, obtained by allowing the gels to come to equilibrium with an applied stress field, and is not to be confused with the 5 potential. We further note that the value y j o = 70 mV yields a good qualitative interpretation of the position ofthe (c,T) phase boundary in the n-butylammonium vermiculite system.z4 (25) Huizenga, J. R.; Grieger, P. F.; Wall, F. T. J . Am. Chem. Soc. 1960,72,2636. (26) Ito, K.; Ise, N.; Okubo, T. J . Chem. Phys. 1985,82, 5732. (27) Low, P. F. Langmuir 1987,3,18.
Electrical Theory of Clay Swelling
Langmuir, Vol. 10,No. 9, 1994 2887 CLAY
CLAY
Figure 2. Schematic illustration of the swelling of n-butylammonium vermiculite: (a) the unexpanded crystal in a 1.0 M n-butylammoniumchloride solution; (b-d) the gels formed in 0.1, 0.01, and 0.001 M solutions, respectively.
4. The Three-Component System The n-butylammonium vermiculite system is an example of a three-component system of a monodisperse colloid, electrolyte, and solvent. There are four constituents in the macroionic solution, the negatively charged clay plates, n-butylammonium ions (counterions), chloride ions (co-ions), and water, but these may not vary independently because they are subject to the restriction that
[ n - B u f l = [plate-]
+ [Cl-]
in a n obvious notation. Hence the number of components is 4 - 1= 3. In the following we shall refer to the solvent as water and to the electrolyte as salt. The raw phenomenon of the clay swelling in the n-butylammonium vermiculite system is represented schematically in Figure 2. In the cases studied in refs 18-23, v* was always much greater than V, the volume occupied by the macroions. We now define V,,, to be the volume occupied by the macroions in the coagulated state, as in Figure 2a in the vermiculite system. This is a n experimentally controlled variable. We define the sol concentration r by
r = VmiV where v* is the total volume of the condensed matter system. In the case of swelling illustrated by Figure 2a,b v* decreases by approximately 0.1%.21 This is avery small fractional volume change compared to that observed in V, so in the following we ignore the electrostriction of the solvent which accompanies swelling; that is, we take V* = constant. Although the phase boundary has been investigated with respect to temperature and hydrostatic pressure,2l we now restrict attention to P and T constant, so that we can represent the phase behavior of the system on triangular graph paper. For the n-butylammonium vermiculite system, the molecular weight of the salt is 109.5. We use the order of magnitude figure of 100 to make the plotting of the phase diagram simpler. The partial molar volume of the salt is nearly independent of the salt concentration, its average value being equal to 110 mL/mo1.28The density of the salt is therefore approximately 1g/cm3, so volume fractions and mass fractions are identical for the simple electrolyte solution in this case. n-Butylammonium chloride salts out from simple electrolyte (macroion free) solutions a t about 4.5 M.24 This value does not depend significantly on the presence of macroions in the solution
//+ SALT c/m= 4.5
4.
:
+ 0.20.1
0.01
0.001
WATER
Figure 3. Phase diagram of the three-component system of clay (n-butylammoniumvermiculite),salt (n-butylammonium chloride),and water at T = 4 "C, P = 1atm: (a)a straight mass fraction plot; (b) the mass fraction plot when the molecular weight of the salt is re-scaled by 1000. The curved phase boundary is calculated in section 6, with the dotted line representing the re-appraisedphase boundary. (c)A schematic plot. The r = 0.1, c = 0.01 M point has been placed at the center ofthe triangle, and the scale has been distorted to show all four regions clearly. The labeling of the regions is explained in the text. The crosses indicate the points studied by Williams et a1.24
and so is independent of the sol concentration r. This enables us to draw in the left-hand wedge in Figure 3a, which gives the phase diagram in mass fractions. For the density of the n-butylammonium vermiculite crystals, whose known mass used in the experiment determines V,, we have taken the figure of 1.86 g/cm3appropriate for the Eucatex samples studied in ref 24. The left-hand wedge of Figure 3 represents a threephase region of crystalline clay, solid salt, and saturated salt solution and is labeled region IV in the schematic Figure 3c. In Figure 3c, the r = 0.1, c = 0.01 M point has been placed a t the center of the triangle and the scale has been distorted to show all four regions clearly on the same plot. (28) Desnoyers, J. E.;Arel, M. Can. J. Chem. 1967,45, 359.
Smalley
2888 Langmuir, Vol. 10, No. 9, 1994
As the salt concentration is decreased below 4.5 M, the solid salt phase disappears, but the clay crystals do not swell until c is decreased below 0.2 M (at T = 4 “C, P = 1 atm.). To a first approximation, this value is also independent of r, which enables us to draw in the central wedge (region 111) in Figure 3. This represents a twophase region of salt solution and crystalline clay. In both regions 111 and IV,the macroions are in their primary minimum (crystalline, coagulated, flocculated) state. To the right of region 111, in electrolyte concentrations c < 0.2 M, the clay absorbs water macroscopically and swells “osmotically”into the secondary minimum (gel, sol) state, giving us the one-phase (I) and two-phase (11) regions of colloid stability. In the one-phase region, the clay soaks up all the salt solution, reflecting the osmotic repulsion between the plates under high r, low c conditions. Under low r, low c conditions, the clay gel exists in equilibrium with a practically macroion-free supernatant fluid (the two-state structure), reflecting the long-range electrical attraction between the plates. If we plot the phase diagram in ordinary weight percentages, as shown in Figure 3a, the most interesting chemistry, that of gel formation, is confined to too small a region on the right-hand edge. We gain more insight into the phase behavior of the system if we re-scale the molecular weight of the salt by a factor of 1000, as shown in Figure 3b. This has the effect of fanning out the plot around the c = 0.01 M line and shows clearly the curved phase boundary between the one-phase and two-phase regions of colloid stability. It been calculated using the method described in sections 5 and 6. 5. The Salt Fractionation Effect The central prediction of the Sogami theory is that there is a weak attractive tail in the thermodynamic electrostatic interaction potential between colloidal particles in electrolyte solution^.^-'^ In the linearized theory, which we now pursue, the position of the minimum in the pair potential between macroionic plates is given in the low c, low r limit by the equation
amin =4
(8)
where xminis the equilibrium separation of the particles. This localizes the plates a t a distance of four Debye screening lengths, where for monovalent ions in water at 25 “C the inverse Debye screening length is defined by eq 9, where K is expressed in units of A-1 and c is the K’
= 0.107~
(9)
concentration of the electrolyte solution in mom. Equation 9 suffices to define K in a simple ionic solution, but for macroionic solutions the situation is not so simple. To quote Schmitz,16 “There is some ambiguity in polyelectrolyte and colloidal systems as to whether or not the macroion contributes to the calculation of the screening length.” The confusion surrounding the many possible methods used for calculating K has even led Yamanaka et al.29to raise the question ofwhether the Debye-Huckel screening length as defined by simple electrolyte theory has any meaning a t all in polyion systems. Here we adopt the position used by Schmitz of treating the Debye-Huckel screening length as a scaling parameter, for the purpose of characterizing the solution properties of charged particles that differ in ionic concentrations. In calculating it, we shall adopt the simplest possible procedure, that of assuming that neither the macroions nor the counterions contribute to the screening length. Under this condition
Table 1. Some Illustrative Values of the Fractional Co-ionDefect @) as a Function of the Surface Potential (VO) 0 25 50
70 75 100 200
0.50 0.38 0.27 0.20 0.18 0.12 0.02
0.50 0.35 0.24 0.17 0.16 0.09 0.01
u... the Debye-Huckel
screening length is calculated solely from the added electrolyte concentrations, and it follows that the macroion-macroion interaction potential is independent of the concentration of macroion.”16(Schmitz’s italics). With this simplification, the minimum in the pair potential always lies a t 4 / ~ . Even within this approximation, there is no reason why the average salt concentration in the gel phase should be equal to that in the supernatant fluid. We now calculate out the salt distribution between the gel phase and the supernatant fluid in the two-phase region of colloid stability. The ion exclusionfrom the gel phase is described in the manner of F. W. Klaarenbeeke30 The distribution of ions is described as the expulsion of a certain amount of ions of the same sign as the colloid. For a single flat double layer on a negatively charged wall, the ratio (9) ofthe defect ofnegative ions to the total doublelayer charge is given by the integra130
which is solved by
where @O is the surface potential, lz is Boltzmann’s constant, and T i s the temperature. It is easy to see that g must be equal to ‘12 as ~0 tends to zero by expanding the exponentials in the linear approximation (Debye limit). For small values of the double-layer potential ~ 0 the , quotient is equal to l/2, but for large negative values of Vo, the quotient is smaller even in the limit of low sol concentration. As a n example, we take Iqol = 70 mV, for later comparison with the n-butylammonium vermiculite gels.23 In this case, eq 11gives g = 0.20, where it is only worth keeping 2 significant figures because this represents the limit of the accuracy of all the relevant colloid experiments. Some values of g which cover the experimentally accessible range are given in the second column of Table 1. As these results are obtained, it has been assumed that the potential tends to zero a t an infinite distance from a single charged plate immersed in a n electrolyte solution. We now consider the more interesting case oftwo charged plates immersed in an electrolyte solution. The geometry of the two-plate problem was shown in Figure 1. In the inner region, Ri, the limits of the integral in eq 10 are no longer appropriate because of the finite value of the midplane potential, as shown in Figure IC. In keeping with our use of the linear approximation, we take the potential in the left-hand half-region to be the (29)Yamanaka, J.; Matsuoka, H.; Kitano, H.; Ise, N.; Yamaguchi, T.; Saeki, S.; Tsubokawa, M. Langmuir 1991, 7, 1928. (30)Klaarenbeek, F. W. Ph.D. Thesis, Utrecht, 1946.
Electrical Theory of Clay Swelling
Langmuir, Vol. 10,No.9,1994 2889
Table 2. Some Illustrative Values of the Salt Fractionation Factor ( 8 ) as a Function of the Surface Potential ( ~ 0 )
vo(mV)
S
0 25 50
1.0 1.6 2.3
70
2.8
qdmV)
S
75
2.9 3.1 2.3
100 200
exponentially decaying function
v = v,,exp(-m)
(12)
where x is the distance from the plate surface. The potential a t the midplane, vd, is then given simply by v0 exp(-2) = 0.135v0, because the midplane is two Debye screening lengths from the surface. The integral given in eq 10then has to be redone from v d to rather than from 0 to ~ 0 but , this has little effect on the ratio g, as shown in the third column of Table 1. In our example case with v0equal to 70 mV, the potential a t the midplane is 10 mV and g = 0.17. The calculation of the salt fractionation effect is now extremely straightforward. Let Nde be the total number of co-ions per unit area expelled from the half-region between the midplane and the surface. This defect of negative ions is given by
(13) Let ride be the average number density of the defect, given by
ride = gZdd
(14)
where d is the half-separation of the plates. Since the surface charge is proportional to K (eq 7) and d is inversely proportional to K (eq 8), ride is proportional to K ~ .We can convert the term on the left-hand side into a defect molarity, Cde, using
ride = 6.02
1 0 -Cde ~
where n is expressed in A-3 and c in mol/L, and we can convert the right-hand side into a molarity using eq 9. This yields Cde
= 0.64
(16)
for I)O equal to 70 mV and, generally, that the defect molarity is proportional to the molarity ofthe salt solution. The simple linear relation (eq 16) leaves a certain ambiguity in how we should define c, the salt concentration. Following ref 7, we begin with the limit of low sol concentration and take c = cex,the salt concentration in the supernatant fluid. Since cgel
= cex - Cde
(17)
the ratio of the salt concentration in the supernatant fluid to that in the gel phase is seen to be constant (for a given constant surface potential). This is expressed by Cedcgel = s
(18)
where we have labeled the salt fractionation factor by s, which is consistent with the notation used in ref 24. The results for the Coulombic attraction theory prediction for s are given in Table 2. An interesting feature of Table 2 is that s has a turning point as a function of q o . This may be a n artefact of using
the nonlinear relation of the surface potential to the surface charge, eq 7, in a n otherwise fully linearized theory, but it may also be a genuine effect. It is noteworthy that for the n-butylammonium vermiculite system s lies close to the maximum possible ratio. For @O = 70 mV, s = 2.8, which renders it straightforward to measure experimentally.24 The experimental results on the n-butylammonium vermiculite gels are in good agreement with this prediction. The ratio is approximately constant across the range of salt concentrations between 0.001 and 0.1 M, its average value being equal to 2.6 f 0.4.24 The idea presented here that the salt ratio should be constant with respect to the added electrolyte concentration appears to be a novel one. A major advantage of such a simple result for the salt distribution is that it enables us to calculate the d-spacings in the two-phase region of colloid stability as an analytic function of r and c. This in turn will enable us to calculate the complete phase boundary between the one-phase and two-phase regions of colloid stability. 6. The Sol Concentration Dependence of the d-Spacings The Coulombic attraction theory result x,in = UKonly yields a n experimentally testable prediction ofxminvs c if we know the relation between K and c. In ref 7 we assumed Kgel = Kex
= Kb
(19)
where the symbols represent the inverse Debye screening lengths in the gel, supernatant fluid, and globally, respectively. This gives a n unambiguous experimental definition of K because the supernatant fluid is for all practical purposes a simple electrolyte solution. Its experimentally controlled, known concentration then determines K~~ via eq 9. However, in view of relation 18, eq 19 is no longer tenable. We continue to adopt the simplest possible procedure, that of assuming that neither the macroions nor the counterions contribute to the screening length. Since K is then determined solely by the electrolyte concentration, it seems logical to choose K~~~~ = 0
.107~~~~
(20)
in accordance with eq 9. An immediate benefit of this choice is that it gives us a new prediction for thed-spacings as a function of c in the limit of low sol concentrations (r < 0.01). In this limit (see Figure 21, v* >> V,, so the salt which is excluded from the gel phase when V , expands to V has a negligible effect in the large volume v* - V which remains as the supernatant fluid. For the nbutylammonium vermiculite gels, we then have the relationcb = cex= 2 . 8 ~ ~ ~Together 1. with eqs 9 and 20, this implies K~~ = 1 . 7 ~ ~and ~ 1 substitution , of Kgel into eq 8 gives the new prediction: K,S:,~~ = 6.7. This gives quantitative agreement with the observed interlayer spacings in the low r, low c limit. For example, for r < 0.01, cex= 0.001 M ( K =~ 0.0103 ~ A-l) the prediction is Xmin = d e. = 650 A, and the observations lie between 550 and 680 ik22,23 Note that equatingx,i, with d,~, the d-spacing in the gel phase, is consistent with the approach used in ref 7 and has been used implicitly in the preceding section. For the case r > 0.01, the equation Cb = cexis no longer applicable, because the salt excluded from the gel phase has a significant feedback effect on the concentration in the supernatant fluid. However, the problem is solved easily using eqs 8, 18, and 20 together with two simple conservation principles and the assumption that the
2890 Langmuir, Vol. 10, No. 9, 1994
Smalley
Table 3. Sol Concentration Effect in n-Butylammonium Vermiculite Swelling in (a) 0.001,(b) 0.01, and (c) 0.1 M Soaking Solutions
Table 4. (r,c)Phase Boundary in n-Butylammonium Vermiculite Swellinp 10-5
(a) c = 0.001 M
650 580 530 390
-0 0.01 0.02 0.05
I
m
1940 970 390
1
(b) c = 0.01 M 210 200 190 180 150 130
-0 0.01 0.02 0.05 0.10 0.15
m
1940 970 390 190 130
65 61 57 53 49
m
194 97 65 49
swelling in clay minerals is perfectly homogeneous, so that we can relate the microscopic d-spacing directly to the volume of the macroion phase. The equation of the conservation of salt is VgelCgel+ vexce, = (V* - V,k,
(21)
where ca is the concentration of the added salt and we have assumed that the macroions are initially salt-free. Vgel and Vex are the volumes of the gel phase and supernatant fluid at equilibrium, and cgeland ceXare their respective (average) salt concentrations. The equation of the conservation ofvolume of the condensed matter system (electrostriction effects ignored) is (22) and the equation relating the volume of the gel phase to the d-spacing in the gel phase is (23) where 2a is the c-axis repeat distance of the crystalline mineral, a n experimentally determined quantity. It is straightforward to show that the six equations 8, 18, and 20-23 lead to a quadratic equation for dgelin terms of r and c. For n-butylammonium vermiculite, 2a = 19.4 Azz and the result is conveniently expressed as
where c is the concentration of the added salt in mol/L, r is the sol concentration, and d,l is the d-spacing in angstroms, given by the positive (physical) root of the equation. Some illustrative values of dgelas a function of r and c are given in Table 3. As a n example of the complete set of parameters specifylng the solution a t an (r,c) point, we consider the addition of 99 cm3 of 0.001 M solution to 1 cm3 of pure (salt-free) clay. In this case, Vgel = 30 cm3, cgel= 4.4 x M, Vex= 70 cm3, and cex= 1.2 x M. Note that Cex is significantly higher than the value of the added electrolyte concentration, illustrating the feedback effect clearly. The result for the d-spacing therefore differs significantly from that obtained in the limit of infinite sol dilution (r 0, c = 0.001 M, d g e l = 650 A): solution of eq 24 gives dgel= 580 A, corresponding to an approximately
--.
0.03 0.1 0.2 0.3 0.4
0.0050 0.016 0.050 0.084 0.15 0.24 0.39 0.50 0.57 0.62
3900 1200 390 230 130 80 50 39 34 31
4.0 4.0 4.0 4.1 4.2 4.6 5.1 5.7 6.1 6.5
I1
1
The block designates the best characterized experimentalrange.
(c) c = 0.1 M
-0 0.1 0.2 0.3 0.4
10-4 10-3 0.003
30-fold expansion of the crystal, in excellent agreement with the results of Williams et al.24 The third column in Table 3 has been calculated as d, = 19.4/r, which defines the maximum obtainable spacing, when the clay has soaked up all the solvent. Both dgeland d, are monotonically decreasing functions of r, and the boundary with the one-phase region is defined by dgel= d,
= d* = 19.4lr*
where r* is the sol concentration a t the phase boundary between the one-phase and two-phase regions of colloid stability. For the three example cases given in Table 3, namelyc=0.001M,c=0.01M,andc=0.1M,r*=0.05, 0.15, and 0.39, respectively. The calculation can be repeated for any arbitrary value of c, which gives a complete calculation of the (r,c) phase boundary. Some illustrative results are given in Table 4. At the high salt concentration end of Table 4, for c 0.1 M, we are really pushing mean field theory beyond its limits. This is shown by the third column of the table, which gives the d-spacing a t the phase boundary. For c = 0.4 M, d,l= 30 A, which is the approximate thickness of the clay plates with two adsorbed layers of nbutylammonium ions.24,31This is the region of crystallization into the primary minimum, which occurs a t c = 0.2 M a t 4 "C, where the phase diagram in Figure 3b has been plotted. The predicted increase in swelling between 0.4 and 0.2 M soaking solutions is 8 A. This contains an insufficient number of water molecules for double-layer theory, treating the solvent as a dielectric continuum, to be appropriate. The results for c > 0.1 M have therefore to be regarded with caution: c = 0.1M should be considered the upper limit of reliability in Table 4, and the results have been boxed off to reflect this. The range of salt concentrations below M has been experimentally unobtainable because of the effect of salt leaching out of the crystals, even after they have been subject to a lengthy washing procedure.24 It is highly doubtful that we could ever achieve an ionic strength M uni-univalent electrolyte as low as that given by a solution in a solvent in contact with a real clay mineral. This is a pity because, according to the theory presented here, such a material would diffract visible light. It is apparent from columns 1and 3 of Table 4 that d* varies approximately as the inverse square root of c, the electrolyte concentration of the soaking solution. This is made explicitly clear in the fourth column of the table, which gives N,, the number of Debye screening lengths between the plates a t the phase boundary. N , has been , K~ = 0.107c, with c the calculated as N , = ~ d * where global salt concentration. This is an easily controlled experimental variable, and we can use it because all the (31)Rausel-Colom, J . A. Trans Faraday SOC.1964,60, 190.
Langmuir, Vol. 10, No. 9, 1994 2891
Electrical Theory of Clay Swelling Table 5. €&-appraised(r,c)Phase Boundary c
(MI
0.001 0.003
0.01 0.03 0.1
4/K
(A)
flu+)
(A)
400
2
231 126 73 40
3 6
9 15
%in
(A)
NX
402 234 132 82
4.0
55
5.5
4.1 4.2 4.5
salt is trapped inside the clay in the one-phase region of colloid stability. In this case, cin = cex= Cb and we can use K~ = 0.107cb to define K unambiguously. The equilibrium separation of the particles in the twophase region is roughly inversely proportional to the concentration of the salt solution and so is the phase boundary. Equation 8, which contains this dependence, is only the small K approximation to the equilibrium separation of plate macroions, the full equation' being
The two terms in eq 26 are given separately in the second and third columns of Table 5, where the second term has been labeled f ( a , K ) . In column 4 of Table 5 we have given the predictions for xminas a function of c, and in the fifth column we have expressed this as the appropriate number of Debye screening lengths, N,. The right-hand columns of Table 5 and the blocked part of Table 4 show a remarkable agreement. This is no accident. The essential feature of the function given by eq 26 forxminis that, for separations greater than about UK,the plates attract each other, so that the clay does not soak up any more solvent beyond this point: osmotic swelling stops, and the two-phase region begins. This suggests an easy method for calculating directly the position of the phase boundary by using xmin= d*. The re-appraised phase boundary obtained by this method has been plotted as the dotted line in Figure 3b. I t only deviates from the solid curve, that obtained by approaching the phase boundary from the two-phase region, a t the high salt concentration end. Around c = 0.1 M t h e p d t fractionation effect will deviate from the calculated value of 2.8, because this has been derived from the approximate eq 8, so we should not expect quantitative agreement here. It has already been remarked that this salt concentration anyway marks the upper limit of applicability of mean field theory. Of course, we could have guessed from the outset that the position of the secondary minimum in the Coulombic attraction theory would define the phase boundary, but then we would have missed out on some interesting results in the two-phase region.
7. Discussion The preceding considerations show that the Coulombic attraction theory is well adapted to explain the existence, extent, and properties of the two-phase region of colloid stability. Such considerations also show that the n-butylammonium vermiculite system is governed by electrical forces. It is highly unlikely that we could accurately predict both the d-spacings and salt fractionation effect in the two-
phase region if any of the other forces which are commonly introduced into colloid theory, such as hydrophobic forces or van der Waals forces, played any significant role. The reason why we see the electrical phenomena so clearly in this model system is because the n-butylammonium ion approximates as closely as any ion does to ideal behavior: the enthalpy of solution of simple n-butylammonium salts is nearly equal to zero,32and their partial molar volumes are nearly independent of concentration,2s implying that there are no special ion-solvent effects between nbutylammonium ions and water. In DLVO theory the two-phase region can only be created by the van der Waals force, which is independent of the salt concentration across the concentration range 0.001 M < c < 0.1 M.2 This force has to be balanced with a force which decays exponentially as a function of K , which means that it decays by a factor exp(- 10)across this range. The unhappy consequence of this prediction is that the position of the secondary minimum, and therefore the position of the phase boundary, varies very rapidly as a function of K , in contradiction to the experimental results. A further unhappy consequence of this balance is that it always produces a primary minimum much deeper than the secondary minimum. This renders it unable to explain the raw phenomenon of osmotic swelling, in which a primary minimum material develops spontaneously into the secondary minimum, and unable to explain the thermodynamic character of this transition.21,22Such subtle effects as the salt fractionation effect discussed here are way beyond its scope. It is noteworthy that in the one-phase region of colloid stability the net interaction between the plates is a repulsive function which decays approximately exponentially between the plates. This is the prediction of DLVO theory.lS2 Furthermore, in this region all of the added salt penetrates between the clay sheets, co-ions alike with counterions, so the ion distributions given by DLVO theory are also approximately correct. In this region, therefore, the electrostatic part of DLVO theory still applies. In this sense, the DLVO theory can be seen to be a limiting case of the Coulombic attraction theory. In the two-phase region of colloid stability, neither the interaction between the clay plates nor the ion distribution between them are at all as imagined by DLVO theory. Although the DLVO theory remains adequate for calculating the properties of charged colloidal suspensions in the one-phase region, it contains no hint of its own limitations and is seen to be but the limiting case of the Coulombic attraction theory a t high sol concentrations. At low sol concentrations, many important practical problems, such as sedimentation problems in lakes and the rheology of drilling muds, will have to be re-appraised because the interaction between the charged particles is not that envisaged by DLVO theory, which is unfortunately still common currency amongst many experimental workers in the field. Acknowledgment. I thank the SERC for provision of a n Advanced Fellowship to support this work and the ERATO Project of the JRDC for their support during its completion. I also thank Dr. R. K. Thomas and Professor S. Levine for helpful suggestions and Professors N. Ise and I. S. Sogami for their encouragement. (32)Krishnan, C. V.;Friedman, H.L.J.Phys. Chem. 1979,74,3900.
