4308
Langmuir 1996, 12, 4308-4310
Comments Interaction of Highly Charged Plates in an Electrolyte. An Attempt To Clarify Issues and Correct Mistakes
Introduction According to Sogami and co-workers1 the electrochemical contribution to the interaction between two identical overlapping double layers which conform to the PoissonBoltzmann equation is under some circumstances attractive. This conclusion contradicts most previous work in this area and has been criticized.2-4 In reply Sogami and Smalley5,6 have attempted to refute these criticisms. In particular Smalley6 has argued that the work in ref 2 contains two mistakes. Firstly he claims that it is incorrect to take the interaction force as the distance derivative of the electrochemical contribution to the free energy under the constraint that the surface charge density is held constant. Secondly he claims that eq 39 of ref 2 does not follow from the arguments and results that precede it. It is shown below that these two claims are groundless. Also the opportunity is taken to comment on some issues concerned with calculations of interaction free energies based on charging processes. The comment is structured as follows. Firstly the fundamental thermodynamic arguments which underpin the calculation of electrochemical interactions are summarized. A comparison of two charging processes is then given. Finally the flaws in the arguments forwarded by Sogami and Smalley5,6 are identified and discussed. Summary of the Thermodynamic Background Consider a closed system consisting of two parallel flat plates immersed in a solution of a 1:1 electrolyte. Let the area of the plates be sufficiently large that edge effects are negligible, and let their thickness be such that any electric field in the plate interiors is quite negligible. Also, let there be one ionic species i which adsorbs specifically only to the opposing surfaces of the plates. Let the amount per unit area adsorbed be Γ, and let the amount of solution be such that any change in the bulk concentration due to adsorption is insignificant. Let f be the total Helmholtz free energy of the system divided by the plate area. At constant T and V, f may be regarded as a function of the amount per unit area specifically adsorbed and the plate separation x. It follows from eq 33 of ref 7 that
df ) (µ j σi - µ j bi ) dΓ + X dx
(1)
j σi and µj bi , respectively, are the electrochemical where µ potentials of adsorbed and bulk i and X is the mechanical force per unit area required to maintain the plates at separation x. The two terms on the RHS of eq 1 denote the chemical and mechanical work associated with infinitesimal changes in Γ and x. At chemical equilibrium
j bi µ j σi ) µ
Let feq denote the subset of f for which eq 2 applies. It follows from eq 1 that
X) X)
(∂x∂f )
Γ
( ) ∂feq ∂x
(3a,b)
(µ j σi ) µ j bi )
Equations 3a and 3b describe the only situations where the mechanical force is given by a derivative of f with respect to distance. It is wrong therefore to make such an identification under any other circumstances. Also eq 3a applies both at and away from chemical equilibrium. To proceed further, it is useful to suppose that the double layers concerned can be divided into inner and diffuse regions and that the ion distribution in the latter is governed by the Poisson-Boltzmann equation. It is also j σi - µ j bi ) into electrical and chemical parts useful to split (µ by writing
(µj σi - µj bi ) ) (µσi - µbi ) + νieφ
(4)
where φ is the electrical potential difference between the boundary separating inner from diffuse regions and the bulk solution remote from the plates and
µbi ) µθi (T,p) + kT ln nbi
(5)
where nbi is the concentration of i in the bulk solution remote from the plates. Equation 4 may be taken as defining (µσi - µbi ). Substituting the RHS of eq 4 into eq 1 gives
(
)
µσi - µbi dq + φ dq + X dx df ) νie
(6)
where q ) νieΓi. Let fel be defined by
fel )
∫0qφ dq
(7)
where the integration is performed at constant T, V, total composition, and x. Under the conditions of interest in a given bulk solution, fel may be regarded as a function of q and x so that
dfel ) φ dq +
( ) ∂fel ∂x
q
dx
(8)
Subtracting eq 8 from eq 6 gives
(2)
(1) Sogami, I. S.; Shinohara, T.; Smalley, M. V. Mol. Phys. 1991, 74, 599; 1992, 76, 1. (2) Levine, S.; Hall, D. G. Langmuir 1992, 8, 1090. (3) Overbeek, J. Th. G. Mol. Phys. 1993, 80, 685. (4) Ettelaie, R. Langmuir 1993, 9, 1888. (5) Smalley, M. V.; Sogami, I. S. Mol. Phys. 1995, 85, 869. (6) Smalley, M. V. Langmuir 1995, 11, 1813. (7) Hall, D. G. J. Chem. Soc., Faraday. Trans. 2, 1972, 68, 2169.
S0743-7463(96)00244-2 CCC: $12.00
d(f - fel) )
(
) [ ( )]
µσi - µbi ∂fel dq + X νie ∂x
dx
(9)
q
One of the main results proved in ref 8 is that when the approach developed therein applies and i is the only ionic (8) Hall, D. G. Adv. Colloid Interface Sci. 1991, 34, 89.
© 1996 American Chemical Society
Comments
Langmuir, Vol. 12, No. 17, 1996 4309
q ) (nj - ni)x νie
species common to the bulk solution and the inner regions
(
∂(µσi
∂x
)
µbi )
q
)0
(10)
Hence (X - (∂fel/∂x))q does not depend on q at constant x and is equal to X0, the interaction force when q ) 0. In other words
( ) ∂fel ∂x
q
) X - X0
(11)
which expresses electrical neutrality and
ninj ) n2
∂fel ∂x
)
q
ni -λνieφ ) exp n kT
dq ∫0q(∂φ ∂x )q
∂fel ∂x
eq
()
) (X - X0) + φ
∂q ∂x
eq
(13)
When the Poisson-Boltzmann equation applies φ(∂q/∂x)eq is positive. Consequently
( ) ∂fel ∂x
eq
nj -λνjeφ ) exp n kT
(12)
and |φ| can be expected to decrease as x is increased at constant q, it is apparent that (X - X0) is negative so that when X0 is zero X is repulsive at all separations. Consider now the dependence of fel on x at chemical equilibrium. According to eqs 8 and 11
( )
(16)
which expresses the equality of ion products for the solution between the plates and the bulk solution. The equality of the electrochemical potentials leads to the expressions
Since eq 7 gives
( )
(15)
g X - X0
Hence when νi + νj ) 0, it follows from eq 15 that
λνieφ q ) 2 sinh νienx kT
Comparison of Two Charging Processes The quantity fel in eq 7 refers to the process whereby fully charged ions are transferred from the bulk solution to the surface when the plate separation is x. According to this viewpoint the solution between the plates when q ) 0 is the same as that remote therefrom for all x. Consequently for this charging process X0 ) 0. An alternative process which is more amenable to calculation uses a coupling parameter λ, 0 e λ e 1, and may be described as follows. Take the initial state of the system as that in which all the ions are discharged and the plate separation is x. Adsorb ions onto the two surfaces in amounts corresponding to the final charge q. Then calculate the quantity f ′el given by eqs 1 and 2 of ref 2 whilst maintaining the condition that the charge on the plates is strictly balanced by that of the solution between them even when λ ) 0. It turns out that f e′ l * fel. To show this, consider the situation where λ , 1. Under these circumstances the concentration of ions between the plates is uniform, as is the electrical potential φ, which is also equal to the potential at the plate surfaces. Let ni and nj, respectively, denote the concentrations of coions and counterions between the plates and let n denote the concentrations of i and j in the bulk solution, which are, of course, equal. ninj and n are related by the following equations
(18)
Let
f ′′el ) lim λf0
∫0qφλ dq
(19)
where the integration is performed at constant λ. It follows from eq 18 that f ′′el is given by
(14)
and if it is wrongly assumed that (∂fel/∂x)eq ) (X - X0), then the force calculated on this basis is more attractive than the correct value given by eq 11. Equations 3a and 11 justify the viewpoint adopted in ref 2. The above arguments constitute a total refutation of Smalley’s first criticism. They are not confined to situations in which the plates interact at constant potential and are readily extended to cover the specific adsorption of several species.
(17a,b)
f ′′el ) 2kTnx[z sinh-1 z - x1 + z2 + 1]
(20)
where
z)
q 2νienx
(21)
Equation 21 shows that f ′′el f 0 as x f ∞ but is nonzero for finite x. It appears therefore that the electrical free energy of charging the plates does not vanish in the limit that λ f 0. Consequently fel * f ′el but is given by
fel ) f ′el + f ′′el
(22)
An alternative view of the equilibrium between the bulk solution and that between the plates is to regard it as a Donnan equilibrium. In this case, as λ f 0, the Donnan osmotic pressure π is given by
π ) ni + nj - 2n kT
(23)
When eqs 15 and 16 are used to eliminate ni and nj from this equation, the result is
π ) n[x1 + z2 - z + (x1 + z2 - z)-1 - 2] ) kT n[2x1 + z2 - 2] (24) and it is straightforward to demonstrate by differentiation of eq 20 that
( )
π)-
∂f ′′el ∂x
q,n
(25)
This shows that f ′′el is equivalent to the work done against osmotic pressure in the discharged state.
4310 Langmuir, Vol. 12, No. 17, 1996
Comments
An alternative expression for the electrochemical contribution to the force between identical charged plates is8-11
X)
∫0φF dφ
(26)
where the integration is performed at the midplane so that the field strength is zero. The LHS of eq 26 may be evaluated in the limit that λ f 0 and, as might be expected, is equal to -π as given by eq 23. An embarrassing feature of the above argument is that |φ| f ∞ as λ f 0. This is a consequence of imposing the constraint that the charge of the solution between the plates must balance the surface charge whilst at the same time insisting that this solution is in equilibrium with bulk. These conditions cannot be met by any model of a finite system because the Debye length becomes infinite and the nonbound ions distribute uniformly. In other words the distance over which edge effects matter becomes greater than the plate dimensions however big the plates are. Despite this difficulty the argument leads to conclusions which make good physical sense. The reason for this is that the equilibria are governed by λeφ/kT, which remains well behaved and finite as λ f 0. The charging process adopted by Verwey and Overbeek9 avoids the conceptual difficulties associated with an infinite value of |φ| but at the cost of allowing ions to adsorb and desorb according to rules which are somewhat artificial. Discussion The main conclusions to emerge from the two preceding sections are as follows. (1) The interaction force X is given by eq 3a both when the system is at chemical equilibrium and when it is not and is equal to the derivative with respect to distance of f if and only if either Γ is fixed or eq 2 applies. (2) When eq 10 applies, (X - X0) is given by eq 11 and always makes a repulsive contribution to the total interaction. To put X - X0 ) (∂fel/∂x)eq is clearly wrong. (3) fel is given by eq 7 and includes both the electrical term in ref 2 and the term designated previously therein as ‘chemical’. The criticism by Ettelaie4 that eq 35 of ref 2 is not met by the model therein arises from a confusing statement near the end of p 1093 of ref 2. Fc(l,Z) itself is not determined. What is determined is the dependence of Fc(l,Z) on l at constant Z. To write
Fc(l,Z) ) Fic(l,Z) + Foc (l,Z)
(27)
is incorrect because the RHS of this expression should also include a term involving adsorbed ions which leads to eq 35 of ref 2 being satisfied at equilibrium. However this issue has no effect whatever on the conclusions reached by Levine and Hall concerning the sign of X because, as eq 3a shows, eq 36 applies both at and away from equilibrium in the case under consideration. The arguments presented in refs 4-6 indicate that there is some misunderstanding of the difference between chemical and electrochemical potentials. The distinction j σi and µ j bi in eq 1 recognizes that species i may between µ be found in two different environments, an adsorbed state (9) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (10) Derjaguin, B. V. Russ. Chem. Rev. 1979, 48, 363. (11) Sanfeld, A. Thermodynamics of Charged and Polarised Layers; Monographs in Statistical Physics and Thermodynamics; North Holland: Amsterdam, 1962; Vol. 10. (12) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans Green Co. Ltd: London, 1966; p 40. (13) Guggenheim, E. A. J. Phys. Chem. 1929, 33, 842. Thermodynamics, 5th Revised Edition; North Holland: Amsterdam, 1967; p 300.
and a nonadsorbed state. Indeed it is hard to see how a surface charge can arise spontaneously without a difference of this kind. In some situations the adsorbed state may even be covalently bonded. The difference (µ j σi b µ j i ) is an adsorption affinity.12 It is as well defined as any other reaction affinity in a nonequilibrium system and is zero at equilibrium. The term (µσi - µbi ) is defined by eq 4 and is clearly nonzero at chemical equilibrium if φ is nonzero. The viewpoint expressed by Guggenheim13 and parroted by Sogami and Smalley that (µ j σi - µ j bi ) cannot be split in this way seems to deny the existence of φ and by so doing leads to a position that is hard to reconcile with using the Poisson-Boltzmann equation. Although Sogami and co-workers5,6 appear to deny the existence of an adsorbed state in their model, it is hard to envisage how such a view can be sustained, since some process involving adsorption or desorption must be deemed to occur if the surface charge is allowed to vary. A further point not considered by Sogami and coworkers1,5,6 is that expressions for X may be obtained by methods other than differentiation of the free energy. According to these expressions the interaction between identical plates is always repulsive. Also when the Poisson-Boltzmann equation applies, they are the same as those obtained via the free energy.9 The remaining issue to address is, “Where have Sogami and co-workers gone wrong?” This issue has been discussed at length by Overbeek3 and by Ettelaie4 and will be considered only briefly here. The first point to note is that the treatment of Sogami and co-workers is based on statistical mechanics whereas most previous work has a thermodynamic foundation. Their strategy is to construct a Hamiltonian which is the sum of a kinetic term and an electrical term and to obtain from this a Helmholtz free energy F. This free energy has the property that when it is evaluated at different values of x for systems whose charge on the inner surfaces is adjusted to maintain the electrical potential at the inner surfaces of the plates constant, it exhibits a minimum. As the above arguments show, the derivative of this free energy with respect to x gives the mechanical force if and only if either the surface charge is held constant or chemical equilibrium is maintained as x is varied. For this to be the case, it is sufficient and necessary that (∂F/∂q)x ) 0. However, as Ettelaie has shown,4 this condition is not met by the model of Sogami and co-workers. Finally it is appropriate to address Smalley’s second criticism of ref 2. This criticism is absurd. That eq 39 of ref 2 follows from eqs 37 and 38 is readily verified. One simply obtains (∂u/∂l)z from eq 38 and then substitutes back into eq 37 and cancels terms where appropriate. It is Smalley who makes a mathematical mistake. Consider eqs 6, 9, and 10 of his Note.6 It is apparent that, for all three equations to hold, either Zi or Z0 or both must be allowed to vary. Otherwise two forms of q(u) result which are incompatible. The partial derivatives in his eqs 1113 are taken under the condition that Z0 is held constant. Consequently his claim that eqs 16 and 17 are the same is invalid. When the correct derivative of u is taken (i.e. at constant Zi not constant Z0), it follows straightforwardly that the RHSs of his eqs 16 and 18 are equal so that eq 39 of ref 2 is valid. Denver G. Hall
NEWI Plas Coch, Mold Road, Wrexham Clwyd LL11 2AW, U.K. Received March 14, 1996 In Final Form: June 10, 1996 LA960244Q
