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Exact Description of Aqueous Electrical Double Layers S. Marcˇelja* Department of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia Received February 24, 2000 We describe the exact steps needed to reduce the problem of planar electrical double layers in aqueous solvent to two simpler problems: (i) a reference system consisting of solvent between the surfaces and (ii) ions interacting with other ions and with surfaces via the potentials of mean force evaluated in the reference system. The mapping is achieved by extending the McMillan-Mayer transformation to the nonuniform fluid between the surfaces. The results indicate approximate additivity of short- and long-range effects in ion density and pressure, as observed earlier in numerical work. After two approximation steps, we arrive at a practical and accurate method for calculating properties of planar aqueous double layers.
Introduction While the majority of practical calculations relating to electrical double layers are still performed with the Poisson-Boltzmann equation, theoretical developments1 are usually based on the primitive model of aqueous electrolytes: hard core ions in a dielectric continuum. However, the primitive model is only a convenient approximation without a rigorous basis. Because of the molecular nature of the solvent, it is obviously inaccurate for small ion-ion or ion-surface separations. This causes serious discrepancies between theory and experiment, particularly in cases where surface charge is relatively high and the counterion is monovalent.2,3 As we have shown earlier,4,5 problems relating to the so-called secondary hydration force result from the reliance on the primitive model. The macroscopic Coulomb formula for ion-ion interaction in a dielectric continuum, q1q2/R, is an exact asymptotic law, valid at large ion-ion separations. The corrections due to the explicit structure of the aqueous solvent affect only the short-range part of the ion-ion potential. We could then expect that calculations involving electrical double layers can be performed by considering ionic fluid interacting via suitably renormalized ion-ion potentials and a reference fluid that will describe geometrical packing effects. This would be similar in spirit to recent works on simple and aqueous fluids6,7 where short-range forces are described via a suitable reference fluid and longer-range attractive interaction is treated within a mean-field theory. Remarkably, an exact description of planar aqueous electrical double layers is possible with such a two-step theory where a short-range, neutral aqueous reference system is considered in addition to the ionic fluid in a continuum dielectric interacting with renormalized potentials. The renormalized potentials turn out to be manyparticle potentials of mean force between the ions evaluated in aqueous solution in the presence of surfaces. * E-mail:
[email protected]. (1) For a review, see, for example: Kjellander, R. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 894. (2) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531. (3) Israelachvili, J. N.; Pashley, R. M. Nature 1983, 306, 341. (4) Marcˇelja, S. Nature (London) 1997, 385, 689. (5) Marcˇelja, S. Periodicum Biologorum 1998, 100 (Suppl. 2), 7. (6) Weeks, J. D.; Selinger, R. L. B.; Broughton, J. Q. Phys. Rev. Lett. 1995, 75, 2694. (7) Lum, K.; Chandler, D.; Weeks, J. D. J. Phys. Chem. B 1999, 103, 4570.
In this Letter we consider all steps required in the calculation of the double-layer interaction in aqueous solvents, starting with the exact formulation based on the McMillan-Mayer theory8,9 and fully defining the approximations. Formation of the Double Layer Electrical double layers must be described as a nonuniform fluid, with at least two componentssthe counterions and the solvent. While we present a detailed discussion only for the case of planar surfaces, a similar description can be constructed for other geometries. However, a nonplanar geometry requires more approximations. Establishment of double layers near surfaces or between interacting surfaces will be analyzed as two separate steps. In the first step we consider neutral surfaces in a reference fluid: For the reference fluid we can choose just the solvent (normally water) or the background electrolyte. Molecular packing in the reference fluid rearranges in response to the presence of surfaces and associated short-ranged potentials. In the second step the surfaces are charged and the double layers form. Introduction of surfaces changes the water (or electrolyte) in the restricted volume between the plates into a nonuniform fluid and creates perpendicular pressure acting on the surfaces. The theoretical problem of solving the structure of the nonuniform water system between the plates is difficult and therefore most often relegated to simulations. In experiments, oscillatory pressure profiles at small surface separations have been measured in the surface force apparatus.10 In the second step the surfaces are charged, resulting in a situation where a relatively low concentration of ions is maintained between the plates due to the requirement of electroneutrality. This situation is a nonuniform counterpart to the calculation of osmotic pressure, and we propose to use the McMillan-Mayer theory adapted to the nonuniform fluid between the surfaces. In a formal description of fluid between planar surfaces, it is practical to imagine the space divided into many layers parallel to the surfaces.11 The fluid in each of the layers (8) Mayer, J. E. Equilibrium Statistical Mechanics; Pergamon: Oxford, 1968. (9) Hill, T. L. Statistical Mechanics; McGraw-Hill: New York, 1956. (10) Pashley, R. M.; Israelachvili, J. N. J. Colloid Interface Sci. 1984, 101, 511. (11) Kjellander, R.; Marcˇelja, S. J. Chem. Phys. 1985, 82, 2122.
10.1021/la000266j CCC: $19.00 © 2000 American Chemical Society Published on Web 06/22/2000
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Letters
has a uniform density. Molecules within a layer may be understood as being different “species”, and the system of many uniformly dense layers is isomorphic to a manycomponent two-dimensional fluid. This formal device makes it possible to carry all of the uniform fluid results over to the nonuniform case with minimal modifications. In the limit of very fine division into layers, it is an exact description. McMillan-Mayer Step We select the state of the water or the background electrolyte between uncharged surfaces as the reference state for the application of the McMillan-Mayer exact expression. To avoid confusion between the coordinate z and the activity, the direction perpendicular to the surfaces is denoted x and the notation R ≡ {x, r} is used. The layer widths, which do not all have to be the same, are denoted as ∆xj. The grand partition function of the nonuniform electrolyte between the surfaces is then11
Ξ(z) )
∑∏ mg0 s,j
[
]
(zs∆xj)ms,j ms,j!
∫e-βU
m({m})
d{m}
(1)
z ≡ (zi1, zi2, ..., zw) is the set of activities for the ion species and for water and is assumed to include the effect of the electrostatic potential difference between the bulk solution and the space between interacting surfaces. The set {m} consisting of the numbers ms,j specifies the number of particles of species s in the layer j. s can be water or one of the ion species, and the layer index j ) 1, 2, ..., N. z is the same in all layers, and the layer widths ∆xj do not depend on species. Um({m}) is the total potential energy in the configuration with m particles, β ) 1/kBT, and the integration over all coordinates for the state m is performed over two dimensions parallel to the surfaces and, in case of water molecules, the orientational degrees of freedom. This expression for Ξ is formally the same as that for a two-dimensional fluid, except that the activities in the layer j are given as zj ) z∆xj, reflecting the fact that integration in the x direction (normal to the surfaces) is replaced by a discrete sum. When there are no ions, the expression for Ξ(z) contains only a sum and product over water “species” in different layers. Proceeding to the second step, surfaces are charged and the McMillan-Mayer formula is applied to the now formally uniform fluid system. The grand partition function of the charged state Ξ(z) is expressed in terms of the potentials of mean force and the grand partition function Ξ(z*) of the reference state
Ξ(z)
)
Ξ(z*)
[ ( 1
∑∏m
mg0 σ,j
σ,j!
) ]∫
(zσ - z/ )∆xj σ γ/ σ,j
(1)/
-βWσ,j F/ ) F/ σ,j σ,0 e
(3)
is the density of ion species σ in the reservoir, where F/ σ,0 then
) γ/ σ,j
z/ σ F/ σ,j
)
z/ σ F/ σ,0
(1)/
(1)/
eβWσ,j ) γ/ eβWσ,j σ,0
(4)
γ/ is the activity coefficient for the ion σ in the reservoir. σ,0 (1) becomes In the continuum limit the potential Wσ,j (1) Wσ (x) describing the affinity of ions in the reference system for approach to a given position x between the surfaces. In solving the structure of the fully charged double layer, the major influence of the walls on the ions is described through the potential of mean force evaluated for the reference system with uncharged walls. This potential, which indirectly includes solvent packing near a surface, causes some short-range variation in ion density which is only weakly linked to the behavior caused by interaction between the ions. If the only ion species are counterions with the activity zc, the grand partition function reduces to
Ξ(z) Ξ(z*)
)
[ ( ) ]∫ 1 zc∆xj
∑∏ mg0 j m ! j
mj
γ/
/
e-βw(m)({m}) d{m} (5)
0,j
z/ c ) 0, since there are no counterions in the reference state, mj is the number of counterions in the layer j, is the activity coefficient of the counterion in and γ/ 0,j the layer j determined in the reference state at zero concentration. Pressure
mσ,j
e
the activity coefficients γ/ σ,j, and the factors ∆xj (which are absent in the uniform fluid formula) all depend on the position in the space between two surfaces. For water, zw ) z/ w, since it can freely exchange with the reservoir. All terms in the sum which contain water in one or more of the layers are equal to zero. Ions, on the other hand, are held in the double layer by the electrostatic potential difference required to maintain electroneutrality, and the activities are changed from their reservoir values. The surviving terms describe only ions, and the numbers in the set mσ,j give the number of ions of species σ in the layer j. The activity coefficients in the reference system γ/ σ,j are different from layer to layer due to the influence of surfaces. This effect is only significant in layers near the surfaces. If we introduce the potential of mean force for the interaction of an ion with the surfaces W(1) as
/ ({m}) -βw(m)
d{m} (2)
γ/ ) z/ /F/ are the activity coefficients for species σ in the σ,j σ σ,j ({m}) are the many-particle potentials of layer j, and w/ (m) mean force, both determined in the reference state. If the charge on the surfaces is uniform, it does not create local fields and the activities zσ remain independent of the position. While the nonuniform fluid expression above is formally very similar to the standard McMillan({m}), Mayer formula, the potentials of mean force w/ (m)
The double-layer system has fixed area and is free to change surface separation d. The pressure between the surfaces in the double layer (the normal component of the pressure tensor) is given as p ) -kBT ∂ log Ξ(z)/∂d. This is now explicitly seen to consist of two separate contributions, arising from the derivative of the partition function for the reference system Ξ(z*) and from the sum over ions in the double layer. In other words, the pressure difference between the reference state and the background solution needs to be added to any subsequently calculated pressure due to excess ions, and we have
Pd.layer ) Pref + Pions - Pbackground
(6)
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If the background electrolyte is chosen as the reference system, the potentials of mean force between the two ions appearing in eqs 2 or 5 are screened and decay as exp(-κr)/r, where κ is the inverse of the renormalized decay length. This is a consequence of the finite electrolyte concentration acting to screen the many-body potentials of mean force between excess ions. To avoid working with screened potentials, we can choose water between surfaces as the reference fluid. A change in pressure between the background electrolyte and the reference fluid is then split into two steps, and instead of the simple path background electrolyte f electrolyte between surfaces f final double layer, we have background electrolyte f bulk water f water between surfaces f final double layer. The net pressure in the double-layer system is then
Pd.layer ) (Pref water - Pbulk water) + (Pions - Pbackground ions) (7) In this form, the change of pressure is split into separate contributions from water and from ions. Approximations Superposition Approximation. To proceed further, we need to approximate the potentials of mean force (or the correlation functions) involving many ions between the surfaces. The normal procedure is to use first the Kirkwood superposition approximation
gR,β,γ,...,ω(rR,rβ,rγ,...,rω) ≈ gR,β(rR,rβ)gR,γ(rR,rγ)...gψ,ω(rψ,rω) (8) R, β, ... denote both species and layers for ions. The approximation is of course more accurate at lower densities. After this step, the pressure between the surfaces is given by the previously described formalism,11 where pair potentials and ion-surface potentials are replaced with the corresponding potentials of mean force. The second step normally replaces correlation functions for the nonuniform fluid with their uniform counterparts bulk (|Rσ - Rτ|) g(σ,j),(τ,k)(rσ,rτ) ≈ gσ,τ
(9)
Here the species and layer indices σ and j, respectively (or τ and k), are indicated explicitly. The correlation bulk or the equivalent potentials of mean force functions gσ,τ for different ion species in bulk are available from simulations (e.g. ref 13). The problem is now reduced to the form where accurate solutions are available using the hypernetted chain (HNC),11 the reference HNC,12 or the simulation with only ions in the system. The scope for improvement in the theory rests with two approximation steps (steps 8 and 9). Effective Potentials. A recent method based on effective ion-ion potentials in aqueous electrolyte solutions introduced by Lyubartsev and Laaksonen13 presents an alternative, which may be an improvement over the standard Kirkwood superposition. Effective potentials are obtained in a reverse Monte Carlo procedure involving only ions and designed to reproduce ion-ion pair correlation functions obtained in molecular simulations. (12) Greber, H.; Kjellander, R.; Åkesson, T. Mol. Phys. 1997, 92, 35. (13) Lyubartsev, A. P.; Laaksonen, A. Phys. Rev. E 1997, 55, 5689.
The Coulomb potential between two ions in a solution is reduced by the solvent dielectric response, which depends on the overall ion concentration. The asymptotic behavior of the Coulomb interaction in aqueous solutions q1q2/R is a known exact result. The effective potential procedure retains this exact asymptotic form of the ionion potential and evaluates the short-range part by matching known pair correlation functions. To the extent that interaction in electrolytes can be split into pairwiseadditive terms, this appears to be the best procedure. Effective potentials decay with pure Coulomb law and do not show a strong concentration dependence. When the electrolyte concentration goes to zero, the effective potentials become identical to the potentials of mean force. Application to Arbitrary Geometries. It is easy to imagine the process described in previous sections carried out for an arbitrary geometry of the surface. However, terms involving water in the McMillan-Mayer formula are no longer exactly zero because charging of surfaces creates electrical fields which influence the local activity of the solvent. The solution step, which for the planar geometry was feasible within the anisotropic HNC approximation,11 also becomes more difficult and may have to be done through simulation. Nevertheless, considerations regarding the additivity of short- and long-range effects and the interaction of ions with the surface via the potential of mean force remain approximately valid. In any geometry, the uncharged surface of the same shape represents the natural reference system, just as it does for planar surfaces. Conclusion We presented here an exact formulation that simplifies the problem of electrical double layers by considering separately the solvent reference system and the ionic fluid in a continuum dielectric. With the use of the superposition approximation, practical calculations with aqueous solvent become very similar to those in the primitive model. Hard core and ion-surface potentials in the Hamiltonian are replaced by the respective potentials of mean force calculated in the uncharged reference system, and the oscillatory water or electrolyte pressure from the reference system is added to the result for pressure. The near additivity of short- and long-range electrostatic effects in the structure of the ionic density near charged surfaces in a model aqueous solution was noted earlier by Patey, Torrie, and co-workers on the basis of numerical work.14,15 We have shown here the theoretical basis for this additivity in ion density. Moreover, we found that the contribution to the double-layer pressure arising from the aqueous solvent is also approximately additive to the ionic contribution. The short-range interaction of ions with surfaces was shown to be predominantly described by the potential of mean force for the interaction between the ion and uncharged surfaces. Acknowledgment. I am indebted to David Andelman, Yoram Burak, Siewert-Jan Marrink, Johan Ulander, and most of all to Roland Kjellander and Alexander Lyubartsev for many valuable insights into this problem. The support of the Mortimer and Raymond Sackler Institute of Advanced Studies of the Tel Aviv University is gratefully acknowledged. LA000266J (14) Torrie, G. M.; Kusalik, P. G.; Patey, G. N. J. Chem. Phys. 1989, 91, 6367. (15) Torrie, G. M.; Patey, G. N. Electrochim. Acta 1991, 36, 1677.
