136
Lesser Blum
Theory of Electrified Interfaces Lesser Blum Physics Department, College of Natural Sciences, University of Puerto Rico, Rio Piedras, Puerto Rico 0093 1 (Received July 2 1, 1976)
A new theory for the electrified interfaces, using the mean spherical and exponential approximations as the starting point, is proposed. This theory is based on a general result of Blum and Stell. It treats the exclusion volume effects exactly and consistently, but the long-range electrostatic forces are treated in an approximation; for the mean spherical case it is equivalent to the linearized Debye-Huckel theory. General expressions for the differential capacitance are given; in the limiting cases of zero potential, equal size, and low concentration they agree with the classic result of the linear Guoy-Chapman theory. The exponential approximation result with no image forces has a parabolic dependence in the applied potential for low potentials, but, unlike the classical result, it remains finite as the applied potential grows.
1. Introduction
The theory of the thermodynamics and structure of simple fluids and solutions has seen a dramatic change in the last 15 years; very precise yet simple equations have been derived that account quantitatively for the observed properties of fluids in equilibrium. Most of the successful equations (Percus-Yevick, hypernetted chain, mean spherical approximation, optimized cluster expansions, etc.) derive from what Percus’ calls linear response theory, since they can be derived from a functional Taylor expansion, keeping only linear terms in the disturbance. This progress entrained a substantial improvement in the theory of electrolytic solutions. Ever since Arrenhius proposed his theory of electrolysis, physical chemists have been fascinated by the peculiar thermodynamic and electric properties of ionic solutions. This interest is quite justified by the widespread applications of ionic systems in chemical technology and biochemistry. Historically, the first milestone on the way toward a quantitative understanding of electrolytes was, no doubt, the Debye-Huckel (DH) theory.2 However one must not forget that the basic concept of the DH theory, the ionic cloud, was also present in some form in the GuoyChapman (GC)3theory for flat electrodes (interfaces), and that preceded the DH theory by some 15 years. Strangely enough, while the bulk of electrolytic solutions have been extensively studied, the theory of electrodic interfaces has seen little change in the basic approach since Guoy, Chapman, and Stern.4 The DH theory is a limiting theory for very dilute solutions. It is in good agreement with experiment only for very dilute solutions (“slightly contaminated water”, as is often quoted). Therefore, a great deal of theoretical effort has been dedicated to the task of getting workable and sound theories for more concentrated ionic solutions. Basically there’aretwo kinds of approaches that have been considered: the first, older and perhaps more popular, is to try and improve on the DH theory itself. This approach has a serious fallacy in that the DH theory is basically inconsistent in its treatment of the hard exclusion core. However one does obtain easy equations with good accuracy, if perhaps the meaning of the adjustable parameters is not always clear. The more fundamental approach was pioneered by the work of Mayer? Friedman, and Stell and Lebowitz: who showed the precise relation between the cluster expansions and the DH theory, and how the theory could be improved. It was the hypernetted chain equation (HNC) of Allnatt7that the very extensive work of Friedman and collaborators8 showed to be such a successful equation for both the so-called primitive model The Journal of Physical Chemistry, Vol. 81, No. 2, 1977
of charged hard spheres and also the more realistic models that included soft core interaction and also the very important solvation effects. As was mentioned above, one of the assumptions of the DH theory is that the ionic cloud that surrounds a given charge is effectively formed by point charges with no repulsion core, although the central charge is of finite size. This is the origin of the inconsistencies that are also responsible for the poor performance of this theory. However it also predicts that net charge density surrounding a given ion decays exponentially as a function of the distance to the ion. For concentrated solutions this is in violation of the so-called second moment condition of Stillinger and Lovettgwhich predicts charge oscillations in the ionic cloud. In addition there is also another theory, the mean spherical approximation (MSA) of Lebowitz and Percus,lothat gives exactly the second moment, as was first shown by Groeneveld.ll Furthermore, the MSA that is essentially a version of the DH theory with the exclusion core treated exactly and consistently was solved analytically for quite a few models of interest and relevance. The first analytic solution for the restricted primitive model (a neutral mixture of hard charged spheres of equal size and opposite charge) is due to Waisman and Lebowitz,12 who showed that this theory was in excellent agreement with computer simulations and hypernetted chain equation results that are numerical and quite hard to obtain. This initial success promoted the search for analytic solutions of the MSA for a number of models with central and noncentral forces. Recently we found a solution for the primitive model of ionic solutions for the general mixture of unequal charge and size ions,13which is in reasonably good agreement with the HNC and also turns out to be a convenient way to represent the data of real solutions of simple electr01ytes.I~Another relevant case that has a known solution is the mixture of hard spheres with charges and point dipoles.16 This case should be interesting in view of the role that is assigned to the solvent by some current theories of the inner Helmholtz layer. Last, but not least, the MSA can be corrected systematically using a sequence of approximations that should converge to the exact answer; these approximations will be discussed below. Quite recently, the distribution of hard spheres in the neighborhood of a hard wall was found in the PercusYevick approximation (Henderson, Abraham, and Barker,17 Percus18), and was found to be in excellent agreement with machine computations (Lidg) except perhaps for small deviations in the vicinity of the wall. The small remaining discrepancy was later eliminated using the
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Theory of Electrified Interfaces
metallic); and (b) the surface charge force which is derived generalized MSAZ0by Waisman, Henderson, and Lefrom simple electrostatics. bowitz.21 Since the MSA does not yield the correct low density One of the obvious shortcomings of the GC theory is the limiting form of the binary correlation function the results neglect of hard core exclusion effects. In a rigorous staare then used to calculate the density profile in the exp tistical mechanical theory a system without repulsions approximation, which is the next order correction in the would just collapse22because of the infinite attraction of optimized cluster expansion of Andersen and Chandler3' the image charges right at the wall. The occurrence of this (the MSA is the lowest order approximation in this hicatastrophe is prevented by assuming the existence of a erarchy). distance of closest approach (Stern4)that is equal to the We have also considered the functional Taylor series repulsion core of the ion (GrahameZ3)and that can be expansion of Ste1129-31which is also formally exact. accounted for by an equivalent parallel plate capacitor.24 However in the present work we will not include this This takes us to the dual model of the electric interface, approach which will be left for future publications. We in which the double layer is subdivided into two regions: should remark, however, that the mean spherical apthe compact, inner Helmholtz layer, in which the ions are proximation used in this article is by no means a natural at fixed positions near the electrode wall, and the diffuse limitation of the theory, but rather a result that is at hand region where the GC theory is valid. To explain the which treats the exclusion effects exactly and gives a fair observations of the dependence of the capacitance on the representation of the properties of bulk electrolytes. It is applied voltage and ionic concentration, it has been frenot excluded that, in the future, numerical methods will quently assumed that the structure of the inner layer plays be at hand to compute the Wiener-Hopf factor functions the dominant role and also that the orientation of the for more elaborate representations and more elaborate solvent (water) molecules in this inner layer causes some approximations, such as, for example, the HNC equations of the anomalous shapes seen in some cases. This last with hydration shell corrections (see, for example, ref 25 assertion seems very doubtful in view of the relatively low and references cited therein). hydration energies of many ions25but, apart from that, the In section 2 we give the basic equations for the MSA and general scheme of the dual theory seems not to be very exp approximations of the density profile of the individual consistent since it does introduce a phase separation in a ions. In section 3 some general formulas for the relevant rather ad hoc manner. Recently, Cooper and Harrisonz6 properties, charge and potential profiles, capacitance, and showed that this division into inner and outer regions will excess energy, are given. Section 4 is devoted to the lead to gross inconsistencies, such as the necessity of discussion of several limiting cases for the restricted assuming negative capacitances for the inner region. These primitive model of electrolytes. authors have also criticized the role assigned to the solvent adsorption. 2. Mean Spherical and Exp Approximations Furthermore, all the theories up to this point have To introduce the notation, let us restate the main results treated the ionic clouds using the DH approach. This even of Blum and Ste1129for the case of primitive model applies to the more satisfying and statistically sound electrolytes. The system under consideration is therefore approach of Buff and Stillinger.27 It will be amusing to a neutral mixture of n ionic species, which are charged hard notice that the numerical solution of the Buff-Stillinger spheres of number density p i , diameter u,, and charge z, equations that we performed about 10 years ago28showed (in electron units) (1 5 i 5 n). The correlation function them to produce even higher accumulation of ions near the for the wall particle pair is go,(x),where x is the distance electrode than the GC theory. While this is not unexfrom the particle to the wall. Clearly p,go,(x) is the pected, it just underlines the necessity of including the probability of finding an ion of species i at a distance x correct form of the excluded volume effects in a workable from the wall. The indirect correlation function theory. Let us mention also in this connection that none (2.1) of these theories satisfies the second moment c ~ n d i t i o n , ~ h d x ) = goj(x) - 1 and would therefore fail to exhibit the charge oscillations together with the wall-particle Ornstein-Zernike equation that these moment conditions predict. One is therefore defines the corresponding direct correlation function naturally led into the necessity of formulating a theory of c01(x)17J8 electrified interfaces that incorporates the modern theory of electrolytic solutions (in our present work, the h,i(x) = c O ~ ( X ) 2~ZpjJo"dtJ,- t X + tds I MSA'0,12J3~'6)to the recent spectacular progress in the X t c i j ( t ) hol(s) (2.2) theory of hard i n t e r f a c e ~ , l ~since J ~ , ~both ~ theories are in good general agreement with the computer experiments the quantity cJt) in this equation is the bulk direct done on the same Hamiltonian systems. Such a theory is correlation function of the ionic pair i,j.32 When the bulk implicit in the recent general solution of the Ornsteinproperties of the fluid phase are studied, and we have to Zernike equation for hard interfaces found by Blum and solve the OZ equation for the pair correlation function St~3ll.~' In this work, which is summarized at the beginning h&), we need to know the value of the function c,(t) inside of next section, a very general formula for the density the hard repulsion core of the pair. Finding c,,(t) in this profile is given as a function of the direct correlation region is always the difficult part of the problem, to the function outside the hard wall (the closure) and the point that only a very few approximations, in which the Wiener-Hopf factor correlation function for the bulk direct correlation function is either zero or a combination phase. In the MSA of the primitive model of electrolytes of exponentials, can be solved explicitly. In the flat wall the factor correlation functions are explicitly k n ~ w n ' ~ J ~ case, however, the problem is of the classic Wiener-Hopf and the closure is just given by the total interaction potype that is extensively discussed in many textbooks. It tential of the ions with the electrode. This interaction is turns out that the wal-particle direct correlation function divided into two contributions: (a) the image charge col(x) inside the hard wall just drops out of the solution, contribution, which is a complex, many body force, and which is a most welcomed feature since it allows for a which depends on the ionic concentration around the ion solution of the very general case in which the direct and on the nature of the solid surface (dielectric or correlation function is any arbitrary function of the dis-
+
The Journal of Physical Chemlstw, Vol. 8 1,
No. 2, 1977
138
Lesser Blurn
tance xSz9A simpler way of putting it is to say since only hoi(x)for x > 4 2 is required, then to solve (2.2) we need to know
Using the convolution property of the Laplace transforms we can obtain the useful form of Agi(x) in terms of the factor correlation function Pl,(r)
Coj(x) = $j(x) (2.3) for x > uj/2 only. Our major approximation is then the assumed form of 6j(x) which, as was said above, is simply the interaction potential for the MSA. The central result of ref 29 was the reduction of the OZ equation (eq 2.2) to the ordinary integral equation
Agi(r) = - 14Zluipr&~,uwd~ @j(x+ Y )
=
XGlj(0)- Fj(x) 1 I
= - Qjrr-
277
Fj(x)
(2.4) (2.5)
Here, (2.4) is completely general, while (2.5) is valid for the MSA and GMSA.20 A summary of the main results and properties of the Baxter factor correlation function Qrj(r)are given in Appendix A. We did use also the definitions 1 hlj = -(ut - Uj) (2.6) 2
Alternatively, using Parseval's formula, we obtain Fj(x) as a real space integral
Fj(x) = 7~h1ydr$1(x + r ) Plj(r)
(2.9)
where P&) is the factor correlation function for the bulk pair correlation function hij(t), and is defined by (see also appendix A)
(2.10)
It is apparent that an explicit solution for goj(x)can be obtained by Laplace transformation, since (2.4) and (2.5) are convolutions. If we write
-@j(s){tj(is)}ij-l J
The Journal of Physical Chemlsty, Vol. 81, No. 2, 1977
Equations 2.13 and 2.15 are the basic results of our previous work (although in a somewhat different form).29 It can also be verified that for the single component case, in the exponential closure 6(x) = Ae-PX,our result agrees with that of Waisman et a1." which was obtained by a different method. Consider now the MSA for the interface. Then
(2.16) 6 j ( x ) = + P Uoj(x) where uo.(x)is the wall ion potential interaction. Perhaps we shouid mention at this point that although the MSA does not go to the right limits at low dilution concentration, it should however be a reasonably good theory for the more dense systems, and especially for the difference correlation functi~n~~l~~
Stell and collaborators have shown that the results for the bulk are quite good. It is perhaps not justified to extrapolate these conclusions to the flat wall case, but we should notice that the quantities studied in the next section are functions of precisely this correlation function difference. Therefore it is only reasonable to expect fair agreement in the more concentrated solutions. Let us now analyze the potential that goes into (2.16) for our model system. It will be the sum of two contributions, the first one due to the excess charge at the surface of the wall and the second due to the image chdrges (2.17) The first term is simply the potential energy of a charge in an uniform field35
$ y T ( ~= )(zj/eo)PeEg
(2.18)
(2.11)
(2.12)
(2.19)
(2.13)
where
F ~ ( s=) /oj,zwdxe-sx Fj(x)
(2.15)
where Eo is the bare uniform field caused by the excess surface charge at the electrode, p = l/k,T is the Boltzmann thermal factor, to the dielectric constant, and e the elementary charge. The contribution to the potential of the image charges is much more complex. If the wall is flat and homogeneous with dielectric constant E # eo, then every charge in the bulk, zit will have an image of charge
which is the Precus-Yevick hard core contribution to the wall particle pair correlation ftmction.lJ7J8 Equation 2.12 is the generalization to mixtures of the results of Henderson et al.17 and Percus.18 The external or soft core contribution in (2.11) is given by
=
x Plj(Y) PjiT(r - X )
(2.14)
located at the point
rir= (-xi,y i , z i )
-
If the electrode is metallic, then the dielectric constant is 1 ~ 1 m so that zi' = -zi and the interaction is attractive. If the wall is a biological membrane, then E < to and the image force is repulsive. Notice that the very general nature of (2.13) and (2.15) makes it possible to study more realistic interfaces in which the dielectric constant does depend on the distance to the wall x. This will be the case even for a realistic representation of a metallic interface, because the electron density is not uniform at the surface because of the surface states. It will be the case also for the technologically interesting layered electrodes, or
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Theory of Electrified Interfaces
semiconducting electrodes, and in biological membranes. In all of these cases (2.19) is replaced by a sum over infinitely many images. A very elegant method to deal with this problem has been proposed by Buff and Stillinger.27 Again here we will have to restrict our treatment to the case of a single image, leaving the other cases for some future work. Now in our system, every ion interacts not only with its own image, but also with the images of all the remaining ions in the system $i'M(Xi) =
+
However from (2.23) the integrand in (2.26) clearly tends to unity and therefore Gcj(sclx')does not seem to exist because the integral of the right-hand side of (2.26) is divergent. This difficulty is resolved when we consider the potential of a charged sheet of constant surface density
Pb)
6(x: - XI)
$(x) = JdYi b
i
1 Ir - r l /
xi)^
-
zi-zzjl :2j
1
(2.20)
In this equation, the first term represents the interaction of the charge i yith the images of the remaining ions in the system. p,(r,lr;) gives the condition21 probability of finding ion j at rj, knowing that ion i is at ri. For simplicity we will take 2
r j = (xi,0, 0)
The second term in (2.20) is just the interaction of i with its own image. Consider now the conditional probability pJ(;jl;i). When the particles i and j are far from each other, then it should be true that for a fluid interface A A
+ (x XI)^]'/^
= lim 2np(x1)[[~' L-+-
- Ix - x,I]
(2.27)
Clearly, the first term diverges like L. However it does not contribute to the field because it is constant, and merefy amounts to a shift in the potential. Therefore it can be supressed with no further consequence. Then
$(x) = -2np(x1) Ix - x1 i
(2.28)
Just as a check let us differentiate (2.28) twice. We obtain the correct result
(2.21)
lim pj(rjIri)+ p j ( x j )
r.. U
R 2 = y:
-+co
where A 4 rr.j -- r r. - r.J
-
It will be convenient at this point to define the position dependent binary correlation function
where 6(x) is the Dirac delta function. It is now clear that the same kind of constant infinity should appear in (2.25). We can therefore proceed to remove it using the above procedure. First, we replace 2
(2.22) From (2.21), the asymptotic behavior of this function is 2 -
lim gij(ri,rj) = 1
A
2
gij( ri, rj) = hij(rj, rj) + 1 -
in (2.26). Hence
(2.23)
Irijb-
Clearly also if the pair of molecules is far from the wall, then the correlation function must be that of the bulk. Then lim
x i ,xj - t
+
A
gij(rj,rj) = gij( r j j )
(2.24)
(2.30) Eliminating the constant infinity, we obtain
We remember also that ~ j ( x=) P j g o j ( x ) is the unknown quantity of our problem. It is then only natural to rewrite the integral of the first term in (2.20) as
Jd< pj(
