Ionic adsorption from a primitive model electrolyte—nonlinear treatmentSteven L. Carnie and Derek Y. C. Chan Citation: The Journal of Chemical Physics 75, 3485 (1981); doi: 10.1063/1.442458 View online: http://dx.doi.org/10.1063/1.442458 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/75/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Restricted primitive model for electrolyte solutions in slit-like pores with grafted chains: Microscopic structure,thermodynamics of adsorption, and electric properties from a density functional approach J. Chem. Phys. 138, 204715 (2013); 10.1063/1.4807777 New integral equation theory for primitive model ionic liquids: From electrolytes to molten salts J. Chem. Phys. 100, 9147 (1994); 10.1063/1.466669 Primitive model electrolytes. II. The symmetrical electrolyte J. Chem. Phys. 72, 5942 (1980); 10.1063/1.439093 Debye Model and the Primitive Model for Electrolyte Solutions J. Chem. Phys. 56, 3382 (1972); 10.1063/1.1677708 Computations for Higher Valence Electrolytes in the Restricted Primitive Model J. Chem. Phys. 56, 3071 (1972); 10.1063/1.1677643

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Ionic adsorption from a primitive model electrolyte-nonlinear treatment

Steven L. Carnie and Derek V.C. Chan Department of Applied Mathematics, Research School of Phycical Sciences, Institute of Advanced Studies. Australian National University, Canberra, A.C. T. 26()(), Australia (Received 12 September 1980; accepted 29 October 1980)

In order to investigate the effects of both fmite ion size and specific adsorption on the electrical double layer, we consider a restricted primitive model elecrolyte against a uniformly charged nonpolarizable surface, in which the ions adsorb on the wall via Baxter's sticky potential. Wall-ion interactions are treated in the hypernetted-chain (HNC) approximation and bulk interactions are treated in the mean spherical approximation (MSA). In the HNC/MSA, this model does not exhibit any saturation in the adsorbed surface charge density although one can be produced as an artifact of linearization. The expression for the potential of mean force possesses the term previously identified as responsible for the discreteness-of-charge effect, although there is a quantitative difference due to the absence of a low inner layer dielectric constant in our model. Results for the surface potential and distribution functions are presented for the case of anionic

adsorption on the surface.

I. INTRODUCTION

In recent publications there has been considerable effort devoted to studying the behavior of model elec­trolyte solutions near charged surfaces. This clas­sical problem of the electrical double layer is being re­examined using the integral equation formalism that has been developed to study the behavior of fluids in the presence of an external field. Two superficially dif­ferent but essentially equivalent paths have been tried: the potential approach developed by Buff, Stillinger, Bell, Levine, and Outhwaite! and the integral equation approach adopted by Blum, Henderson, and Smith2•

3

and Carnie, Chan, Mitchell, and Ninham. 4 In all cases the primitive model for the electrolyte was employed. In this model the ions are taken to be in a dielectric continuum and the real molecular nature of the solvent is ignored except possibly through the assignment of effective ion sizes to account for the presence of hy­dration shells. Very recently the model of an ion­dipole mixture at a charged surface has been used to elucidate the effects of the mOlecular nature of the sol vent on the double layer problem. 5 All the above studies are relevant to an indifferent electrolyte at a charged surface because specific or non-Coulombic interactions between the ions and the surface (except for a hard sphere exclusion) are absent from the models.

The question of specific ionic adsorption is not new. It was first addressed by Stern6 in 1924 as an improve­ment over the Gouy-Chapman or Poisson-Boltzmann picture of the electrical double layer. Although con­ceptually correct, the Stern treatment of ionic adsorp­tion is more of an afterthought than an integral descrip­tion of the electrical double layer. A self-consistent treatment of specific ionic adsorption from a primitive model electrolyte onto a neutral surface has been con­sidered recently in the mean spherical approximation (MSA).7 Although ion adsorption and discrete charge effects emerge naturally from this calculation the model still has two rather serious omissions: ' the es­sential nonlinear behavior of the electrical double layer and the ability to descr~be adsorption at charged as well as neutral surfaces.

In this paper we wish to consider, under a single model, aspects associated with a primitive model elec­trolyte at a charged surface including specific ion ad­sorption, discrete charge, and nonlinear effects. We employ the restricted primitive model (RPM) for the electrolyte; the case of unequal ion sizes is a straight­forward extension. SpecifiC (non-Coulombic) interac­tions between the surface and the ions are modeled by the sticky potential of Baxter6 to mimic the short-ranged nature of the specific adsorption process. Moreover, a parametrization of the specific adsorption process by a sticky potential allows the hypernetted chain (HNC) closure to be retained for the surface-ion correlation functions-a procedure we know will give the essential nonlinear behavior. We note that inclusion of specific adsorption leads to an additional term in the equation for the contact value of the ionic distribution functions (Sec. II). In Sec. III we make explicit the connection between our model and the Gouy-Chapman-8tern picture and indicate how our model may be used to describe sur­faces with ionizable chemical groups. We also show that linearization of the equations (which reduces to the model of Ref. 7) produces an unphysical saturation of adsorbed surface density. We then examine the addi­tional terms in the potential of mean force due to the adsorbed layer of ions and compare our adsorption isotherm with that of Levine et al. 9 in order to show that the discreteness-of-charge effect is inherent in the HNC/MSA for this model.

Numerical results for both thermodymamic (surface potential) and structural (distribution functions) quanti­ties are presented in Sec. IV, chiefly for the situation analogous to anion adsorption onto a mercury electrode ..

II. THE MODEL

In the restricted primitive model, the interaction po­tential between ionic species i and j with charges z e and zJ e is given by i

UiJ(r) =00 , r<R,

Z Z e2 =~

E:r r>R, (2.1)

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3486 S. L. Carnie and D. Y. C. Chan: Ionic adsorption from a primitive model electrolyte

where E is the dielectric constant of the inert solvent and R is the ionic diameter-the same for all species.

The surface or wall is taken initially as a large spherical particle of diameter Ro and carries a charge Zo e. The planar result is obtained in the limit Ro - o(), which is to be taken at constant surface charge density of the wall: <To = Zo e/ 41T~. We shall call <To the in­trinsic charge density on the wall. The interaction po­tential between an ion of species i and the wall is written as

u,o(r) :=u:o(r)+u~o(r) ,

where

~ ~o(r):= , O<r<oo, (r

(2.2)

(2.3)

is the Coulomb potential, and ufo(r) is the non-Coulombic or specific interaction taken to be of the form

ufo(r) =0(), r< RiO =i(R+ Ra) ,

(2.4)

The structure of the electrolyte at the wall is ob­tained by solving the Ornstein-Zernike (OZ) equations for the ion-wall indirect correlation functi.on hiO(r)

For the moment, we take the bulk direct correlation functions cu(r) to be known functions. The wall-ion HNC closure, whose status has been considered else­where, 4,10 may now be used to provide an additional relation between hin and c,n (13 = 1/kT):

1 + h/o(r) = exp[ - {3u,o(r) + h/o(r) - c/o(r)]

=exp[ - i3uto(r)] exp[ - {3~o + hw(r) - cw(r)) •

(2.6)

We parametrize the Boltzmann factor associated with the short-range specific potential by a Dirac 6 function

exp[-!3ufo(r))=0 , r<R/o ,

where

41TR~oa/= f (e-a,z,t(r) -1)dr. r>RtO

(2.7)

(2.8)

Equations (2.5)-(2.7) require hiO and c/o to have the form

hiO(r):= bi 13(r - RIO) + hio(r) ,

c/o(r)=bj 6(r-R,o)+cro(r),

with

hio(r)=-1, r<RiO

and

(2.9)

(2.10)

1 + hio (r) = exp [ -- i3z4o( rl+ hro( r) - cro( r)], r> R,o .

(2.11)

The "starred" functions contain no delta functions and bi is a constant that is determined by matching coef-

ficients of the delta functions in Eq. (2.6):

bi := a/ [exp( - i3zf,0 + hto - C10)]hRjo

:= a/ [1 + hto(Rio)] , (2.12)

where the second equality follows from Eq .. (2.11).

We proceed to obtain a set of one-dimensional integral equations for ionic distribution functions at a plane sur­face. We begin by separating explicitly the long-ranged part of the bulk direct correlation function

) i3 Zi Z J e2

0 ) cjJ(r =- - +ciJ(r, O<r<oo, Er

(2.13)

where c~J(r) is a short-ranged function. Combining Eqs. (2.5), (2.9), and (2.11) it is straightforward to show that (r> Rjo )

In[1+hin(r)]=- !3z,e {Zoe+41TR~n<T6

(r

+4rre f'" dS~LPJZjh'Jo(S)} Rjo J

- .Bzi eifJ(r) + ~ Pj f de C~j(/ r - s / ) hJo(s) ,

(2.14)

where

<Ta= L Pj Zj ebj J

is the adsorbed surface charge density and

ifJ(r) = 41T f'" ds S (1 -~) LPJ Zj ehJo(s) Err J

(2.15)

(2.16)

is the mean electrostatic potential. The three terms in braces in Eq. (2.14) mutually cancel because of electro­neutrality since they represent, respectively, the in­trinsic charge on the wall, the adsorbed charge, and the charge in the diffuse part of the double layer.

We can now take the planar limit (Ra - 00) USing the substitutions

x = r - RiO, Y = s - RiO ,

hi(x)=hi'o(X+RIO)=hi'o(r) .

Equations (2.10) and (2.14) simplify to

(2.17)

(2.18)

h i (x)=-1, x<O, (2.19)

1n[1 + hi (x)} = -j3zj eifJ(xl+ ~PJ f.: dye~J<I x- y/ )hJ(y)

(2.20)

where

e~J(x) = 21T f'" drrc~J( r) , x

(2.21)

(2.22)

Equations (2. 19}-(2. 22) have to be solved subject to the boundary conditions

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s. L. Carnie and D. Y. C. Chan: Ionic adsorption from a primitive model electrolyte 3487

6

o

/ /

A I I

10 20 c.(lle/em 2 )

30

FIG. 1. Surface potential at the plane of closest approach, 1J!/f as a function of the surface charge density, in the absence of ion adsorption, at electrolyte concentration; (A) 0.01 M, (B) 0.1 M. -- HNC/MSA, --­modified Gouy-Chapman (I Monte Carlo results of Torrie and Valleau).

(2.23)

(2.24)

The adsorption parameter a, is related to the non­Coulombic adsorption potential <J>,(x) by [cf. Eq. (2.8)]

(2.25)

and the potential at the charged wall l/Jo is given by

21TR l/Jo = l/Ja + -- (10,

E:

where

(2.26)

(2.27)

is the potential at the plane of adsorbed charges. Once the bulk direct correlation functions c~J(r), the adsorp­tion potentials <1'> I (x), and the intrinsic surface charge density (10 are specified, the properties of the electrical double layer can be determined by solving Eqs. (2.19)­(2.27).

We observe that the inclusion of ionic adsorption gives rise to an additional term in the potential of mean force between an ion and the charged wall-the last term in Eq. (2.20). It is easy to see that this term repre­sents hard sphere and short-range electrostatic interac­tions between ion i in the diffuse layer and all the ad­sorbed ions. The presence of this additional term will alter the equation for the contact density of the ions. 4

To derive the new contact condition, we begin with the identity

L: P, [1 + h,(O)] = L P, - f" dx L: P, [1 + h,(x)] , I 0 I

dln[l + h,(x)] x dx (2.28)

and use Eq. (2.20) for the logarithm in the integrand to­gether with the manipulations detailed in Ref. 4 to get the desired result

~P,[1+h,(O)] = 2;/3 (0'0+0',,1+ i {~P,+{3/x} +21T ro dxx LP,-[l+h,(x)]

o ,

(2.29)

where

X =/3 [~PI - ~ P,PJ f dr ~J(r)r1 (2.30)

is the isothermal compressibility of the electrolyte. The result given by Eq. (2.29) serves as a useful check on the internal consistency of numerical calculations. Although the derivation of this result invokes the HNC closure, a general derivation of exact contact condi­tions which includes effects such as images and the molecular nature of the solvent has been given else­where. 24

So far, we have made no assumptions about the bulk properties of the electrolyte. In principle, we could treat the bulk interactions exactly by using the results of computer simulations. However, since such simula­tions do not explicitly calculate the direct correlation functions, the best chOice for e~J(x) must be found from the more successful of the integral equation theories. Undoubtedly the most accurate of these theories uses the HNC closure,11 whereas the MSA which has the ad­vantage of being analytically soluble also gives reason­able values for the thermodynamic properties of 1 : 1 RPM electrolytes. 12

We have previously used the MSA for the bulk interac­tions in a model for the double layer in the absence of ionic adsorption. 4 The analytic nature of the MSA e~J(r) simplifies the numerical solution of the HNC/MSA equa­tions, and has the advantage of allowing analytic extrac­tion of the extra contributions to the potential of mean force which are due to ion size effects. Although one may expect the HNC/HNC treatmene to be more accurate, both give similar results for the surface potential. In Fig. 1 we present a comparison between the HNC/MSA results of l/Ja vs (10 for 0.1 M and 0.01 M electrolyte against a nonadsorbing, uniformly charge, nonpolariza­ble surface and recent Monte Carlo results for the same system. 1S The HNC/MSA is in reasonable agreement with Monte Carlo results in the regime of surface charge densities and ionic concentrations considered in Fig. 1.

The HNC/MSA treatment of the double layer including ionic adsorption is then given by Eqs. (2.19)-(2.24) together with e~J(x) given by4.12

(2.31 )

where the first term comes from the solution of the Percus-Yevick equation for hard spheres14

ePHYS(x) = (11T~~)4 {- i 11(1 + 211)2 (1 - ~ )

+17(11+2)2(1-~) -(1+2111(1- ~)} x<R (2.32)

=0, x>R,

where

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3488 S. L. Carnie and D. Y. C. Chan: Ionic adsorption from a primitive model electrolyte

and the electrostatic term is of the form

c 27r(3tf {B2 ( r) ( ~) ( X)} e (x) = -E: - R "3 1 - K - B 1 - 11 + 1 - R '

x<R,

=0, x>R. (2.33)

Here

B=[1 + KR - (1 + 2KR)1/2] /KR , (2.34)

where K is the usual Debye-Hiickel inverse screening length, given by

.-2 _ 41f(3e2

" ..2 1\ - L... PI Zj •

E: I (2.35)

In the next section we compare this model of the double layer with existing theories of ionic adsorption.

III. COMPARISON WITH EXISTING THEORIES OF IONIC ADSORPTION

In order to clarify the approximations inherent in the HNC/MSA equations, we first consider two simplifica­tions: (i) the point-ion limit (R - 0) and (ii) the limit of small potentials (I (3elJlal < 1) but R* O. In the third part of this section, we examine the contributions to the po­tential of mean force in the HNC/MSA equations and com­pare with earlier "discrete-ion" theories of ionic ad­sorption.

A. The point·ion limit

The neglect of ion size within the HNC/MSA is equiva­lent to setting

e~j(r) =0 (3.1)

so that Eq. (2.20) gives

In [1 + hi (x)] = - (3ZI elJl(x) (3.2)

which, together with Poisson's equation, gives 'the Gouy -Chapman theory. For simplicity we consider a 1: 1 electrolyte with one adsorbing species, which we take to be species 2. Therefore in the point-ion limit, the adsorbed surface charge density is

(3.3)

We can compare this result with the Stern isotherm15

Z;! eNs (3.4)

where Ns is the surface density of adsorption sites, NA

is Avogadro's number, M is the molecular weight of the solvent, and <P2 is the specific adsorption potential. As <Pz becomes more negative, the adsorption becomes stronger and eventually the adsorbed surface charge density saturates to the value

(3.5)

as (3<P2 - - 00. At low coverage, we can simplify Eq. (3.4) to

Us =P2 Z;! e ( ~~) exp( - (3<pz) exp( - (3Z;! elJls)

which is of the same form as Eq. (3.3) with

(3.6)

If we use the Gouy-Chapman result

(3elJl/l =2 sinh-1[ 2:~e (uo+ US)] ;: 2sinh-1(at)

in Eq. (3.3), we find

Us = P2 Zz e lltl [at + (0'; + 1 )1/2]-2.2

(3.7)

(3.8)

(3.9)

from which it is easy to verify that Us does not saturate in the strong adsorption limit lltl- 00. In our model, this lack of saturation in the point-ion limit is not unexpected since there is no extra parameter corresponding to Ns,

the adsorption site density. Therefore, in the point­ion limit, the HNC/MSA theory of ion adsorption is equivalent to the Gouy-Chapman-Stern model at low surface coverage.

In .the absence of an intrinsic surface charge Uo = 0, our model can be made to mimic a neutral surface that acquires a charge by a surface reaction of the kind

(3.10)

which can be regarded as the adsorption of, say, hy­drogen ions. The reaction is governed by a dissocia­tion constant K

K - [A] [W]s - [AW] (3.11)

where [H+]s is the concentration of adsorbing ions at the surface. In the point-ion limit, we have

(3.12)

In this mass-action model, the adsorbed surface charge density iS16

Us = Zz eNs/ [1 + P: exp«(3z2 elJls)]

which reduces to the weak adsorption result

ul! = Zz epz i exp( - (3z2 elJls) ,

provided K» Pz, or equivalently,

PH>PK,

(3.12)

(3.14)

(3.15)

where PH = -logloP2, pK = -log10K. Hence by comparing Eqs. (3.3) and (3.14) we can identify lltl with Ns/Kas long as Eq. (3.15) is satisfied. Thus ion adsorption in the point-ion limit of the HNC/MSA at Uo =0 is equivalent to the weak adsorption regime of the mass­action model.

B. Linearized equations (R,*O)

For the case of small potentials (I (3elJls l <1), the left­hand side of Eq. (2.20) can be linearized and we re­cover the MSA for an interface with both intrinsic and adsorbed surface charges. This can be solved analytical­ly by the techniques in Ref. 7. The result for the sur­face potential is

(3.16)

which is just the sum of the potentials due to the in-

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S. L. Carnie and D. Y. C. Chan: Ionic adsorption from a primitive model electrolyte 3489

trinsic and adsorbed surface charges.,2,7 The adsorp­tion isotherm is

_ _ Pa Zz eaz[ (1 + 21l)/ (1 - Tj)2 - 47r/3zze(1o/ EK] •

(1/3 -Pz Zz ebz - 1 + pz az[7rR2/(1 -1) + (47r~z~ e2 1 EK)(l- B)]

This expression is only valid if

4.,r!3Zz e(1o "" 1 + 21) . EK (1-1))2'

(3.17)

(3.18)

otherwise the distribution function (1 +!za(x)] will be negative at contact.

The diffuse layer charge density is given by

where

47r!3~ 0 = - -- + Qu(x) , x<R

EK

and

o () 47r!3~ Qu x =-­

EK

=0, x>R.

x<R

Equation (3.19) can be rewritten as

(3.19)

(3.20)

(3.21)

pCh(X) = _ «(10 + (18) K+ ~ Pj ~ <18 Q~l(X) - K (' pCh( y) dy

(3.22)

so that Eqs. (3.16)-(3.22) are the generalizations of Eqs. (2.20), (2.32), and (2.39) of Ref. 7 to the case of an interface with both intrinsic and adsorbed surface charge.

We note that the linear result Eq. (3.17), like Eq. (2.39) of Ref. 7 [which can be obtained from Eq. (3.17) by setting (10 = 0], implies that the adsorbed surface charge denSity (18 will saturate in the strong adsorption limit liz - 00:

[(1 + 21)/(1-1)2 - 47rj3zze<1o/EK]

<1a- Za e [7rJl2/(1-1))+(41Ti3z~ez/EKXl -B)l ' (3.23)

This saturation is still present in the limit R = O. More­over, we observe that the linear result in the R = 0 limit can also be obtained by linearization of Eqs. (3.3) and (3.8) which in their nonlinear form do not show any saturation in <18 as liz - 00. Thus the form of saturation of (J8 implied by Eq. (3.23) and Eq. (3.29) of Ref. 7 is an artifact of linearization.

In the present model of ion adsorption there is no parameter corresponding to the adsorption site density N s • Nevertheless, excluded volume interactions in the adsorbed layer can result in saturation when the adsorp-

tion density approaches the two-dimensional close packed lim it. However, lateral interactions are not handled properly in the HNC/MSA so we do not expect a natural upper bound in the adsorption density to emerge from our calculation. Even so, as we shall see in the next section, at low surface coverage where the HNC/MSA is expected to be applicable we can see interesting and Significant differences between the pOint-ion Gouy-Chap­man-Stern theory and the HNC/MSA.

C. HNC/MSA equations

We now examine the contribution to the HNC/MSA potential of mean force due to the adsorption of finite size ions. Following earlier work4 we extract analytical­ly correction terms to the Gouy-Chapman result in the limit of low ionic density. Using the density expansion of e~rBA in Eq. (2.20) we can write the potential of mean force of species i in the form

Wj (x) = Zj eq;(x) + w,2a) (x) + Wj(Zb) (x) + •••

(3.24)

where only terms up to linear order in the ionic density are exhibited. The Gouy-Chapman theory only keeps the first term on the right-hand side of Eq. (3.24). The terms W~za) and WlZb) which are of the same order (linear) in ionic density have been considered in detail earlier. 4

To recapitulate, they represent, respectively, finite ion size contributions to the excluded volume and elec­trostatic interactions in the diffuse double layer. The ion size corrections to the interaction between an ion of species i located at x with the adsorbed layer are given by the terms

W~d(za)(X)=R{~ PZ[l+ hz(O)] kTJ7r(R Z-x2) , x<R

=0, x>R, (3.25)

Wf(Zb)(X):= - zjR {Z2 }; pz[ 1 + !za(0)]}~ 27r(R - x) ,

x<R

=0, x>R. (3.26)

These two terms are of the same order in the bulk den­sity of adsorbing ions (species 2) and they originate from the last term of Eq. (2.20). A Simple phYSical interpretation of w:d(za) and Wf(Zb) is as follows.

The first term wf'(za)(x) can be interpreted as R times the x component of the force exerted on a sphere of radius R centered at x (that is, the co-sphere of an ion of diameter R) as it is inserted into a layer of thick­ness R of ideal gas ofdensity{(lIz/R)Pz[l+!za(O)]}, located adjacent to the charged wall (see Fig. 2). Con­sequently this term will act to repel all ions from the surface and increase the extent of the diffuse layer.

The second term, Wf'{Zbl(X) is R times the electro­static force between a point ion at x and a uniform vol­ume charge of density {- Zz e(lIz / R) [1 + hz(O)]} contained in the conical section subtended by the interaction of the co-sphere and the plane x=R (see Fig. 2). A more ap­propriate physical picture of this contribution should only involve the spherical cap bounded by the plane

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3490 S. L. Carnie and D. Y. C. Chan: Ionic adsorption from a primitive model electrolyte

""-R-

FIG. 2. Schematic illustration of the overlap between the co­sphere of an ion in the diffuse layer and the layer of adsorbed ions.

x:; R. At contact x = 0, both interpretations give the same result. W~(2b) is always attractive for the ad­sorbing species and repulsive for the indifferent ion. When the adsorbing species is also the counterion this term acts to decrease the extent of the diffuse layer.

The separation of the potential of mean force into separate contributions from the diffuse and adsorbed layers and the interpretation of the adsorbed layer con­tribution as R times a force is a consequence of the delta-function nature of our adsorption potential. Even so, it is still valid to conclude that as a result of ion adsorption ion size corrections to the potential of mean force consist of ~o types: hard sphere excluded volume terms which tend to increase the extent of the diffuse layer and ion size corrections to the electrostatic interaction (self-atmosphere terms) whose effect de­pends on the relative signs of the adsorbing species and the net surface charge. From Eqs. (3.25) and (3.26) it is easy to verify, at least to lowest order in density, that ion size corrections to the electrostatic interaction are more important. The same is true for the diffuse layer terms w~2a) and W~2b) which have been considered in detail for the case with no ion adsorption. 4

We now compare the HNC/MSA equations with earlier theories of ion adsorption. 9,17 Using Eqs. (2.15), (2.20), (2.24), and (3.24) we can write our adsorption isotherm as

<18=PZ ~ eZ2 exp{-.B[Zz e1P8+ w~2a)(0)+ W~2b)(0)+ •••

+ W~(2a)(0)+ W~(2b)(0)+ ••• ]} . (3.27)

This can be compared with the expression of Levine et al. 18

O! (Ns Z2 e - Po (18)Pa {[ ] } <18 == no (NsZze'fa-i exp - (3 <1>2 + Zz elPB + Z2 e¢B ,

(3.28)

where O! is the activity of the species 2 in the bulk electrolyte, Ns is the maximum adsorption site density, no is the density of water molecules in the aqueous phase, <1>2 is the real specific adsorption potential, and ¢8 is the self-atmosphere potential of the adsorbed ion. The parameter Pa specifies the number of sites that an ad­sorbed ion occupies. 18

Since our model has no fixed adsorption sites, we do not have the same pre-exponential term, except for ~ which can be clearly identified with exp( - (3<1>2)' The term due to the mean electrostatic potential is common to both Eqs. (3.27) and (3.28) but terms like w~2a)(0) and W~2b)(O) do not appear in Eq. (3.28) since the dif-

fuse layer is treated in the pOint-ion limit by Levine et al. 9 Adsorbed layer repulsive terms such as W:S(2a)(0) are also absent from Eq. (3.28). The self-atmosphere term which had been invoked to explain various experi­mental anomalies can be identified with W~(2b)(0). However, a comparison of W:S(2b)(O) with the expres­sion for the self-atmosphere term18 will reveal that the latter is larger by a factor - E:iE:1 where E:1 is the di­electric constant of the inner layer (or Stern layer). The reason for the difference is that our model takes the dielectric constant of the solvent to be the bulk value E: up to the surface rather than making a further assump­tion about the presence of a Stern layer. A consequence of this is that, when using Eq. (3.28), much smaller values of <1>2 (viz. ~) are required in order to obtain dramatic deviations from Gouy-Chapman-8tern theory. Indeed, this larger value for the self-atmosphere term is crucial for an explanation of experimental results.

IV. RESULTS

As in earlier work,4 the HNC/MSA equations (2.15), (2.19)-(2.27), and (2.31)-(2.33) were solved by direct iteration with parameters E: =78. 5, T:;298 K, R=4.25 A for a range of concentrations (0.01 M - 0.5 M). The contact condition Eq. (2.29) was satisfied to within 5% for O. 5M electrolyte and to within 1% for the lower con­centrations.

Most of our calculations have been performed with a view to represent a physical system such as anionic adsorption on mercury. In this case, the adsorbing species (species 2) are anions, the intrinsic surface charge <10 has a range of values (-30-+40 ILC/cm2

) and ~ /R is varied from 0.1 to 100. We also briefly discuss some calculations where the adsorbing species are cations, <10 = 0, and ~ is fixed by the identification men­tioned beneath Eq. (3.15). This is meant to represent an interface that acquires surface charge by a surface association reaction.

For the purpose of comparison we also present re­sults in the pOint-ion limit R = O. This limit is tanta­mount to replacing Eq. (2.20) by Eqs. (3.1) and (3.2).

Numerical results for the surface potential-surface charge (IPB - (10) relationship are presented in Figs. 3-5 for 1:1 electrolytes of concentrations O.OlM, O.lM, and 0.5M, respectively. Allowing for the change in scale from Fig. 3 to Fig. 4, one can see that as the electro­lyte concentration is decreased, differences between the HNC/MSA and the point-ion limit for <10 < 10 ILC/cm2

decrease; for <10> 10 ILC/cm2, these differences in­

crease.

4

~ kT Of-+--+--+--'-+--+--t--I

-4

-- 0 -10 o 10 20

FIG. 3. Surface potential ¢8 as a function of intrinsic sur­face charge 000 at 0.01 M for adsorption strengths a2/R: (a) 0.1, (b) 1. 0, (c) 5. O. -­HNC/MSA, --- point-ion limit.

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S. L. Carnie and D. Y. C. Chan: Ionic adsorption from a primitive model electrolyte 3491

6

I. e+p

kf2

0

-2

-4

-6 --4 -20

' '0 ,

0 20 o,(IlCfcm')

40

FIG. 4. Surface potential lJ!8 as a function of intrinsic sur­face charge (10' at 0.1 M for adsorption strengths a2/R: (a) 0.1, (b) 1.0, (c) 5.0. -­HNC/MSA, --- point-ion limit. The vertical bar marked A in­dicates the parameters appro­priate to case A discussed in the text.

The potential at the point of zero charge ('I'pzo) ap­pears to change more rapidly with concentration for the HNC/MSA than for the point-ion limit-the difference depending on the parameter~. This is in accord with the usual invocation of discrete charge effects to ex­plain the Esin-Markov effect, namely that the experi­mentally measured values of d1/JpZc /dlnc are two or three times higher than that theoretically attainable in the point-ion limit. 19-.'11 Our results show only small deviations from the point-ion limit, and the reason is that our adsorbed layer self-atmosphere term [Eq. (3.25)] is smaller than the usual one9 by a factor EiEI

because we have not postulated the a priori existence of the Stern layer.

The general form of the results in Figs. 3-5 can be explained using the results of Sec. III. For negative 0"0, where the adsorbing anions are co-ions and are therefore repelled from the surface as a result of elec­trostatic interactions, there is little adsorption except for the largest value of~. so that the 1/J8 vs 0"0 curves in this regime are almost the same as the case of no adsorption. 4 For positive values of 0"0 where the ad­sorbing anions are now the counterions, there is ap­preciable adsorption as the effects of the new terms Wr

in the potential of mean force become significant [see Eqs. (3.24)-(3.26)]. In fact, these terms are responsi­ble for the increasing differences between the HNC/MSA and point-ion limit result as ~ is increased. The above comments can be applied to most of the surface charge vs surface potential curves shown in Figs. 3-5. The case ~ / R dOO, O. 5M (Fig. 5, curves d) is interesting and deserves further comments. At 0"0 =0, the HNC/ MSA potential at zero charge is larger in magnitude than the point-ion result. This is because adsorption is en­hanced by the dominant finite ion size correction to the electrostatic interaction between an ion and the adsorbed surface . charge- W~d(2b) in Eq. (3. 26)-which always at­tracts the adsorbing ions towards the surface. As <To

FIG. 5. Surface potentiallJ!8 as a function of intrinsic sur­face charge ao at 0.5 M for adsorption strengths a2/R: (a) 0.1, (b) 1. 0, (c) 5.0, (d) 100. --HNC/MSA, --- point-ion limit. The vertical bars marked Band C indicate the parameters appropriate to cases Band C discussed in the text.

b

C / / / /

~~~O~~O~~10~~~~~3~O-L~40 o,(pClcm1j

FIG. 6. Adsorbed surface charge (18 as a function of in­trinsic surface charge (10 at 0.5 M. -- HNC/MSA, point-ion limit.

becomes negative, the contribution from Wf(Zb) to the adsorbing species is counteracted by a similar term W~2b) in the potential of mean force [cf. Eq. (3.24)] which accounts for the ion size correction to the in­teraction between ions in the diffuse layer. 4 The term W~2b) repels cO-,ions from the surface-for ao < 0, the adsorbing anions are co-ions. Thus the mutual can­cellation of W~2b) and Wr<2b) for the adsorbing anions is responsible for the crossover between the HNC/MSA and point-ion limit result where in the latter case both W:Zb ) and W ~(Zb) are assumed to be zero.

An interesting and important common feature among the results given in Figs. 3-5 is the maximum in 1/J8

as a function ao. In the absence of ion adsorption the maxima in 1/J/l occur at around + 40 llC/cm2 in the con­centration range between O.lM and O. 5M.4 As a result of ion adsorption the maxima move to lower values of <To as the adsorption strength ~ increases. Indeed, for ~ /R = 100, 0.5M, the maximum occurs at a small nega­tive value of 0"0 (-10 MC/ cm2)_a feature that appears in Grahame's experiments on the adsorption of r on mer cury.22 However, the maximum in Fig. 5, curve d, is shallower than that given by existing theories9,22 that have been constructed to explain the experimental observations; the reason for the difference is the ab­sence of an inner of Stern layer of low dielectric con­stant.

The adsorbed surface charge density <Til as a function of ao is shown in Fig. 6. The small differences between the HNC/MSA curves and the point-ion curves show that <Til is less sensitive than 1/J1l to finite ion size effects. These differences become smaller as the electrolyte concentration is lowered. In all cases where the ad­sorbing species is the counterion, the magnitude of all is increased when finite ion size effects are included. This is to be expected in view of the effects of W)2b)(X)

and. W~(Zb)(X) which had been discussed above.

An advantage of integral equation formulations of the double layer is that structural as well as thermodynamic information can be obtained. We now examine the waH­ion distribution functions for three cases of interest. The relevant properties of these cases are summarized in Table I as well as being indicated in Figs. 4 and 5. We note that in all cases the surface potentials are low to moderate, I e1/J8 /kTI ::3, and the surface charges <To are in the regime where we sllould have some con­fidence in the HNC/MSA-ao:520 MC/cm!, see Fig. 1. To illustrate the interplay between ion adsorption and ion size effects we compare the HNC/MSA results with those obtained in the point-ion limit where ion size ef­fects are ignored.

The wall-ion distribution functions for case A are

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3492 S. L. Carnie and D. Y. C. Chan: Ionic adsorption from a primitive model electrolyte

TABLE I. Parameters for the distribution functions shown in Figs. 7-9.

HNC/MSA Concentration az :!1 RZ '!A R2

Case (M) If Uo (~C/cm2) e Us (~C/cm2) e el/!JkT

A 0.1 5 16.0 0.181 -13.6 - 0.154 0.95

B 0.5 5 10.7 0.120 -8.8 - 0.131 - 0.52 C 0.5 100 3.2 0.036 -15.0 - 0.17 -3.04

shown in Fig. 7. From Fig. 4 we can see that ion size effects reduce l/is and therefore must increase the amount of ion adsorbed, L e. 1 I (Js I must increase. Conse­quently the contact value of the total correlation func­tion ~(O) of the adsorbing anions (which are also the counterions in this example) must be higher in the HNC/ MSA. This result is consistent with the observation that for (Jo> 0 both ion size electrostatic corrections to the potential of mean force, W~2b) and Wr<2bl, which are the dominant correction terms, act to attract the counter ion and adsorbing ions towards the surface. In the contact condition Eq. (2.29), the third term on the right hand side is now of the same magnitude as the second term. Because of the lower net surface charge, the adsorption excess

rj =Pj fo''' hj(x)dx (4.1)

of species 2 is decreased and ~(x) is a shorter-ranged function than that in the point-ion limit. As for species 1 the nonadsorbing cation (co-ion), all ion size correc­tion terms in Eq. (3.24) serve to repel these ions from the surface. The contact value of ht(x) therefore be­comes more negative and the crossover between HNC/ MSA and the point-ion result is due both to the smaller net surface charge and to the need to preserve electro-

-02 h,(x)

-04

-10 0

6

hix)

\ \

2

, /

/

/'

/

4 x/R

, "

2 xlR

6 8 FIG. 7. Case A. Wall-ion total correlation functions for the indifferent species (1) and the adsorbing anions (2) at 0.1 M, a2IR=5. --HNC/MSA. ---point-ion limit.

4

neutrality. The cusps in h,,(x) and ~(x) at x=R are artifacts of the use of the MSA for the bulk direct cor­relation functions-c~lsA is discontinuous at X= R.

The distribution functions for case B are given in Fig. 8. In the point-ion limit the surface has a small posi­tive net surface charge. However, ion size effects, as discussed in case A, are sufficient to increase ! (Je! to make the surface become net negative. The increase in I (Jsl implies that the contact value of adsorbing anions hz(O) must also increase. Since the surface is net nega­tive the adsorbing anions, species 2, are also the co­ions so ~(x) must eventually become negative. The repulSion of the nonadsorbing species (1) (counterions) from the surface by W~<2b) and the constraint of electro­neutrality leads to the form of ht(x) shown in Fig. 8. The cusps at x = R are again artifacts of using e~rSA. We note that in case B the adsorption excesses in the HNC/MSA are of the opposite sign to those in the point­ion limit.

In case C, Fig. 9, the adsorbing anions, species 2, are now the co-ions because the surface is net negative. However, adsorption still takes place against the electro­static repulsion of co-ions because of the large adsorp­tion parameter (aa / R = 100). The indifferent cation (species 1) is repeUed from the wall, predominantly

02 h,(x)

o~~~----~==~

-06

0 2 3 4 xlR

02

01 h~x)

0

2 3 t. x/R

FIG. 8. Case B. Wall-ion total correlation functions for the indifferent species (1) and the adsorbing anions (2) at 0.5 M. a2/R=5. -- HNC/MSA. --- point-ion limit.

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S. L. Carnie and D. Y. C. Chan: Ionic adsorption from a primitive model electrolyte 3493

because of the finite ion size electrostatic correction W:a<26l(X). This is reflected in a lowering of ~(O) rela­tive to the point-ion value.

Although the cusps at x=R are artifacts of using e~JMBA, it is tempting to interpret the maxima in ~ (x) at x = R in Figs. 8 and 9 as an ion pairing phenomenon between the adsorbed anion and the nonadsorbing cation in the diffuse part of the double layer.

We reiterate our earlier remarks that in the three cases discussed above, the range of surface potentials and surface charges are in the regime where one would have some confidence in the HNC/MSA. Therefore it seems worthwhile to test the predictions of the HNC/ MSA, in particular the results for the distribution func­tions shown in Figs. 8 and 9, against Monte Carlo ex­periments. To build in an adsorbed layer of ions in the Monte Carlo calculation, it is necessary to specify the value of the intrinsic surface charge 0"0 and the value of the adsorbed charge O"B; thereafter ions in the adsorbed layer are only permitted to move in the plane x=o during the sampling procedure.

In Fig. 10 we show the surface potential versus PH for a surface where the model of ion adsorption is de­scribed by Eqs. (3.11)-(3.14). We would of course require a three-component electrolyte, e. g., HCI/NaCI, in order to fix PH and KR independently. The special properties of e?rSA(x) for a symmetric RPM electro­lyte allow us to keep all equations the same as for a two-component electrolyte, except that l1:! is replaced by 1011 Ns/NA R, where Ns is in cm-2, NA is Avogadro's number, and R is in A. The surface potential is larger for the HNC/MSA than for the point-ion limit, for the same reasons that I/Jpzc is more negative for the HNC/

h,(x) 1

6 1 1

4 \

2 \

0 0

0

-0·2 h 2(x)

-04

-06 /

-08 I

,

2 x/R

/ /

/ /

I /

3

/. /

FIG. 9. Case C. Wall-ion total correlation functions for

4 the indifferent species (1) and the adsorbing anions (2) at 0.5 M, a2/R= 100. -- H:NC/MSA, --- point-ion limit.

-100L---.J'-----!2~----!3e--~4

x/R

~~~~~-.J~O~~~=-~2 pH-pK

FIG. 10. Surface potenttall/iB as a function of adsorbing ion concentration at zero intrinsic surface charge. -- HNC/ MSA, --- point-ion limit.

MSA in Fig. 5. The only difference is that the adsorb­ing species for Fig. 10 is cationic.

V. CONCLUSION

We have presented a model for ionic adsorption from a primitive model electrolyte that treats finite ion size effects in the diffuse and adsorbed layers on an equal basis. Essential nonlinear behavior is retained and comparison with simplifying limits has shown the limita­tions of the model. The model has the essential fea­tures required for the "discreteness-of-charge" ef­fect. 16 However, a major ingredient in USing the primi­tive model to interpret experimental results is the neces­sity to have an inner layer dielectric constant E:l that is substantially smaller than that of bulk solvent. The existence of such a layer in the primitive model is an a Priori postulation. Since there is some evidence that the inner layer properties can be derived from a civilized model electrolyte,S we feel that further work should investigate such models. These models should be comparable to recent molecular models of the sol­vent layer. 23 Another important feature lacking in our model is a proper treatment of lateral interactions. This could be remedied by the use of nonuniform fluid statistical mechanics following the approach of Buff and Stillinger17 in which the two-dimensional nature of the adsorbed layer is taken into account explicitly.

ACKNOWLEDGMENT

We are very grateful to Glen Torrie and John Valleau for making available to us their recent Monte Carlo results prior to publication.

IS. Levine and C. W. Outhwaite, J. Chern. Soc. Faraday Trans. 2 74, 1670 (1978) is an excellent review of the poten­tial approach.

2L • Blum, J. Phys. Chern. 81, 136 (1977). 3D. Henderson, L. Blum, and W. R. Smith, Chern. Phys.

Lett. 63, 381 (1979). 4S. L. Carnie, D. Y. C. Chan, D. J. Mitchell, and B. W.

Ninham, J. Chern. Phys. 74, 1472 (1981). 5S. L. Carnie and D. Y. C. Chan, J. Chern. Phys. 73, 2949

(1980).

60. Stern, Z. Elektrochem. 30, 508 (1924). 7D. Y. C. Chan, D. J. Mitchell, and B. W. Ninham, J. Chern.

Phys. 72, 5159 (1980). 8R. J. Baxter, J. Chern. Phys. 49, 2770 (1968). Ss. Levine, G. M. Bell, and D. Calvert, Can. J. Chern. 40,

518 (1962). lOD. E. Sullivan and G. stell, J. Chern. Phys. 67, 2567 (1977). uD. N. Card and J. P. Valleau, J. Chern. Phys. 52, 6232

(1970).

t2E. Waisman and J. L. Lebowitz, J. Chern. Phys. 56, 3086

J. Chern. Phys .• Vol. 75, No.7, 1 October 1981

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3494 S. L. Carnie and D. Y. C. Chan: Ionic adsorption from a primitive model electrolyte

(1972), 56, 3093 (1972). 13G. M. Torrie and J. P. Valleau, J. Chern. Phys. 73,

5807 (1980). 14M. S. Wertheim, Phys. Rev. Lett. 8, 321 (1963). 15E. J. W. Verwey and J. Th. G. Overbeek, The Theory of

the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948).

16B. W. Ninham and V. A. Parsegian, J. Theor. BioI. 31, 405 (1971).

I1F. P. Buff and F. H. Stillinger, J. Chern. Phys. 39, 1911 (1963).

18S. Levine, J. Mingins, and G. M. Bell, J. Electroanal. Chern. 13, 280 (1967).

190. A. Esin and V. M. Shikov, Zh. Fiz. Khim. 17, 236 (1943).

2oB. V. Ershler, Zh. Fiz. Khim. 20, 679 (1946). 21 0. A. Esin and B. F. Markov, Zh. Fiz. Khim. 13, 318

(1939). 22 D. C. Grahame, J. Am. Chern. Soc. 80, 4201 (1958). 23W. R. Fawcett, Isr. J. Chern. 18, 3 (1979). 248. L. Carnie and D. Y. C. Chan, J. Chern. Phys. 74,

1293 (1981).

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