Article pubs.acs.org/JPCC
Effect of Nonelectrostatic Ion Interactions on Surface Forces Involving Ion Adsorption Equilibria Vivianne Deniz and Drew F. Parsons* Department of Applied Mathematics, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia ABSTRACT: The chemical, or chemisorption, part of colloidal interaction free energy is revisited. Consistent incorporation of nonelectrostatic interactions in the chemical potential for the constant potential and charge regulation boundary conditions is developed. This gives rise to shifted adsorption equilibria, and thereby a shift in the predicted surface electrostatic potential. It also results in an additional component previously unaccounted for in the total double layer interaction force. The altered force leads to the need of recalibrating electrostatic surface potentials and equilibrium constants when fitting to experimental force data. A numerical illustration is presented using ionic dispersion potentials for mica surfaces interacting across NaCl at various concentrations. The new force component due to ionic dispersion is typically repulsive and exceeds entropic repulsion in magnitude. These results suggest that the effect of ionic dispersion is more profound than previously believed, even at low electrolyte concentrations.
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INTRODUCTION Understanding colloidal stability is of vital importance to many applied industries such as chemical, pharmaceuticals, bio- and nanotechnological, and others.1 The net force between the surfaces of any two colloidal particles is derived from the interaction free energy; its description is therefore central to the issue. The DLVO-theory of Derjaguin and Landau2 and independently of Verwey and Overbeek3 brings this down to two separate quests: describing the interaction of overlapping diffuse double layers and the direct surface−surface van der Waals interaction. While this decoupling has proved to be invalid4 and the DLVO-theory is subject to a number of limitations, it has nevertheless provided a useful foundation on which the understanding of colloidal behavior has been built. The formation of diffuse double layers can be dealt with for instance within a Poisson−Boltzmann formalism, such as the Gouy−Chapman theory.5,6 The van der Waals interaction is described by the Hamaker energy7 (via the more refined Lifshitz theory8). Though it is outside the scope of this Paper, it is briefly mentioned for completeness in later sections. Here, we address the question of excluded nonelectrostatic interactions in the process of building double layers and their interactions. Much ground has already been covered on the topic of the role of nonelectrostatic interactions in the context of neglected ion-specific, or Hofmeister, effects. These effects can be captured with an extended DLVO-theory by the inclusion of, for instance, ion-surface dispersion interactions4 in the chemical potential of the Poisson−Boltzmann equation,1 which must be solved self-consistently given some imposed boundary conditions. Common conditions are constant surface charge density (CC), constant surface potential (CP) or regulated surface charge density (CR). Once the solute particle © 2013 American Chemical Society
distributions and potential profiles are found, the free energy change can be estimated and the interaction force deduced. The way in which dispersion potentials have been included in the past was developed only for surfaces held at constant charge, when no particle chemisorption occurs at the interface.9 For the cases of constant potential and charge regulation when chemisorption does occur, the energy of the chemisorption process must be accounted for, and will affect surface forces. The requirement that the chemisorption and physisorption chemical potentials are balanced at the interface has previously been accounted for in conjunction with electrostatic physisorption potentials.10−14 But the relationship between nonelectrostatic physisorption potentials and the chemisorption energy has not previously been considered. Addressing this question is the main aim of this Paper. Although we argue our case here by using ionic dispersion interactions, for which a numerical illustration is presented, the treatment is valid for any general nonelectrostatic interaction potential. Double Layer Free Energy. The change in free energy with the build-up of a double layer has previously3,10,11 been found by considering an imagined process of formation of surface charge; chemically binding or removing potential determining ions from the surface. In this Paper we primarily follow the outlines of Overbeek,11 whose analysis concerns surfaces held at constant charge or constant potential, and that of Chan and Mitchell,10 whose analysis concerns surfaces at constant potential or which are charge regulated. They have in common that they consider a purely electrostatic model, where Received: April 25, 2013 Revised: July 2, 2013 Published: July 10, 2013 16416
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neglecting ion correlations and other possible nonelectrostatic interactions. Chemical Potential and Total Free Energy. It is the difference in chemical potential of the potential determining ions i in the bulk reference state to that of their bound state in the surface that drives the process of the build-up of the surface charge.26 The overall change in the free energy however also involves a contribution due to the difference in chemical potential of the site itself with the ion bound to it, relative to its unoccupied state.27 The difference in chemical potential between surface and bulk reference states can thus be described as the sum of the difference of the chemical potential of the site and ion respectively. We split the process that the ion undergoes into two separate and subsequent steps; the ion being pulled out of bulk (B) up to a physisorbed (ps) state at the interface (I) and then bound into a chemisorbed (cs) state onto a surface site (S). We will here let S refer to a site rather than the surface to remind that the surface by virtue of its sites can be found in two different states; unpopulated S or q q populated SPi i, where Pi i is the ith type of potential determining ion (pdi), with valency charge qi. The total change in chemical potential of the entire system comprising of species i and the site it adsorbs to then follows as
ions are drawn out from bulk and bound on to the surface, corresponding to an association (basic) equilibrium. For consistency, therefore, we likewise take the association equilibrium as an example (the dissociation equilibrium of an acidic surface site is similar and traced out in parallel with the basic case in our analysis). Similar to Chan and Mitchell, our starting point is the description of the chemical potential. Where they separate the electrostatic from all nonelectrostatic components in the total chemical potential, we proceed slightly differently. It is constructive to think of the free energy change of formation of the ion distribution profiles, that is, the diffuse part of the double layer, as a physisorption energy, distinct from a chemisorption energy which describes the formation of the surface charge. We therefore decompose the total chemical potential into a physisorption and chemisorption part. This distinction between contributions to the total free energy has been considered previously only with regards to entropic and electrostatic interactions.12,13 But the question of how these chemisorption terms change when nonelectrostatic ion interactions are present has not been considered. The description of the chemisorption energy, specifically its relationship to the nonelectrostatic part of the physisorption energy of the potential determining ion, is the focus of this paper. We apply the chemisorption energy in the context of a mean field Poisson−Boltzmann description of ion distribution profiles, neglecting ion−ion correlations. But chemisorption is also relevant in computer simulation or at higher levels of theory which do include ion correlations. Monte Carlo simulations by Wu et al.,15 for instance, found a strong enhancement in physisorption of ions due to ion correlations, albeit for a constant surface charge with no chemisorption. This enhanced physisorption, combined with a chemisorption mechanism for the surface charge, would have an additional impact on surface forces via the chemisorption free energy. In principle, this effect can be described using the same formalism presented in this paper, with the effect of ion correlations on the chemical potential of ions treated as a nonelectrostatic contribution to the physisorption energy of the potential determining ion. Indeed, Monte Carlo simulations of Lund and co-workers14,16 found that charge regulation (using electrostatic ion interactions) does play an important role in protein− protein interactions. Nonelectrostatic ion interactions can be included alongside ion correlation effects. Monte Carlo simulations17,18 and hypernetted chain (HNC) calculations19 of colloidal interactions have been performed including ion dispersion interactions but with constant surface charge. One interesting phenomenon predicted both by ion correlation forces20−22 and by ion dispersion interactions23,24 is charge reversal (reversal of the surface potential). There is scope for further work to determine which of the two mechanisms dominates the charge reversal effect under which conditions. In any case, the best general model in the continuum solvent approximation will include all three effects, ionic dispersion, ion correlation and a charge regulated surface. We use ion dispersion interactions to illustrate the impact of nonelectrostatic physisorption energies. Others may be considered. Bratko et al.,25 for instance, found that image forces due to dielectric discontinuities drive a reduction in ion physisorption. For simplicity, in this paper we apply a Poisson− Boltzmann model with only ion dispersion energies, thereby
Δμtot, i = Δμpdi + Δμsite = (μPIqi − μPBqi ) + (μSPqi − (μS + μPIqi )) i
i
≡ Δμps, i + Δμcs, i
i
i
(1)
Chan and Mitchell refer instead to the chemical potential μ of the ion in the surface, taking Δμ = μS − μB. But the two perspectives are equivalent if we allow that their μS equals our qi μSP − μS, and also assume that Δμps,i only comprises i electrostatic elements. The latter point is the reason for our different terminology; with the distinction between physisorption and chemisorption parts of the chemical potential, it becomes apparent that the physisorption part can also comprise of nonelectrostatic elements, which must not necessarily be grouped with the “chemical” or chemisorption component. Assuming that no change in volume partitioning occurs during such a charging process, Helmholtz (F) and Gibbs free energies are identical. We therefore express the free energy change via the succinct formalism of Gibbs free energy of transfer, (adopting the notation of Chan and Mitchell10), summed over all potential determining ions i up to give the total free energy change per unit area as S
F = υ∑ i
∫0
Γ0, i
Δμĩ dΓi
(2)
where dΓi is the amount of ions transferred per unit area. We have here introduced v to cover both the association (basic) case (υ = +1) and the dissociation (acidic) case (υ = −1) in the same formalism. The tilde notation denotes nonequilibrium states. This is to emphasize that the equilibrium of relevance in this context is the one between the potential determining ions in bulk and in the surface. Although after each adsorbed infinitesimal amount of dΓi a “secondary rearrangement of the charges in the solution is allowed to occur, until, as far as the solution is concerned, ionic equilibrium is reestablished”,28 this subsequent process does not contribute to the free energy change of the system28 (but makes Gauss’ law and Poisson− Boltzmann equation valid at any stage). This follows from the 16417
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fact that Δμ̃ i in eq 2 represents the system once the solution rearrangement has already occurred (when the diffuse layer is in equilibrium with the current nonequilibrated surface charge), that is, it is an equation of state.10 Splitting the chemical potential into a physisorption and chemisorption component allows the same for the total free energy. F=
∑∫ i
0
Γ0, i
Δμps,̃ i dΓi +
∑∫ i
0
Γ0, i
Uel =
εε0 2
∫0
∞ ⎛ dψ ⎞ 2 ⎜
⎟
⎝ dx ⎠
dx ≡ Fel
(4)
where ϵ and ϵ0 are the dielectric constant of the solvent and the permittivity of vacuum, respectively. The change in mixing entropy11,12 is measured as the deviation of the ion distributions in their bulk state to that of their final equilibrium configuration, that is the configuration assumed once the surface charge density has reached it equilibrium magnitude,
Δμcs,̃ i dΓi ≡ Fps + Fcs (3)
−T ΔS = +kBT
We address these parts separately below, the former described briefly for completeness and followed by the latter in detailed analysis as the focus of this Paper. A widely used nomenclature for these two parts is inner and outer, or, compact and diffuse parts of the potential. Presumably, this distinction is inspired from the notion of the double layer as built up from immobile charges at the surface constituting the inner compact part, while the mobile solution charges constitute the outer diffuse part of the double layer. The different parts have been attributed various properties such as the free energies required to build them, capacitances, potentials etc.29−34 While the physical characteristics are retained irrespective of the labels used, we prefer physisorption and chemisorption respectively, at least when dealing with the chemical potential and the free energy. It removes any ambiguity of meaning. In particular it does so for the chemical potential, which for instance has been described as the “conjugated electrostatic potential [that] plays the role of an electrochemical potential... and should not be confused with the electrostatic surface potential...”34 With the notation of physisorption and chemisorption potential, the meaning and origin of the two parts become evident. So does the fact that in the general case, the physisorption chemical potential can be comprised of electrostatic as well as nonelectrostatic, or “chemical”, elements. The contributions to the chemisorption part are purely nonelectrostatic since an adsorbed species at the interface has the same electrostatic potential energy, qiψ0, in both the chemically bound and unbound states, such that its contribution is canceled in Δμcs. Physisorption Free Energy. In a purely electrostatic description, Fps is often referred to as the electrostatic work done in charging up the system or creating the double layer. The ions are modeled to interact with the surface only electrostatically, i.e. the physisorption process is driven only by the electrostatic surface potential; Δμ̃ps = qψ̃ 0, where q ≡ ze is the charge of the adsorbed species of valency z, assuming only one potential determining species is present. Since the surface charge density can be expressed as σ = qΓ, the electrostatic work assumes the familiar form Fps = ∫ σ00ψ̃ 0dσ,10,12 where σ is the magnitude at some intermediate state away from the equilibrium value σ0. Likewise, ψ represents the mean-field electrostatic potential profile at equilibrium, with the value ψ0 at the interface. Overbeek11 showed, by a direct transformation, that in the electrostatics-only model, this is equivalent to the difference in the change in free energy associated with the electrostatic field energy and the entropic change as the surface is charged up, Fps = Uel − TΔS, (Helmholtz free energy). The physisorption part of the internal energy is then approximated with the electrostatic field energy35 given from the electrostatic potential, ψ(x) and generated by surface charges and ionic charges in solution. In one dimension it is
⎡
⎛
cj(x)
⎢⎣
⎝
cj ,B
∫ ∑ ⎢cj(x)⎜⎜ln j
⎤ ⎞ − 1⎟⎟ + cj ,B⎥dx ⎥⎦ ⎠
≡ Fent
(5)
Here, kB and T are the Boltzmann constant and the temperature. The distributions cj(x) and cj,B represent the activities of ion j at position x and in bulk, respectively. For simplicity we neglect nonideal activities, taking ion activities equal to ion concentrations. Note that the index j here refers to all ions in solution, not only the potential determining for which we exclusively use the label i. The activity cj(x) and electrostatic potential ψ(x) are expressed in terms of distance x from the interface, and integrated from zero to infinity for free surfaces or to L for interacting surfaces separated by a distance L. In contrast to Overbeek, and Chan and Mitchell, we will here permit the ions to also interact with the surface nonelectrostatically, corresponding to a physisorption chemical potential as Δμ̃ ps = qψ̃ 0 + Δμ̃ NES ps , in which case an additional nonelectrostatic (NES) contribution to the physisorption energy appears UNES = −∑
∫ Δμps,NESj (x)cj(x)dx ≡ FNES (6)
j 9
as previously outlined by Edwards and Williams for the specific case of ion-surface dispersion potentials. Here, ΔμNES ps,j (x) is to be understood as the nonelectrostatic contribution to the chemical potential difference between position x and bulk reference at equilibrium. The nonelectrostatic chemical potential μNES ps,j (x) includes but is not limited to the ionic dispersion energy. It may also include a cavity energy, image force, steric hindrance, etc. In totality, the general description of the physisorption free energy change of a double layer is Fps = Fel + Fent + FNES.9 Chemisorption Free Energy. The chemisorption component of the total free energy, Fcs = ∫ Γ0 0Δμ̃ csdΓ, quantifies the change in free energy associated with reactions that bring adsorbed particles between the physisorbed and chemisorbed states. The free energy will be subject to the functional conditions imposed on the equilibrium surface charge density σ0 and that of Δμ̃cs. Typically Δμ̃cs is assigned to be either zero (CC), or independent (CP) or dependent (CR) on σ, the amount of charge already adsorbed to the interface during the charging process. In the case of a static constant surface charge (CC), there is by assumption no chemisorption and Fcs is zero. In order to evaluate Fcs for CP or CR, it is evident that the function Δμ̃cs, or rather, the precise details and mechanisms of the chemisorption, need to be known. These are in general unknown except at equilibrium when the total change in 16418
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Table 1. Generalized Langmuir Adsorption Isotherm and Corresponding Chemisorption Free Energy per Surface for Various Types of Ionizable Surface Sites site type acidic/basic zwitterionic amphoteric
chemisorption free energya
adsorption isotherm υ
σ0(Δμps) = υqN/1 + (Kυ/Hs) σ0(Δμps) = ∑υυqNυ/1 + (Kυ/Hs)υ σ0(Δμps) = qN(Hs/K+ − K−/Hs)/(Hs/K+ + K−/Hs + 1)
−ψ0σ0 − ΔμNES ps σ0/q + kBTN ln(1−σ0/σmax) υ −ψ0σ0 − ΔμNES ps σ0/q + kBT∑υNυ ln(1−σ0,υ/σmax,υ) + kBT∑υNυ ln (1−[D/Kυ] ) −ψ0σ0 − ΔμNES ps σ0/q + kBTN ln(1−σ0/σmax) − kBTN ln (2K−/Hs + 1) + kBTN ln
(1 + D/K+ + K−/D)
Short hand notations: 2D ≡ −K−( f−1)/f + ([K−( f−1)/f ]2 + 4K−K+/f)1/2, where f ≡ N+/N− with f = 1 in the amphoteric case, and D = 0 for monofunctional surfaces. a
Charge Density Dependent Chemisorption Free Energy: Charge Regulation. A charge regulation36 model more realistically accommodates for the physical restrictions that there are a limited number of adsorption sites on the surface to which the adsorbing ions compete, the density of which is also finite at the interface. With a different number of sites to the number of adsorbing ions, different configurations can be constructed with the same energy state resulting in an entropic degeneracy37 that alters the free energy. The chemical potential therefore becomes dependent on σ, its precise functional form given by the choice of adsorption isotherm. Separation dependence of the regulated surface charge enters the equation through the modulation of the interfacial density of the potential determining ions via the Boltzmann factor. We use here the isotherm due to Langmuir, corresponding to ideal adsorption, that is, where the adsorbed species, as well as the adsorption sites, are isolated subunits whose chemical potential is not altered by the state of occupancy of other sites. Alternatively, there are no additional interactions between the subunits.37 For a monofunctional surface the Langmuir isotherm gives the following relationship between the surface charge and the physisorption chemical potential
chemical potential must be zero, from which we obtain the equilibrium condition for the potential determining ions (see eq 1)
Δμcs, i = −Δμps, i
(7)
We derive below the chemisorption free energy for the charge density independent and dependent cases respectively. Charge Density Independent Chemisorption Free Energy: Constant Chemical Potential. When Verwey and Overbeek3,11 derived the chemisorption free energy change of an isolated surface at constant potential, they assumed that in each step of the charging process “a constant amount of free energy is gained”28 per adsorbed particle. To achieve this, Δμ̃ cs must be independent of Γ such that for each increment of adsorbed particles, the same amount of free energy is added to the total free energy until equilibrium is reached. It then follows trivially that Fcs = υ
∫0
Γ0
Δμcs̃ dΓ0 = υΔμcs Γ0 = −υΔμps Γ0
(8)
which with σ0 = υqΓ0 becomes Fcs = −ψ0σ0 − ΔμpsNES
σ0 q
(9)
1+
That is, in the general case there is a nonelectrostatic contribution to the chemical part of the free energy from the term −ΔμNES ps σ0/q, alongside the term −ψ0σ0, which is the wellknown result in the electrostatic picture.10 Further, in regards to two interacting surfaces, Verwey and Overbeek also “tacitly assumed...that the double layer potential is independent of the distance of the surfaces”.26 If we here take the double layer potential to mean the double layer chemical potential, it need not be assumed that it is independent of separation, but it must be independent of separation. The chemical potential difference is determined by surface and bulk chemical potentials (eq 1). The bulk chemical potential is constant in open systems with a reservoir providing particles at some chemical potential. The surface chemical potential is fixed by the magnitude of the binding energy, which remains constant given the assumption that Δμ̃cs is not dependent on Γ. Therefore, constant potential in fact refers to constant chemical potential, that is Δμps = qψ0 + ΔμpsNES = const
σmax
σ=
(
Kυ ̃ / k BT HBe−Δμps
υ
)
(11)
where σmax = υqN with N the total number of adsorption sites per unit area. υ = +1 for associative (basic) sites, υ = −1 for dissociative (acidic) sites. The fraction σ/σmax then represents the fraction of charged surface sites. We have here taken H+ as the sole potential determining ion with bulk concentration HB, and with acid equilibrium constant Kυ. Inverting the relationship between σ and Δμ̃cs results in qψ0̃ + Δμps̃NES = kBT ln
HB ⎛σ − σ ⎞ ⎟ + υkBT ln⎜ max ⎝ ⎠ σ Kυ
(12)
It is obvious that for a given Kυ, the resulting ψ0 is offset by the nonelectrostatic component when it is included. With this, the chemisorption free energy can be evaluated as Fcs =
1 q
σ0
Δμcs̃ (σ )dσ σ0
=−
1 q
=−
⎛ σ0 σ ⎞ Δμps + kBTN ln⎜1 − 0 ⎟ σmax ⎠ q ⎝
(10)
In the electrostatic model, it necessarily follows that ψ0 = const. But in a general model, if ΔμNES ps varies with separation, then so too must ψ0 vary with separation, in order to maintain a constant total chemical potential. Constant chemical potential therefore does not in general imply constant electrostatic surface potential.
∫0
∫0
Δμps̃ (σ )dσ
= −ψ0σ0 − ΔμpsNES 16419
⎛ σ0 σ ⎞ + kBTN ln⎜1 − 0 ⎟ σmax ⎠ q ⎝
(13)
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In the absence of nonelectrostatic physisorption, the electrostatic result of Chan and Mitchell10 is retrieved. Note that the first two terms −ψ0σ0 − ΔμNES ps σ0/q are precisely the same as the chemisorption free energy for the case of constant chemical potential, eq 9. The configurational degeneracy in the case of a surface charge dependent chemical potential manifests through the third term, ⎛ σ ⎞ FcsCR = FcsCP + kBTN ln⎜1 − 0 ⎟ σmax ⎠ ⎝
Pps = −
εε0 ⎛ dψ ⎞2 ⎜ ⎟ 2 ⎝ dx ⎠
j
⎡
−
∑ ⎢⎢∫ ⎣
j
+
(14)
∫x
(15)
(16)
Pcs =
where Ftot = Fps + Fcs + FHam is the total free energy of the two surfaces, made up by the double layer components Fps + Fcs, and the surface van der Waals energy FHam (Hamaker energy). A practice of convenience from a computational point of view to ensure numerical stability39 is to substitute the free energy as Ftot = −∫ Ptot dL as it follows form the thermodynamic definition of pressure, giving
∫L
PHam
0
L
cj(x)
cj(x)
NES d(Δμps,2 )
dL
NES d(Δμps,1 )
dL
dx
⎤ dx ⎥ ⎥⎦
(19)
Δμps, t dσ σ d(Δμps, t ) t + t q dL q dL
NES Δμps, dσt t dσt = ∑ ψ0, t + q dL dL t = 0, L
(20)
In the last step we have used the fact that the variation of a constant chemical potential is zero. Note how the electrostatic term cancels against the equivalent term in eq 19, but the nonelectrostatic term, (ΔμNES ps,t /q)dσt/dL, remains. We shall refer to it as PNES cs,t to distinguish it from its physisorption counterpart, PNES ps , in eq 19. Its corresponding force will be NES named - cs . Charge Regulation. The free energy change given by eq 14 for a charge regulated monofunctional surface gives rise to the following chemisorption pressure,
(17)
where the total pressure is Ptot = Pps + Pcs + PHam. We describe the physisorption and chemisorption parts of the double layer pressure separately in the sections following below. The nonretarded Hamaker pressure is40
−H = 6πL3
∑ t = 0, L
∞
P dL
xp
The first and second terms result from the variation of the electrostatic field energy, Fel. By definition, the second term vanishes for constant charge conditions. For constant potential and charge regulated conditions, as we shall see later, it cancels against an equivalent term arising from the chemisorption pressure Pcs (see next section). The first term as well as the third, which is due to the entropic component, are to be evaluated at any position xp between the surfaces, conveniently at the midplane xp = L/2 if the interacting surfaces are identical, in which case dψ/dx at x = L/2 vanishes. The last term is generated from the nonelectrostatic part of the chemical potential. Its form here follows from the assumption that the interaction of an ion with one surface can be separated from its interaction with the other surface. Chemisorption Pressure. Constant Chemical Potential. With the chemisorption free energy for a single surface given by eq 9, the pressure contribution from two interacting surfaces t becomes
where = is defined as the difference of the free energy change of the surfaces interacting a distance L apart to that of the two free noninteracting surfaces, infinitely separated. That is
- = 2π 9
dσt dL
+ kBT ∑ [cj(xp) − cj ,B]
A similar analysis can be made for an amphoteric site. Langmuir adsorption isotherms generalized to include nonelectrostatic interaction potentials for surfaces with various site types and the corresponding double layer free energies are listed in Table 1. Derivation of Interaction Force. The force between a sphere of radius R and a flat surface (or two cylindrical surfaces of radius R) can be related to the interaction free energy ν between two flat surfaces by Derjaguin’s approximation38
ν(L) = Ftot(L) − Ftot(∞)
t = 0, L
xp
p
- = 2πR =(L)
− ∑ ψt
Pcs =
∑ t = 0, L
(18)
Δμps, t dσ kBTNt dσt σ d(Δμps, t ) t + t + σmax , t − σt dL q dL q dL (21)
The variation of the chemical potential with respect to separation need not be zero here. However, a direct differentiation of σ (eq 11) leads to the relationship
where H is the Hamaker constant specific to the medium and geometry of the two surfaces. Physisorption Pressure. The contribution from the physisorption free energy, Fps, to the total pressure between two interacting surface t located at x = 0 and x = L respectively follows from the variation of the free energy with respect to separation
kBTNt dσt σ d(Δμps, t ) =− t q dL σmax , t − σt dL
(22)
by which we obtain the same formula for the pressure as in the constant chemical potential case, 16420
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∑ t = 0, L
ψt
NES Δμps, dσt t dσt + dL q dL
Article
contributes to the physisorption process. Dispersion is however switched off for H+ with respect to the adsorption equilibrium. It is thus not included in the adsorption isotherm, and consequently, the nonelectrostatic (dispersion) chemisorption NES force component, - cs , is not included in the total force. This model, labeled “OFF”, is only presented in the charge regulated case. In the full model, labeled “ON”, dispersion is switched on for H+, with respect to the chemisorption equilibrium. In this model, the dispersion potential therefore influences the actual regulation of the surface charge since it is included in the adsorption isotherms for both constant potential and charge regulated conditions. The total force also includes the additional chemisorption component. Ionic Dispersion. The chemical potential due to an ion’s dispersion interaction with a single surface is described by
(23)
Again, the electrostatic term in eq 23 cancels against the equivalent term from Pps in eq 19, but the nonelectrostatic term, PNES cs,t , remains. The result here has been derived for a monofunctional surface, but it is easily verified that eq 23 is also valid for amphoteric and zwitterionic surfaces. Numerical Illustration using Ionic Dispersion Potentials. Using ionic dispersion potentials for the nonelectrostatic part of Δμps, we present a numerical illustration of the effect of the new developments compared to an electrostatic model. The precise origin and detailed description of the dispersion potentials are accounted for elsewhere1,41−43 and only briefly outlined below. Since charge regulation is considered the more realistic of the common boundary conditions, we set it as the focal point of this comparative study. A model system is constructed using acidic mica surfaces immersed in an NaCl electrolyte at concentrations from 10−4−1.0 M. The acidic mica is chosen due to its prevalence in experimental set-ups. We compare the new theory against a reference curve, which we take to be the classic charge regulated force curve with only electrostatic interactions. For simplicity, the pKa of the electrostatic reference is kept constant over the concentration range, here chosen as 5.6 consistent with a pH of 6.0,44 which we hold fixed through out this study. The site density at the mica surface N is taken as 2.18 nm−2,44 the dielectric constant of water is ε = 78.36,45 and the mica-water-mica Hamaker constant is estimated to be 0.94 × 10−20 J, taking the dielectric function of water from ref 45. and of mica from ref 46. The temperature T is set to 298.15 K. To this charge regulated electrostatic reference, we compare other boundary conditions and models. We design two different models with dispersion included. The motivation for considering two different ways of including the dispersion potentials is to enable us to compare the new theory developed here with the way dispersion has been handled in the past. This was done either by only applying dispersion to the indifferent salt ions in solution, in which case ionic dispersion interactions only affected the physisorption free energy. That condition gives a consistent treatment for the case of constant charge,47−53 but not for systems modeled under constant potential54,55 or charge regulation.56−59 Even where dispersion was applied to the potential determining ion (H+), such that the adsorption isotherms were correctly described, the additional chemisorption force component was still neglected.60−63 It may be argued that whether or not the dispersion potential is switched on for the potential determining ion H+ does not greatly influence any results since the density of H+ is generally much smaller than the density if other salt ions. This holds true for the constant charge case. Including dispersion will however, as already pointed out, alter the equilibrium between the physisorbed and the chemisorbed states, as described by the adsorption isotherms, eq 10 for constant potential and eq 12 for charge regulation. Omitting the dispersion interaction for H+ in these equations therefore has the consequence that a different set of surface parameters are predicted even for just a single surface. To mimic previous practices, our first model that includes dispersion corresponds to when dispersion is applied to all ions, including the potential determining H+, such that dispersion
ujt (djt ) =
Bjt djt3
g (djt , rj) (24)
where djt is the distance of ion j to surface t, rj is the Gaussian ion radius,64 Bjt the dispersion interaction coefficient,62 and ⎞ −d 2 / r 2 ⎛ 2d ⎛ 2d 2 4d 4 ⎞ − − + 1 e 1 ⎟ ⎜ ⎜ ⎟ 2 ⎝ ⎠ r4 ⎠ π 1/2r ⎝ r ⎛d⎞ erfc⎜ ⎟ ⎝r⎠
g (d,r ) = 1 +
(25)
which in the limit of d→0 gives the finite value ujt = 16Bjt/ 3π1/2r3j for the dispersion potential. In the limit d→∞ it follows that g→1, thus for large values of d the dispersion potential takes the asymptotic form ujt = Bjt/d3jt. The total dispersion interaction of an ion in the gap between two surfaces can be approximated by a linear combination of the interactions with individual single surfaces Ujt(djt ) = ujt (djt ) + ujk(djk)
(26)
where k = t − (−1)t. An ion located at the left interface (t = 1) gives the following contribution Uj1(0) = uj1(0) + uj2(L) =
16Bj1 3π 1/2r j3
+
Bj2 L3
g (L , r j ) (27)
Thus Ujt is always a function of separation, even when the ion is located right at one of the interfaces. With this we can identify ΔμNES ps,it = Uit(0), if we let the dispersion potential, just like the electrostatic potential, drop to zero at the bulk reference state. Table 2 summarizes the ion radii64 and dispersion coefficients used in this study. The dispersion coefficients were estimated using dynamic polarizabilities of the ions53 and dielectric functions for water45 and mica46 from the literature. Analysis of Chemisorption Pressure Component. To better understand the influence of the additional chemisorption Table 2. Ion Gaussian Radii and Dispersion Interaction Coefficients species +
H3O OH− Na+ Cl− 16421
r (Å)
B (10−50 Jm3)
0.970 1.259 0.607 1.693
−0.385 −1.018 −0.040 −1.210
dx.doi.org/10.1021/jp404086u | J. Phys. Chem. C 2013, 117, 16416−16428
The Journal of Physical Chemistry C
Article
pressure component we briefly reflect upon its analytical behavior before presenting the main results. Whether it is attractive or repulsive is governed by the relationship of the signs of dσt/dL and B, but also of their relative magnitudes as NES Pcs, t =
Bk g (L) ⎞ dσt 1 ⎛ 16Bt ⎜ 1/2 3 + ⎟ q ⎝ 3π r L3 ⎠ dL
(28)
For symmetric systems, this term will be repulsive if sign(dσ0/ dL) = sign(B) and attractive otherwise. For asymmetric systems, the dispersion interaction of an ion with one surface may have a different polarity to its interaction with the other surface, that is, sin(Bt) ≠ sign(Bk), in which case their relative magnitudes will influence the nature of PNES cs,t . Anticipating the results of this study, we find that PNES cs,t is always repulsive here in the charge regulated (CR) case. The sign of dσ0/dL is not easily determined in a general model. In the CR case, it can be expressed in terms of the gradients of ψt and Ut according to eq 22. The latter of these is simply given by eq 27 dg k dUt = −Bk dL dL
Figure 1. Total charge regulated force for different models, comparing the electrostatic reference (ES) with two models including ionic dispersion interactions in physisorption, omitting (model OFF) and including (model ON) ionic dispersion in the chemisorption step. Salt concentrations are (a) 10−4 M and (b) 0.1 M NaCl. Even at low concentration (10−4 M), where no recalibration of the pKa is required, it is evident that models OFF and ON differ. While the inclusion of the dispersion potential hardly makes any difference under model OFF as compared to the ES reference, in model ON the effect is significant.
while omitting (model OFF) and including (model ON) ionic dispersion in the chemisorption step. Forces for the concentrations 10−4 and 0.1 M are shown. The difference between the electrostatic reference model and the model OFF is not large in this system, especially at low concentrations (
