J. Phys. Chem. B 2000, 104, 6837-6842
6837
A Simple Model for the Diffuse Double Layer Based on a Generalized Mean Spherical Approximation W. Ronald Fawcett* Department of Chemistry, UniVersity of California, DaVis, California 95616
Douglas J. Henderson Department of Chemistry and Biochemistry, Brigham Young UniVersity, ProVo, Utah 84602 ReceiVed: February 9, 2000; In Final Form: May 11, 2000
A semiempirical method of including the effects of ionic size in a simple analytical model of the diffuse double layer is described. Guided by earlier work in the hypernetted-chain approximation, the ion-wall correlation functions are modified from their Gouy-Chapman approximations by addition of two new parameters. The first is a constant related to the osmotic pressure and thus to the volume of the ions. The second is related to the effect that ion size has on the thickness of the ionic atmosphere in the electrolyte system, and depends on the electrical field at the wall. The exact form of the potential-dependent correction is found using Monte Carlo estimates of the potential drop across the diffuse layer φd. In this way, an analytical expression for the potential drop is derived and used to estimate φd as a function of electrolyte concentration and charge density on the wall for 1-1 electrolytes assuming that the component ions have a diameter of 425 pm.
Introduction The theory of the double layer was a subject of intense theoretical interest in the early 1980s.1 Until then, the widely used model for the diffuse layer was that by Gouy2 and Chapman.3 This well-known theory treats the problem at the primitive level and ignores the finite size of the ions. One approach to improving the description of the diffuse layer4-9 is based on the use of integral equations derived from the Ornstein-Zernike equation. The simplest of these is based on generalized mean spherical approximations (GMSA) which lead to an analytical expression for the potential drop across the diffuse layer.4,6,9 However, this approach met with limited success because the GMSA was used in a way which emphasized linear approximations whereas the variation in diffuse layer properties with electrical field is very nonlinear. Another approach was to apply the hypernetted chain (HNC) integral equation in a simple noniterative method.5,6 This technique provides reasonable results at low charge densities. An important milestone in the development of the theory of the diffuse layer was the Monte Carlo (MC) calculations carried out by Torrie and Valleau.10,11 These calculations show the effects of finite ion size for double layers formed in a dielectric continuum. They were used by Henderson and Blum6 to assess results obtained by the HNC/MSA approach in which the MSA is used to describe ion-ion interactions in the bulk of the solution and the HNC approximation used for wall-ion interactions. In the case of 1-1 electrolytes excellent agreement between the MC results and those of the HNC/MSA model were obtained for a reasonable range of electrode charge densities. The problem with the HNC/MSA approach is that it does not yield an analytical solution to the problem. Therefore, it is
not attractive to experimentalists who prefer to have analytical relationships for the analysis of experimental data. In the present paper, we show that a simple analytical model for the diffuse layer which considers the effects of finite ion size may be derived using the HNC and GMSA results as guides. Furthermore, we show that the results of this model are in excellent agreement with the MC data of Torrie and Valleau for 1-1 electrolytes.10 Noniterative Method for Estimating the Potential Drop Across the Diffuse Layer Using the HNC/MSA Following the earlier work of Henderson and Blum,5,6 the HNC equations describing ion correlation with a charged wall located in a dielectric continuum are
gws(x) ) Ξ(x) cosh{H(x) + fφ(x)}
(1)
gwd(x) ) Ξ(x) sinh{H(x) + fφ(x)}
(2)
where Ξ(x) and H(x) are functions which are defined below, φ(x) is the potential at position x in the diffuse layer, and f ) F/RT. The correlation functions gws(x) and gwd(x) are defined as
gws(x) )
gw1(x) + gw2(x) 2
(3)
gwd(x) )
gw1(x) + gw2(x) 2
(4)
where gw1(x) is the pair correlation function between the electrode wall and the cation of the electrolyte, and gw2(x), that between the wall and the anion. The functions needed to solve
10.1021/jp000507w CCC: $19.00 © 2000 American Chemical Society Published on Web 06/30/2000
6838 J. Phys. Chem. B, Vol. 104, No. 29, 2000
Fawcett and Henderson
these equations are
ξ(x) ) 2πF
∞ scs(s) ds ∫-∞∞ [gws(t)-1] dt ∫|x-t|
(5)
∫0∞ gwd(t) dt ∫|x∞- t| scdsr(s) ds
(6)
Ξ(x) ) exp{ξ(x)}
(7)
H(x) ) 2πF and
The functions cs(r) and cdsr(r) are direct correlation functions defined for the bulk of the solution, and F is the number of ions per unit volume. The coordinate system is chosen so that x is the distance from the wall; x ) 0 corresponds to the distance of closest approach of the ions which all have a diameter of σ, and x ) -σ/2 corresponds to the position of the wall. The problem is solved at the restricted primitive level assuming that the solvent is a uniform dielectric with a relative permittivity of s and that the ions are hard spheres with equal diameters for cations and anions. At first the functions Ξ(x) and H(x) are evaluated at x ) 0. The expression for the first integral involving the direct correlation functions is then
∫ sc ∞
t
s ds ) -
Figure 1. Values of HNC function Ξ(0) plotted against electrode charge density σm assuming an ionic diameter of 300 pm for various concentrations of a 1-1 electrolyte: 0.01 M (O), 0.1 M (0), 0.2 M (4), 0.5 M (3), 1 M (]), and 2 M (").
c1(1 - t2) ηc2(1 - t5) - 2ηc2(1 - t3) , t 1
(8)
where
c1 ) (1 + 2η)2/(1 - η)4 c2 ) -(1 + η/2) /(1 - η) 2
η ) πFσ3/6
(9) 4
(10)
Figure 2. Values of the HNC function H(0) plotted against electrode charge density σm assuming an ionic diameter of 300 pm for 1-1 electrolyte concentrations in the range from 0.01 to 2 M (see legend for Figure 1).
(11)
was assumed that
The expression for the second integral is
∫ t
∞
(
)
3 qΓ Γ (1 - t ) scd (s) ds ) q(1 - t) 1 - t2 , 1+Γ 3 (1 + Γ) sr
(12)
where
(15)
gwd(t) ) sinh(fφ(t))
(16)
tanh(fφ(t)/4) ) tanh(fφd/4)e-κt
(17)
and
0
