1090
Langmuir 1992,8, 1090-1095
Electric Double Layer Interactions of Highly Charged Platelike Colloidal Plates S. Levine' Department of Chemical Engineering, University of British Columbia, Vancouver, British Columbia, Canada V6T 124
Denver G. Hall Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, Merseeyside L63 3JW, United Kingdom Received July 15, 1991. In Final Form: January 22, 1992 The controversy as to whether the electrical forces between highly charged colloidal particles in low salt concentration are repulsive or attractive is investigated. A recent paper by Sogami, Shinohara, and Smalley on the interaction of highly charged parallel platelike colloidal particles in the limiting case where only counterions are situated between the plates is examined. Their prediction of an attractive electrical free energy is confirmed, using two equivalent methods. A consequence of the restriction that co-ions are absent between the plates is the dependence on plate separation of the free energy of the hypothetical completely discharged state. This contributes an osmotic repulsive energy term. Our electrical and chemical terms in the interaction energy are the same as those given in the theory of Sogami et al. However, they define the double layer force by differentiating their energy expression with respect to plate separation at fixed plate potential. This yields non-Coulombic repulsion at small separations between the plates but electricalattraction at large separations, in disagreement with classicalDLVO theory. We present arguments that the plate charge should be held fixed in defining the force. This leads to an electric double layer force in accordance with the DLVO theory.
Introduction In a recent paper Sogami, Shinohara, and Smalleyl have developed a theory of double layer interaction between two parallel highly charged colloidal plates which is in disagreement with the classical DLVO theory. They predict non-Coulombic repulsion at small separations between the plates, but electrical attraction at large separations. Here we wish to examine the double layer interaction by a statistical mechanical theory which was developed by Levine and Bell in a series of papersa2+In the first paper of this series2 the author considered two parallel plates at potentials small enough to allow the linear DebyeHuckel approximation and made use of the familiar charging process. The double layer interaction was found to be repulsive at small separations and attractive at large separations. However the potential became infinite in the hypotheticalcompletely discharged state and the result obtained was clearly incorrect. The approach on which the following analysis is based differs from that of Verwey and Overbeek' in the use of the Debye-Huckel charging process. These authors charged at constant plate potentials, whereas the charging here is performed at constant density of surface ions on the plates. We consider that the latter method is more rigorous from a statistical mechanical point of view and can be applied more generally. It turns out that if the method in the paper cited above2 had been applied to highly charged plates, then the infinity in the potential would not have occurred. (1) Sogami, I. S.;Shinohara, T.; Smalley, M. V. Mol. Phys. 1991, 74, 599. (2) Levine, S. Trans. Faraday SOC.1946, 42B,102; 1948,4, 833. (3) Levine, S.Philos. Mag. 1950, 41 (Series 7), 53. (4) Levine, S. Proc. Phys. SOC.,London, Sect. A 1951,54, 781; 1953, 56, 357. (5) Levine, S. Proc. Cambridge Philos. SOC.1951, 47, 217, 230. (6) (a) G. M. Bell and S. Levine, Trans. Faraday SOC.1957,53, 143; (b) 1958, 54, 785; (c) 1958, 54, 975.
(7) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability o f Lyophobic Colloids; Elsevier: Amsterdam, 1948.
0743-746319212408-1090$03.00/0
In view of the acceptance of the DLVO theory among colloid scientists and the consequent controversy over the existence of long-range electrical attraction between colloidal particles, it seems important to examine further the work of Sogami, Shinohara, and Smalley.' In this paper we shall reproduce their attraction by two different methods in the limiting case where the parallel plates are so highly charged that the concentration of co-ions between the plates can be ignored. The latter restriction is a key feature of their model, a consequence of which is a repulsive non-Coulombic osmotic contribution to the interaction. Experimental observations in the past decade by Ise et al. on ordering formations in suspensions of highly charged latex particles&'5 and by others on the swelling of vermiculite plates16-18suggest an electrostatic attraction at large separations in highly charged colloidal systems and at low salt concentration. To simulate the ordering of the latex particles and vermiculite plates, we consider a one-dimensional cell model of a large number of parallel, charged, identical plates, where the two end plates are in contact with a 1:l electrolyte. Our endeavor to reconcile the theory of Sogami et al. with the DLVO theory indicates that these experiments remain to be explained. (8) Ise, N.; Okubo, T. Acc. Chem. Reu. 1980, 13, 303. (9) Ise, H.; Okubo, T.; Sugimura, M.; Ito, K.; Nolte, H. J. J. Chem. Phys. 1983, 78, 536. (10) Ito, K.; Ise, N.; Okubo, T. Chem. Phys. 1985, 82, 5732. (11) Ito, K.; Nakamura, H.; Ise, N . J . Chem. Phys. 1986,85,6136,6143. (12)Ise, N. Angew. Chem. 1986, 25, 323. (13) Ito, K.; Okumura, H.; Yoshida, H.; Ueno, Y.; Ise, N. Phys. Reu. E : Condens. Matter 1988, 38, 10852. (14) Ito, K.; Nakamura, H.; Yoshida, H.; Ise, N. J. Amer. Chem. SOC. 1988, 110, 6955. (15) Yoshida, H.; Ito, K.; Ise, N. Faraday Trans. 1991, 87, 371. (16) Smalley, M. V.; Thomas, R. K.; Braganza, L. F.; Matsuo, T. Clays Clay Miner. 1989, 37, 474. (17) Braganza, L. F.; Crawford, R. F.; Smalley, M. V.; Thomas, R. K. Clays Clay Miner. 1990, 38, 90. (18) Crawford, R. J.; Smalley, M. V.; Thomas, R. K. Ado. Colloid Interface Sci. 1991, 34, 537.
0 1992 American Chemical Society
Langmuir, Vol. 8, No. 4, 1992 1091
Interactions of Colloidal Plates
of the dispersion medium
I?--[
I
Figure 1. One-dimensionalsystem of 2 u charged, parallel plates at separation 21, in a cylindrical volume of length 2L >> 4u1, containing 1:l electrolyte as the dispersion medium. Crosssection of cylinder is almost identical with plate area.
Electrical Free Energy of Two Charged Plates Consider a large number 21, of identical, parallel, evenly spaced, platelike, charged particles, which are immersed in an aqueous 1:l electrolyte solution. The thickness of a plate equals d and its area is so large that any edge effects can be neglected. The distance between the inner faces of two adjacent plates equals 21. It is convenient to imagine that the edge of any plate is almost in contact with the wall of the vessel containing the system. Alternatively the plate edge reaches the wall and "windows" are introduced to permit the solvent (water) and ions to pass freely between all the plates and the outer solution. The electrolyte solution extends beyond the two end plates, such that the vessel is a cylinder of fixed length 2L and cross section identical with that of a plate, where L >> 2ul (Figure 1). The distance that the electrolyte extends beyond an end plate on each side is therefore L - ( 2 u - 111. Except for the outer faces of the two end plates, each face of a typical plate carries the same negative charge -2e per unit area of the plate, where e is the proton charge. Although 2 in general varies with I , it is convenient to display 2 and 1 separately. It is assumed that there is no potential drop within the region of thickness d occupied by the material of a plate. An end face will have a negative charge -2,e per unit area, where 2, can be assumed independent of 1 for L - ( 2 u - 1)1 much greater than the Debye-Huckel thickness of the double layer. Suppose that all the ions, i.e. those present in the aqueous dispersion medium in which the plates are suspended and the ions constituting the charge on the plates, are completely discharged. We now carry through the fictitious process of charging all the ions uniformly, the fraction X representing the stage in the charging. The electrical free energy per unit area of the inner faces of two typical adjacent plates is given by
F,(l,Z) = 2 t E ( A , l , Z )dX/A
(1)
where E(X,l,Z)is the average electrostatic energy per unit area of the electric double layer between the two plates at stage A in the charging process. This process is carried out at fixed separation 21 and a t fixed number density of ions 2 on each plate. The justification of these two conditions is well established (e.g. refs 2 to 6). The second condition means that a subspace of the configurational phase-space available to the ions is chosen. Since the Helmholtz free energy is a minimum, for large enough total area of the plates, this subspace constitutes most of the configuration space. In the region between the two plates, we choose the x axis normal to the two parallel plates, with origin at the median plane. The mean electrostatic potential in this region is a function of the degree of charging A, position x , the plate separation 21, and number density of plate ions 2, but it is sufficient to denote the potential by +(A,x). Then if dielectric saturation is ignored and c is the uniform dielectric constant
We shall assume the plates to be so highly charged that the concentration of negatively charged, univalent co-ions between the two negatively charged plates is negligible. This implies a high adsorption energy of co-ions onto the plates and also an additional limitation on the subspace of the configuration space available to the co-ions. Apart from the diffuse layers on the outer faces of the end plates, the outer regions on each side of the 2u plates are assumed to contain electrically neutral homogeneous 1:1electrolyte. The equal numbers of univalent cations (counterions) and anions (co-ions) per unit volume in the outer electrolyte regions on each side of the 2u plates can then be assumed independent of position x and are denoted by N(l,Z),a function of the variables 2 and 1, but independent of A. It is convenient to imagine that the charge on the plates consists of adsorbed anions (co-ions) of the same species as in the outer electrolyte regions. Similarly the counterions between the plates are the same species as in the outer regions. Since co-ions are excluded between the plates, by electrical neutrality, the number of counterions between unit area of two adjacent plates equals 22 for all X and 1. In the hypothetical completely discharged state A = 0, the counterions will be uniformly distributed between the plates, and their number per unit volume will equal 2/1 if short-range interactions between the discharged ions and the plate walls are neglected. The length of the cylinder which is occupied by the dispersion medium equals 2L = 2L - 2ud. We can regard the region of the 2u plates and the outer electrolyte regions as two phases, eachof which is electrically neutral. Let 2LN, be the total number of either counterions or co-ions (including the adsorbed co-ions) in our two-phase system per unit area of plates. For given total length 2L of the aqueous medium and given amount of electrolyte present, No is a constant independent of 2 and 1. Conservation of the total number of each species of ions can be expressed as L N , = [ L - l(2u - l)]N(l,Z)+ ( 2 u - l ) Z + 2, (3) where the last term 2, accounts for the double layers on the outer faces of the end plates. It is convenient to write eq 3 as L N , - 2, - ( 2 u - l ) Z (4) L - l ( 2 u - 1) The assumption that N(1,Z)is independent of position x means that the electrostatic potential is constant in the outer electrolyte regions, and since the region between the plates is electrically neutral, this constant can be chosen to be the zero of the potential $(A,x) between the plates. The corresponding number density of counterions between the plates is denoted by n(A,x), although like $(A,x), it is also a function of 2 and 1. On introducing the dimensionless potential N(1,Z) =
O(A,x) = Ae+(A,x)/kT (5) where k is Boltzmann's constant and T is the temperature, we write the Boltzmann distribution in the form
n(A,x) = N(1,Z)exp[-O.(A,x)l (6) Define a Debye-Huckel parameter ~(1,z) in the fully charged state X = 1 by
Recall that the plates are so highly charged that the
Levine and Hall
1092 Langmuir, Vol. 8, No. 4, 1992
X to u. It follows from the second relation in eq 19 that
concentration of negatively charged univalent co-ions between the two plates is assumed to be negligible. A t stage A the Poisson-Boltzmann equation for $0,~) reads therefore
dX/X3 = [Zlne2/(tkT)](tanu + u sec2 u ) / ( u 2tan2u) (22)
d2J,(X,x)/dx2= -[4~N(l,Z)Xe/elexp[-@(X,x)l making use of eq 6. This can be written as
F,(l,Z) = 2kTZJo
d2@(h,x)/dx2 = -[A ~~(1,2)/21 exp[-@(X,x)l (9) making use of eqs 5 and 7. The boundary conditions on @(h,x) are d@(X,x)/dx= 0 at x = 0
d@(A,x)ldx= -X~(l,Z)[exp[-@(h,x)l- exp[-@(X,O)Il'I2 (12) where @(h,O) is the potential at the median plane. Integrating again XK(1,Z)X = -J':$exp[-~~,x)l-
e~p[-@(X,O)ll-'~~ d@(X,x) = 2 exp[@(X,0)/21X
)
{exp[-@(~,x)~ - exp[-@(~,0)11'/~ (13) exp[-@(X,o)/21 on changing the variable of integration to exp[-@(X,x)l. This is rearranged to yield @(X,x) = @(X,O)
(
+ 2 In cos ([X~(l,Z)x/21exp[-@(X,0)/21}
(14) which is equivalent to the formula given by Sogami et al.' although the degree of charging was not shown explicitly. Expressing eq 2 in the form
E( 1, X,Z) = ( E / 47r) ( k T /Xe)
Jo'[d@(h,x)/dx]
dx
+ u sec' u) du
u tan2 u
E(l,h,Z) = (t/7r)(kT/e)2u(tanu - u)/(lX2)
(16)
where
u = X~(1,z)l exp[-@(X,0)/21/2 (17) Equation 16 can be expressed in the form given by Sogami et al.' by introducing (18) The second relation in eq 18 is obtained on writing the boundary condition, eq 11, as
u
[d@(X,x)/dxl,,, = -2u tan u/l = - 4 ~ Z X ~ e ~ / ( d z T ) (19) where we have made use of eqs 14 and 17 to derive the first relation in eq 19. Then we can write eq 16 as
E(I,X,Z)= 2kTZ(1- u2/q) On substituting eq 16 or eq 20 into eq 1
(23)
which, on integration becomes
Fe(l,Z) = 2kTZ(-l+
U tan u
+
In
( c u )-21n cos U ) (24)
where
U = ~ ( 1 , Z )exp[-@(1,0)/21/2 l (25) The condition that Z is held fixed during the charging process described by the integral in eq 21 is embodied in eqs 18and 22. The expression obtained by Sogami et al.' for Fe(1,Z) reads (in the fully charged state X = 1, where u=u)
Fe(l,Z) = 2kTZ[-l+ p/q+ In ( q + p / q ) l
(26) It is readily verified that eqs 24 and 26 are identical. We should keep in mind that the dimensionlesspotential at the median plane l@(l,O)l must be sufficiently large (say >2) to justify the neglect of the co-ion concentration between the plates. Thus the two plates cannot be too far apart. The limiting (maximum) value of U is ~ / 2 .It follows from eq 25 that an upper limit on 1 is estimated atK(l,Z)l= ?rexp(-l) H l,yieldingl= 30Aatanelectrolyte concentration of 10-2 M. Co-ion concentration between the plates cannot be ignored when the plate separation becomes too large and a solution in terms of elliptic integrals becomes necessary. In the completely discharged state X = 0, the counterion distribution between the plates is uniform such that n(0,x)= n(0,O)= Z / l and therefore, from eqs 4 and 6
( 15)
and substituting eq 14, we obtain
q = 2 ~ Z l X ~ e ~ / ( c= k Tu )tan
u (tan u - u)(tan u
(10)
d@(X,x)/dx= -41rZX~e~/(ck7') = -ZX2~2(1,Z)/[2N(1,Z)1 a t x = 1 (11) noting that it is assumed that the potential is uniform within the material of the plates. On multiplying eq 9 through by d@(X,x)/dxand integrating once, we get
arctan
and hence
(8)
(20)
Fe(l,Z)= [ 2 c / ( 1 r l ) l ( k T / e ) ~ ~ ~ 'uu-( tu) a ndX/X3 (21) We apply eq 19 to change the variable of integration from
@(O,O) < 0 or > 0, accordingly as, very nearly, Z > lNo or Z < IN,, which is reasonable since the plates are negatively charged. In the fully discharged state the potential between the plates differs from that outside the plates. It is shown in the Appendix that a potential difference is also found when counterions and co-ions are both present between infinite plates.
Alternative Derivation of the Electrical Free Energy We proceed to show how eq 26 can be derived from a general theory of the free energy of electrical double layers in concentrated colloidal solutions developed by Bell and Levine.6a From eq 18, in the fully charged state X = 1 1
+ q=/v= l/cos2 u
(28)
so that
-2kTZ In cos2 U = 2kTZ[ln ( q / @ )
+ In ( q + p/q)l
(29) It follows from eq 20 at X = 1 and from eq 29 that eq 26 may be written as
Interactions of Colloidal Plates
F,(l,Z) = -E(l,l,Z)
Langmuir, Vol. 8, No. 4,1992 1093
- 4kTZ In cos U - 2kTZ In (q/v)
(30) From the basic formula, eq 1, it is possible to derive an expression for the electrical free energy F, in a colloidal system which takes the familiar general form Fe = E, TS,, where E, is the electrostatic energy and S, is the entropy of the ionic distribution.68 Expressed in the present notation, for the particular colloidal system considered here of parallel plates a t arbitrary sol concentration, this general relation can be transformed to read
F,(l,Z) = -2Ze+(1,1) - E(l,l,Z) + 2kTZ In [N(l,l)/(N)] (31) where we need to identify ( N ) . Equation 31 is an application to the particular geometry in this paper of the general eq 4.10 in ref 6a. Here we can ignore the contribution to the electrical free energy from the co-ions and counterions in the outer electrolyte regions, since these ions would be uniformly distributed and therefore the argument of the corresponding logarithm in eq 31 would in such a way equal 1. We wish to define the quantity ( N ) that eqs 26 and 31 are equivalent. If we consider our colloidal system as consisting of two phases, namely the region occupied by the plates and the two outer regions, then the quantity (N) can be interpreted as the average number per unit volume of counterions in the part of the dispersion medium which contains the plates, i.e. ( N ) = 211. The expression defined by eq 31 is the electrical free energy attributed to unit area of the inner faces of two adjacent plates and the diffuse layer of counterions situated between unit area of the plates. Writing eq 14 a t x = 1and X=las 9(1,1) = O(1,O)
+ 2 In cos U
(32) where U is defined in eq 25, and making use of eqs 5 and 32, eq 31 becomes
F,(l,Z) = -E(l,l,Z) - 4ZkT In cos U - 2ZkT9(1,0) + 2kTZ In [lN(Z,l)/ZI (33) Equation 25 yields O(1,O) = -2 In (2U/[~(l,Z)ll)
(34)
Making use of eqs 7 and 18 at X = 1 and of eq 32, it is readily verified that eq 33 is identical with eq 30. The equivalence of eqs 26 and 31 is thus proven, provided we assume (N) = Z/1. The significance of this result seems to be that by excluding the co-ions from the dispersion medium between the plates, the above two phases have been completely separated. Since each phase is electrically neutral, the zero of the potential between the plates can be arbitrarily chosen.
Force between the Two Charged Plates The total free energy associated with the ions in the electric double layers is the sum of an electrical term F,(l,Z), which we have evaluated and a so-called chemical term Fc(l,Z) which is mainly the free energy of the completely discharged state X 0. The description "chemical term" is misleading since Fc(l,Z) includes the self (hydration) energy of the ions and also the effect of ion imaging in the plate walls. Under equilibrium conditions we expect that the total free energy is a minimum with respect to the number 2 of surface ions a t any given separation 1, i.e.
(35) {d[F,(l,Z) + ~c(~,Z)l/~~)~,,on,~~ =0 which states that Z is a function of 1 and which describes the adsorption isotherm for ions transferring between the plate surfaces and the dispersion medium. The condition that the ions are in thermodynamic equilibrium a t any separation 21 implies that the force between unit area of the plates is simply given by
-({W,(W)+ F c ( ~ , Z ) l / ~ ~ ) / 2 ) ~ i c o(36) n~~~ The justification that eq 36 defines the force is explained, for example, by Bell and Levine.6b We imagine a change in the plate separation taking place in two stages. In stage 1, this change occurs a t fixed number Z and the force is given by eq 36. In stage 2, ions are transferred between the dispersion medium and the surfaces of the plates to restore equilibrium. Assuming thermodynamic equilibrium, there is no free energy change in stage 2. Thus eq 36 rests on the premise that equilibrium exists with respect to transfer of ions between plates and dispersion medium. (Of course, after this derivative is determined, we need to substitute Z as a function of 1, as given by eq 35.) Application of eq 36 to eq 23 yields for the electrical force between unit area of plates
where a positive value signifies repulsion between the plates and a negative value attraction. From the second relation in eq 19, a t X = 1, we have
U tan U = 27re2Zl/(tkT)
(38) Making use of eq 38, and assuming Z independent of 1, eq 37 yields for the electrical force
(39) The right-hand member is obtained by using eqs 6 and 17 a t X = 1, and eqs 7 and 38; n(1,O) is the counterion concentration at a median plane between two plates. The force expression in eq 39 is always negative, Le., the electrical force between the plates will be attractive at all separations. The chemical free energy termFc(l,Z)is usually assumed to be a function of Zonly and hence explicitly independent of plate separation 1, when it would not contribute to the force formula in eq 36. This would indeed be a good approximation for parallel plates of finite area, between which both co-ions and counterions would be present, and the complete Poisson-Boltzmann equation would apply. In that case the regions between the plates and the outer electrolyte regions would have a common zero potential as reference. Also, in the completely discharged state, both co-ions and counterions would be uniformly distributed throughout the entire dispersion medium. This last condition is required if the chemical free energy is to be explicitly independent of plate separation. Consequences of the model of infinite parallel plates are a difference in potential in the discharged state in the region between the plates and in the outer regions and an explicit dependence on both2 and plate separation 21 of the chemical free energy FJ1,Z). In order to determine Fc(l,Z) we consider a simple model of discharged counterions and coions, which is equivalent to that used by Sogami, Shi-
Levine and Hall
1094 Langmuir, Vol. 8, No. 4, 1992 nohara, and Smalley. It is assumed that the discharged ions behave as a perfect gas. The contribution to the chemical free energy from the discharged counterions situated between unit area of two typical plates i d 9
Fci(l,Z)= -2kTZ( In
2rm+kT ((7) + In (%I) 312
e)
(40)
where m+ is the mass of the counterion (anion) and h is Planck's constant. The corresponding force (at constant 2) is (41)
which describes a repulsion between the plates and cancels exactly with one of the terms in eq 39. For two parallel plates in a large volume of electrolyte, we would normally choose ( N )= N(1,l) in eq 31. For our present system, this would alter eq 39 for the electrical force by an amount
for L >> l ( 2 u - 1). Thus, this choice of ( N ) is equivalent to assuming that the chemical free energy Fci(l,Z)in the discharged state is independent of plate separation. Finally we need to consider the contribution to the chemical free energy from the outer electrolyte regions, Fco(l,Z). Sogami et al. subtract the infinity coming from the uniform distribution of ions over infinite outer regions, but this is avoided here by choosing a finite volume for the outer regions. The equal total numbers of co-ions and counterions in the two outer electrolyte regions, each of which is denoted by N = 2[LN0- (2u - 1)Zl + 22,
(43) are independent of plate separation 21 for fixed 2. The volume occupied by either ion species per unit crosssectional area of the parallel plates is given by
v = 2 [ L - l ( 2 u - 1)l
(44)
The (perfect gas) chemical free energy of the discharged ions in the outer regions is divided by the number of pairs of plates 2u - 1 to yield the energy Fc"(l,Z)which corresponds to Fci(l,Z). We obtain (2u - l)F,O(l,Z)=
((
-2NkT( In h2 where m- is the mass of the co-ion and only Vis a function of 1 in eq 45. We find that 1aF,"(l,Z) __---
NkT = - BN(1,Z)kT (46) 2 ai L - l ( 2 u - 1) making use of eqs 4 and 43. This describes an attraction between the plates. Thus combining the three contributions from eqs 39,41, and 46 the force between unit area of two adjacent plates is
-1 a ( F e ( l , Z )+ Fci(l,Z)+ F,"(l,Z))= kT(n(1,O) 2 ai 2N(l,Z)) = kT N(l,Z){exp[-@(l,O)]- 21 (47) substituting eq 6 . Thus the force is attractive or repulsive asexp[-@(l,O)] islessorgreaterthan 2 , i.e. J@(l,O)l isgreater (19)McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976.
or less than 0.693. For this cross-over value of @(1,0)and ~ ( 1 , Z )=l 1, it follows from eq 14 that the potential at a plate is 1@(1,1)(= 1.242. These cross-over values of the potential between the plates are much too small to justify ignoring the concentration of co-ions in the range of I@(1,1)1where the net force between the plates is attractive. Indeed the simple force expression in eq 47 is consistent with the DLVO theory.
Discussion Our results differ from those of Sogami, Shinohara, and Smalley in a number of respects. We need only put u = 1 to describe the model of two parallel plates, as considered by Sogami et al. These authors distinguish the (charge) number density, labeled Zi, on the inner faces of their two plates, from the density 2, on the outer faces. By assuming that 2, is fixed, the potential on the outer faces is fixed, independent of plate separation. Their Zi is determined as a function of 1, by requiring that the potential on the inner face of a plate equals that on the outer face and therefore is constant, independent of 1. On comparing their Zi with our 2, our eqs 26 and 40 for Fe(l,Z)and F2(1,Z) respectively are identical with their corresponding equations for the free energy terms. However, in forming the derivative of the free energy with respect to 1to obtain the force, they assume that Zi depends on plate separation 21, in such a way that the inner face potential is fixed. Consequently they obtain their force by differentiating their free energy expressions with respect to plate separation at constant plate potential, whereas we assume that the plate charge is constant. The result is that they do not obtain our eq 41, which significantly cancels with one of the terms in eq 39 and, indeed, gives a force consistent with the DLVO theory. Their expression for Fco(l,Z) appears to correspond to ours. Their non-Coulombic repulsion at small plate separation is due to a kinetic term, which we identify with a chemical free energy in the discharged state. To summarize, basically the free energy expressions of Sogami et al. are the same as ours, but they obtain their force at constant plate potential whereas we assume constant plate charge. Our result is consistent with the DLVO theory, whereas theirs is not. Their treatment of the force is in apparent contradiction to our eqs 35 and 36 which are generally valid. The issue here is where we can apply eq 36 for the force between two plates, when no co-ions are present between the plates. Equation 36 requires the equilibrium condition, eq 35, which assumes that co-ions can transfer between the plate surfaces and the dispersion medium between the plates. In the complete absence of co-ions such equilibrium is not meaningful. However, it is most unlikely that co-ions will be totally absent. Furthermore, by allowing Zi (or 2) to vary with 1, co-ions must be transferred between the plate surfaces and the dispersion medium between the plates and this is in contradiction to the complete absence of co-ions between the plates. One can argue that a limiting situation is being considered here, where coions are present but their concentration is so small that the limit of zero concentration is a reasonable first approximation. The obvious way to resolve this uncertainty is to extend the theory to smaller Z values, and allow both co-ions and cations to be situated between the plates. Because the plates are infinite, FCo(l,Z)will still depend on I , but eqs 35 and 36 should apply. In such an extension the reason for the difference between our force formula and that of Sogami et al. must still be borne in mind. We would first differentiate the free energy with respect to 1 at constant
Langmuir, Vol. 8, No. 4, 1992 1095
Interactions of Colloidal Plates Zj and then assume that Zj is a function of 1. This dependence of Zi on l could be such that the plate potential is constant, independent of 1, for example. Sogami et al. assume that Zi is such a function of 1 when forming the derivative of their free energy with respect to 1 and this gives a different force formula. It is well-known that for ordinary colloidal particles of finite size the same double layer force can be obtained either by assuming constant surface charge or potential with respect to separation between the particles. However, the free energy expressions, which we denote by F, and AF respectively, from which these forces are derived, are different. We wish to illustrate this difference for two parallel negative finite plates, where edge effects are neglected. Per unit area of the plates the difference between the energies is
F, - AF = -2Ze +(l,l) (48) using the notation in this paper. On differentiating with respect to 1 a t fixed surface potential +(1,1), we obtain
noting that +(1,1)< 0 and > 0. The left-hand member is the difference between the force which would have been derived for finite plates by Sogami et al. and the correct force. The negative sign in the right-hand member implies that an additional, incorrect, attraction has been added. This casts doubt on the validity of the theory of Sogami et al. The model of infinite parallel plates, together with the approximation of zero concentration of co-ions between the plates, greatly simplifies the mathematical problem. However the ions are not uniformly distributed between the two phases in the completely discharged state, and indeed the region between the plates and the outer electrolyte region appear to be completely separated. The calculation of an electrostatic attraction between the plates in the absence of co-ionsis unexpected and the implications are worthy of further investigation. Our results indicate that this attraction is cancelled out by an osmotic repulsion due to the counterions in the discharged state, in disagreement with the conclusions of Sogami et al. In attempting to explain the experimental results in terms of a two-phase system, we had deliberately chosen a model of identical cells of parallel plates. However, our general conclusions also hold for only two parallel plates, as considered by Sogami et al. Although the experimental results on the polymer latex particles by Ise et al. and on the vermiculite clays seem qualitatively consistent with the calculations of Sogami et al., it seems that their explanation of the experiments is unsatisfactory. It is worth noting that the absence of co-ions around charged particles has been assumed in a classical treatment of the electric double layer free energy of rod-shaped particles arranged parallel to one another in a regular array.
The rods were assumed to occupy the whole volume available as a single phase.6a,20,21 In that case N(1,l) would be equated to (N)in eq 29, and the double layer interaction is found to be repulsive.
Acknowledgment. S.L. wishes to thank Dr. M. V. Smalley for having sent the manuscript of his paper before publication and for the extremely helpful correspondence. The present work would not have been undertaken otherwise. Appendix Consider the model of two infinite plates a t separation 21 in an infinite 1:l electrolyte, where both co-ions and counterions are present between the plates. In the completely discharged state the two ion types will be uniformly distributed inside and outside the region between the plates. Let n+ and n- be the number density of univalent counterions and co-ions, respectively,between the plates, N(1,Z) the common ion density outside the plates, and Z the number of univalent anions per unit area on the inner faces of the plates, all in the completely discharged state. If the dimensionless potential between the plates is @(O,O), relative to zero potential outside the plates n, =
exp[-@(O,O)I,
(A.1)
and therefore The condition of electrical neutrality is I t follows that
n, = ([Z2+ 4N(1,2)212]”2f 2)/(21)
(A.4)
and
Thus the model of infinite parallel plates results in a difference in potential between the regions inside and outside the plates in the completely discharged state. In the limit where Z
- -
-
n, Z / l , n- 0, @(O,O,) In [lN(l,Z)/Z] (A.6) which apply in the absence of co-ions between the plates. (20) Alfrey, T.; Berg, P. W.;Morawetz, H. J . Polym. Sei. 1951,7,543. (21) Fuoss, R. M.; Katchalsky, A.; Lifson, S. h o c . Natl. Acad. Sci. 1951, 37, 579.
