Langmuir 1996, 12, 4305-4307
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Stability of an Assembly of Charged Plates Which Interact through Double Layer Forces Denver G. Hall NEWI Plas Coch, Mold Road, Wrexham, Clwyd LL11 2AW, United Kingdom Received June 26, 1995. In Final Form: March 7, 1996
Introduction The electrostatic interaction between charged particles is of fundamental importance in colloid science and continues to be debated.1-7 Recently8 a thermodynamic treatment of the diffuse double layer was forwarded which is more general than the classical approach based on the Poisson-Boltzmann equation and which includes it as a special case. A feature of this treatment is that the interaction force between identical flat plates in equilibrium with a bulk electrolyte solution is always repulsive. For a given solution the dependence of this force on distance is governed by the equilibria which occur at the plate surfaces and within the inner regions of the double layer. Many of the issues have been discussed previously in a series of papers.8-12 One of the main conclusions of this work is that both the free energy of interaction and the interaction force may be lower than those calculated using the constant potential boundary condition.11,12 The argument leading to this conclusion is based on considerations of the stability of an isolated surface with respect to spontaneous disproportion into two coexistent surface phases. In this paper similar arguments are used to discuss stability with respect to continuous changes in plate separation.
Figure 1. X vs x for a stack of parallel plates which undergo a phase transition at X*.
between X vs x and the p,V isotherm of a vapor. According to this figure, as the plate separation is decreased x should jump from x′ to x′′ when the repulsive force reaches X*. The objective of this paper is to establish under what circumstances, if any, the dependence of X on x is as shown in Figure 1 for identical surfaces which conform to the theory described in refs 8-12. Expressions for the Force In ref 8, two expressions for the force are given, namely
Statement of the Problem Consider two identical interacting charged plates in an electrolyte solution. Let x denote the distance between the outer Stern planes (OSPs), which separate inner and diffuse regions of the double layer, and let X denote the interaction force (X is negative for repulsion). Let φ′ and q′ respectively denote the potential at the OSP and the charge per unit area of the inner regions in the interacting system. Also let q denote the charge per unit area of the inner regions for an isolated surface when the potential at the OSP is φ′. For a stack of plates the condition for stability with respect to continuous changes in plate separation is that (∂X/∂x)T,bulkcomposition g 0. In other words if the average plate separation is x, the state in which all plate separations are x is stable if (∂X/∂x) g 0. In contrast if (∂X/∂x) < 0, the state in which all separations are x is unstable. In this case the stack reorganizes itself so that there are two different values of the separation x′ and x′′. At equilibrium these two values are given by the Maxwell construction depicted in Figure 1, which shows the analogy
X)-
2π(q(φm))2
(1)
and
X)
2π [(q′(φ′))2 - (q(φ′))2]
(2)
where φm is the potential at the midplane between the plates and q(φ′) is the charge per unit area at the OSP of an isolated plate when the OSP potential is φ′. Equation 1 shows unequivocally that X is negative for identical plates under all circumstances. Consequently in eq 2 |q′| < |q|. It follows immediately from eq 2 that
[
]
2 ∂(q(∂′))2 ∂X 2π ∂(q′(φ′)) ) ∂x ∂x ∂x
(3)
or alternatively that (1) Sogami, I. S.; Ise, N. J. Chem. Phys. 1984, 81, 6320. (2) Sogami, I. S.; Shinohara, T.; Smalley, M. V. Mol. Phys. 1991, 74, 599; 1992, 76, 1. (3) Smalley, M. V. Mol. Phys. 1990, 71, 1251; Langmuir 1995, 11, 1813. (4) Smalley, M. V.; Sogami, I. S. Mol. Phys. 1995, 85, 869. (5) Levine, S.; Hall, D. G. Langmuir 1992, 8, 1090. (6) Overbeek, J. Th. G. Mol. Phys. 1993, 80, 685. (7) Ettelaie, R. Langmuir 1993, 9, 1888. (8) Hall, D. G. Adv. Colloid Interface Sci. 1991, 34, 89. (9) Hall, D. G. J. Chem. Soc., Faraday Trans. 2 1977, 73, 101; 1978, 74, 1757. (10) Hall, D. G.; Sculley, M. J. J. Chem. Soc., Faraday Trans. 2 1977, 73, 869. (11) Hall, D. G. J. Colloid Interface Sci. 1985, 108, 411. (12) Hall, D. G. J. Chem. Soc., Faraday Trans. 1 1988, 84, 2215.
S0743-7463(95)00515-4 CCC: $12.00
[
]
2 ∂(q(φ′))2 ∂φ′ ∂X 2π ∂(q′(φ′)) ) ∂x ∂φ′ ∂φ′ ∂x
(4)
Equations 3 and 4 enable the behavior of ∂X/∂x under various boundary conditions to be unraveled. Consider first the constant charge condition. In this case q′ is constant and q2 decreases with increasing x because |φ′| decreases. Hence ∂X/∂x is positive. For the constant potential boundary condition ∂[q′(φ′)]2/∂x is positive and ∂[q(φ′)]2/∂x ) 0, because φ′ is constant. Hence again ∂X/∂x is positive. For other situations we make use © 1996 American Chemical Society
4306 Langmuir, Vol. 12, No. 17, 1996
Notes
The line AC denotes the changes in q′ and φ′ with decreasing x when the interactions lie between constant charge and constant potential. The line AE denotes the changes in q′ and φ′ with decreasing x when the interaction is less than constant potential. To satisfy eq 6, it is necessary that the slope of AE is greater than that of AO for all values of q2. If however for a given value of φ the slope of AE is lower than that of AO, then ∂X/∂x is negative. Moreover this state of affairs need not contradict eq 6. When the Poisson-Boltzmann applies to the diffuse regions of the double layer, the lines AC and AE in Figure 2 may be obtainable from electrokinetic and/or potentiometric titration studies of the isolated surfaces concerned. The procedures for doing this and the conditions that must be met are outlined in refs 8-10 and 12. The Sign of (DO/Dq) Figure 2. q2 vs φ for an isolated plate (OA) and for interacting plates under various boundary conditions: AB (constant charge), AC (charge regulation), AD (constant potential), AE (subconstant potential).
of the following inequality12
∂φ e (∂φ′ ∂q′) (∂q′)
(5)
where φ′ and q′ refer to values that occur at the OSP during an interaction and where φ is the potential at the OSP for an isolated surface with charge per unit area q′. For interactions which lie between the constant charge and constant potential boundary conditions, ∂[q′(φ′)]2/∂x is positive because (q′)2 decreases with decreasing x and ∂[q(φ′)]2/∂x is negative because q2 increases with increasing |φ′| and |φ′| decreases with increasing x. Hence ∂X/∂x is unequivocally positive. The remaining case to consider is that where both |φ′| and |q′| decrease as x is reduced. In this case ∂(q′)2/∂x and ∂q2/∂x are both positive. Also if φ′ is positive, then so is ∂φ′/∂x. Consequently the sign of ∂X/∂x is the same as that of the term in brackets in eq 4. Consider now eq 5. For the present case where both terms are positive this equation may be written as
∂q′ ∂q′ g ∂φ′ ∂φ
(6)
At infinite separation when q′ ) q, it follows from eq 6 that ∂X/∂x is positive. This result is consistent with eq 1 as it should be. However, whereas eq 6 compares situations where q′ ) q, eqs 3 and 4 compare situations where q′ * q but where φ′ is the same. Now as the plate separation is decreased, [q′(φ′)]2 and [q(φ′)]2 diverge initially but there is no guarantee that they continue to do so. In other words the possibility that ∂[q′(φ′)]2/∂φ′ < ∂[q(φ′)]2/∂φ′ where |q′| and |φ′| both increase with increasing x cannot be ruled out. The way in which this situation can arise is depicted in Figure 2. The line OA represents the dependence of q2 on φ for an isolated surface and is given by
q2 )
kT 2π
∑i
( ( ) )
ni exp -
νieφ kT
-1
(7)
when the Poisson-Boltzmann equation holds. The line AB denotes the change in φ′ with decreasing x under the constant charge boundary condition. The line AD shows the change in q′ with decreasing x under the constant potential boundary condition.
It has been assumed above that for an isolated surface (∂φ/∂q) g 0. This is certainly true when the ion distribution is governed by the Poisson-Boltzmann equation, but according to Attard and co-workers13 need not always be so. It is appropriate therefore to consider this issue when the diffuse double layer conforms to the theory developed in ref 8. According to this approach the dependence of φ on q is a property of the bulk electrolyte solution insofar as it is governed by the response of such a solution to an applied field. Now it is evident from eq 116 of ref 8 that ∫0φF dφ is necessarily negative and that
(∂φ∂q) ) - 2πq F
(8)
where F denotes charge density. Hence (∂φ/∂q) can only be negative when q and F have the same sign. However, for a diffuse double layer adjacent to a charged surface, overall electrical neutrality ensures that
q(x) +
∫x∞F dx ) 0
(9)
where x denotes distance from the charged surface and q(x) is the total charge per unit area on the surface side of x. It follows that for q(x)/F(x) to be positive F must change sign at some position x+, where x+ g x. Obviously at this position |q(x)| exhibits a maximum. It also follows from electrostatics that
∂φ(x)) 4πq(x) )∂x
(10)
This in turn implies that when |q(x)| exhibits a maximum, there can be two values of φ for the same value of q. Suppose now that q is the charge of the inner regions of a double layer and can be attributed entirely to the adsorption of a single potential determining species. According to refs 8-12 the electrochemical potential of this species is given by
µˇ ) µ(q) + νeφ
(11)
where µ(q) is independent of the nature of the bulk solution. Evidently if q is positive, then so also are ν and φ. If for a given value of q there are two possible values of φ, then that which gives the lowest value of µˇ will prevail because this leads to a lower Gibbs free energy of charging the surface when potential determining ions are transferred thereto from the bulk solution. Consequently if there are (13) Attard, P.; Wei, D.; Patey, G. N. J. Chem. Phys. 1992, 96, 3767.
Notes
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two possible values of φ for a given q, it is the lower value of |φ| where (∂φ/∂q) is positive that is realized. Similar arguments also apply when the inner region charge is attributable to more than one species. Hence it may be concluded that at the boundary between the inner and diffuse regions of an isolated surface (∂φ/∂q) g 0. Conclusions It has been shown above that when the approach developed in ref 8 applies, (∂φ/∂q) g 0 for an isolated surface. Since the force itself is always repulsive, it appears that the situation described above is the only way in which the theory described in refs 8-12 can be consistent with a negative vale of (∂X/∂x). Although not impossible, the conditions under which this state of affairs (14) Gulbrand, I.; Jonsson, B.; Wennerstrom, H.; Linse, P. J. Chem. Phys. 1984, 80, 2221. (15) Kjellander, R.; Marcelja, S. Chem. Phys. Lett. 1986, 127, 402.
can arise seem unlikely. Arguments for attractive electrostatic interactions between charged plates have been forwarded. According to the work described in refs 14 and 15, such interactions arise from fluctuation effects which are not allowed for in theories based on local thermodynamic arguments. Clearly the origin of these attractive interactions is quite different from the effect described above. In contrast the work described and defended in refs 1-4 claims that there is a Coulombic attractive force at large plate separations for systems which conform to the Poisson-Boltzmann equation. This claim, which contradicts most previous work in the area, including ref 8, has been criticized.5-7 The attempts made so far to refute these criticisms are unconvincing. In particular, as will be shown elsewhere, Smalley’s response to ref 5 is groundless. LA950515H
