J. Electroanal. Chem., 111 (1980) 217--222 217 © Elsevier Sequoia S .A, Lausanne -- Printed in The Netherlands

A SIMPLE NON-ITERATIVE METHOD FOR CALCULATING THE POTENTIAL OF AN ELECTRIC DOUBLE LAYER

DOUGLAS HENDERSON

IBM Research Laboratory, San Jose, CA 95193 (U.S.A.)

and

LESSER BLUM

Department o f Physics, University o f Puerto Rico, R~o Piedras, Puerto Rico 00931 (P.R.)

(Received 12th November 1979; in revised form 4th February 1980)

ABSTRACT

A simple non-iterative method for calculating the potential of an electric double layer is developed. The method is presented here within the framework of the hypernetted chain (HNC) approximation However, the basis of the method is general so that applications to other approximations are possible. The method is based on the Henderson--Blum equation for the contact value of the density profile and on the fact that even though the HNC poten- tial differs significantly from the Poisson--Boltzmann (PB) potential, the HNC density and charge profiles differ only slightly from the corresponding PB profiles. The agreement of the HNC potential, calculated by this method, with the results of more elaborate iterative cal- culations is good.

INTRODUCTION

Much of the recent work on the theory of the electric double layer has focussed on the use of integral equations for the density and charge profiles, p(x) and q(x), respectively. For example, we [1--3] and Carnie et al. [4] have used the hypernet ted chain (HNC) integral equation, Outhwaite et al. [ 5,6] have used the modified Poisson--Boltzmann (MPB) integral equation and Croxton and McQuarrie [7] have used the Born--Green--Yvon (BGY) integral equation. Each of these integral equations take into account the finite size of the ions and reduce, for point charges, to the classic Poisson--Boltzmann (PB) theory of Gouy [8] and Chapman [9].

In this note we consider only the HNC equations. However, the non-iterative technique which we develop should be applicable to the other equations. For a model ionic fluid consisting of charged hard spheres of charge +ze and diameter a in a medium of uniform dielectric constant e near a charged hard wall located at x = --a/2, the HNC equations are as follows:

1 ln[hw~(X) --hwd(X ) + 1] 1 ln[hw~(X) + hwd(X) + 1] +

o o o o

= 27rp f hw,(t) dt f sc~(s) ds ( la) --~ Ix --tl

2 1 8

I ln[hws(X ) + hwd(X) + 1] -- 1 ln[hws(X ) _ h w d ( x ) + 1]

= 27rp ? hwd(t ) d t f SCSdr(S)ds 0 Ix -- t l

+ 27rq ; ( I x - - t l + x + t) hwd(t ) d t ( lb ) o

where hws (x) and h w d ( X ) a r e related t o p(x) and q(x) by

p(x) = p [hws(X ) + 11 (2)

and

q(x ) = --zephwd(X ) (3)

where p is the number of ions per unit volume in the bulk ionic fluid, q = [3z2e2/ea, [3 = 1/kT and cs(r) and c~ r are the non-coulombic parts of the bulk fluid direct correlation functions. For convenience, the unit of length used in the calculations is chosen so that a = 1. Image forces are neglected in eqn. (1). In principle, it is possible to realize this situation physically by choosing an electrode made from a material whose dielectric constant is equal to that of the ionic solution. However, metallic electrodes are more usual. For a metallic electrode the contributions of the image forces should be calculated. Before embarking on the development of a systematic theory for these mul t ibody forces, we feel that first it is necessary to develop satisfactory systematic tech- niques for treating the additive Coulomb force considered here.

The functions cs(r) and C~dr(r) are properties of the bulk ionic fluid and are obtained by a separate calculation. For example, they may be calculated in the HNC or mean spherical (MS) approximations. We have adopted the former procedure [3], while Carnie et al. [4] have used the latter procedure. Following Carnie et al. we refer to the two procedures as the HNC/HNC and HNC/MS approximations. One purpose of this note is to examine the difference between the HNC/HNC and HNC/MS approximations.

Equations (1) may be rewritten:

hws(X) + 1 = exp(~(x)) cosh(~?(x) + [3ze¢(x)} (4)

hwd(X) = exp(~(x)} sinh(r/(x) + [3ze¢(x)}

~(x) = 27rp ; hws(t )dt f SCs(S) ds - - ~ I x - - t l

r/(x) = 27rp ? hwd( t )d t f f

and

scg(s) ds Ix--tl

(5)

(6)

(7)

flze¢(x) = 2rrpq ; ([x - - t [ + x + t) hwd(t ) dt 0

(8)

219

To determine hw~(x) and hwd(X) from eqns. (1) or (4) and (5), an initial guess for these functions must be made, the integrals evaluated and new iterates obtained. The procedure is repeated until a solution is obtained. Using our iterative scheme [3] this procedure is very t ime consuming at high fields and low concentrations. No doub t the iterative schemes of others share this problem to a greater or lesser degree. The main purpose of this note is to out- line a non-iterative method for obtaining approximately the diffuse layer po- tential, ~ = ¢(0), from the HNC equations. The method is simple, fast and quite accurate. The penalty paid is that hws(X) and hwd(X) are not obtained. How- ever, ¢ is a quant i ty of greater interest since it is more closely related to experi- ment than either hws (x) or hwd (x). Experimentally, the total potential differ- ence, V, is measured. For our model, V = E a / 2 + ¢. In real physical systems, image forces and solvent effects contr ibute to V so that ¢ cannot be obtained rigorously from V. However, there is some evidence that these extra effects, not considered in this note, mainly affect the inner layer potential so that V = E d / 2 + ¢, where ¢ is the same as that given in our simple model and d is an empirical parameter describing the effective width of the inner layer and includes such solvent effects as the change of dielectric constant near the elec- trode. A further reason why ¢ is o f greater interest than either hw~(X) or hwd(X) is that, within the HNC approximation at least, ~ shows greater deviations from the PB results than do hw~(X) or hwd(X). As a result, ~b is probably of greater interest in the comparison of the results of various approximations.

NON-ITERATIVE METHOD

The basis of our method is the Henderson--Blum result [1,10]

hws(O) + 1 = a + b2/2 (9)

where b = (JzeE/g. is a dimensionless measure of the charge density (eE/4~r) on the wall and ~ = (41rpq)1/2 is the Debye screening length. With the constant, a, given by the osmotic pressure, p / p k T , eqn. (9) is exact. Equation (9) is satisfied by the PB theory when the constant is unity and by the HNC theory when the constant is (1 + {J3p/ap)/2. Previously [3], we had speculated on the basis of numerical results that , in the HNC approximation, the constant might be ({j3p/3p)in and that, even with the value of a, eqn. (9) might not be satisfied exactly by the HNC approximation. However, Carnie et al. [4] have shown both speculations to be incorrect.

Substituting eqn. (4) into (9) gives

a + b2/2 = exp(~(0)} cosh(~(0) + flze¢} (10)

where a is exp(~(0)} evaluated when b = 0. Thus, if ~(0) and ~?(0) can be cal- culated, ¢ can be obtained. In the PB approximation, ~(0) = 77(0) = 0 and ¢ q: 0. Thus, to first order in the difference be tween the PB and HNC profiles, ¢ can be obtained from eqn. (10) by using ~(0) and ~/(0), calculated from eqns. (6) and (7) using the PB hws (x) and hwd (x). Since the difference of hws (x) and hwd(X) from the PB results is small, bo th in the HNC approximation [3,4] and in computer simulations [ 11], this should be an excellent approximation.

220

0 08

0 06

i> 004

-e-

0 02

o / / o . . / o / ~ . o ~

/ .//

/ !

I I I

/ / / / - - o . . . . . _ ~

I I I I I 0 5 10 b 15

Fig. 1. Diffuse layer po ten t ia l d i f ference for a 1 molar ionic fluid (o = 0.276 nm, z = 1, T = 298 K, e = 78.4) near a hard planar wall as a func t ion o f b = •zeE/K. The open and solid circles give the i terative results [ 3 ] for the HNC/HNC and HNC/MS app rox ima t ions respec- tively, while the solid and b roken curves give the results o f the p resen t non-i terat ive m e t h o d for the HNC/HNC and HNC/MS app rox ima t ions respect ively.

RESULTS

In Figures 1--3 values of ¢, calculated by this method, are compared with our iterative results [3] and those of Carnie et al. [4]. At low concentrations the agreement is good. At the higher concentrations the agreement is good for those values of b which are of experimental interest, but deteriorates for large b.

In Fig. 4 we compare ¢, calculated by this method for the HNC/HNC and HNC/MS approximations, with the PB theory results. At low concentrations, the difference between the HNC/HNC and HNC/MS results becomes large. The

0 06

> 004

-8-

0 02

I I I I ~ I

o ~

. / ./

/ I I I I I I I I I

2 4 6 b 8 10

Fig. 2. Diffuse layer po ten t ia l d i f fe rence for a 2 molar model ionic fluid (o =0.276 nm, z = 1, T = 298 K, e = 78.4) near a hard planar wall as a func t ion o f b = ~JzeE/K. The circles and curve give, respect ively, the i terative and present non-i tera t ive results for t he NHC/NHC approx imat ions .

221

01(3

>

005 / i i

o

I I I I I I 0 10 20 b 30

Fig. 3. D i f f u s e l a y e r p o t e n t i a l d i f f e r e n c e f o r a 0 .1 m o l a r m o d e l i o n i c f lu id ( a = 0 . 4 2 5 n m , z = 1, T = 2 9 8 K, e = 7 8 . 5 ) n e a r a h a r d p l a n a r wal l as a f u n c t i o n o f b = ~JzeE/~. T h e c i rc les a n d c u r v e give, r e s p e c t i v e l y , t h e i t e r a t i ve r e s u l t s [4 ] a n d p r e s e n t n o n - i t e r a t i v e r e s u l t s f o r t h e H N C / M S a p p r o x i m a t i o n .

HNC/MS results are close to the PB results for the experimentally reasonable charge densities considered. At higher charge densities the HNC/MS results deviate appreciably from the PB results. In particular, the HNC/MS ¢ also has a maximum. However, the HNC/MS maximum in ~b is at a much larger value of the charge density than is the case for the HNC/HNC approximation. At higher concentrations the difference between the HNC/HNC, HNC/MS and PB results is smaller.

o 31 f r ~ ~ ~ _.__, . . . . . . . . . . .

I " . . . . . . . . . . . ~ S - f o4 ~ _ S _~_~ ~ . . . . . . . . . . . . . . . . . . . . . . . . .

/~ .,~/~..~..~.; ~..~_. O01M

0 1 /~/~

O ~ .-~ "~'~ ~" ~ ~"C- ............ IM

01 02 03 C h a r g e d e n s f t y / C m -2

Fig. 4. D i f f u s e l a y e r p o t e n t i a l d i f f e r e n c e f o r a m o d e l i on i c f l u id (o = 0 . 2 7 6 n m , z = 1, T = 2 9 8 K, e = 7 8 . 4 ) n e a r a h a r d p l a n a r wal l as a f u n c t i o n o f c h a r g e d e n s i t y o n t h e wall . T h e c u r v e s m a r k e d - - - - - - , . . . . . . a n d - - give t h e p r e s e n t n o n - i t e r a t i v e r e s u l t s f o r t h e PB, H N C / M S a n d H N C / H N C a p p r o x i m a t i o n s r e s p e c t i v e l y .

222

C O N C L U S I O N S

The non-iterative method presented here gives potentials which agree well with the results o f more elaborate and time-consuming iterative calculations. Although results are given only for the HNC approximation, the method is also expected to be useful in other approximations.

At low concentrations there are large differences between the PB, HNC/HNC and HNC/MS results. The HNC/HNC is more logically consistent than the HNC/MS approximation and is to be preferred on a purely intellectual basis. The question of which is more accurate cannot be answered until the latest computer simulations of Torrie and Valleau [12] are available.

ACKNOWLEDGEMENTS

The authors are grateful to Dr. D. Chan for a preprint of his paper. L.B.'s research supported in part by the University of Puerto Rico, Center of Energy and Environment.

R E F E R E N C E S

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1 0 D. H e n d e r s o n , L. B l u m a n d J .L . L e b o w l t z , J . E l e c t r o a n a l . C h e m . , 1 0 2 ( 1 9 7 9 ) 315 . 11 G. Torrie and J . Va l l eau , C h e m . Phys . L e t t . , 6 5 ( 1 9 7 9 ) 3 4 3 . 12 G. Torrie and J . V a l l e a n , to be publ i shed.