Explicit formulae for the ion and solvent profiles in the electric double layer using the mean spherical approximation

Chemical Physics 141 ( 1990) 79-86 North-Holland

EXPLICIT FORMULAE FOR THE ION AND SOLVENT PROFILES IN THE ELECTRIC DOUBLE LAYER USING THE MEAN SPHERICAL

APPROXIMATION

D. HENDERSON IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, CA 9512G6099, USA Received 28 March 1989

Previously the mean spherical approximation integral equations have been solved analytically for a model double layer consisting of charged hard sphere ions and dipolar hard sphere solvent molecules near a flat charged hard wall. In that work, results for the double layer potential were obtained. However, no useful expressions for the ion and solvent profiles were given, although a few results for their Laplace transforms were stated, mostly without proof. In this paper we obtain formulae for these. Laplace transforms at low concentrations and invert these expressions to obtain explicit formulae and numerical results.

1. Introduction Until very recently, all theories of the electrochemical interface were based upon what is called the “primitive” model of the electrolyte. In this model, the molecular nature of the solvent is neglected. The solvent is represented by a dielectric continuum whose dielectric constant is that of the bulk solvent. The advantage of such “primitive” models is their simplicity. Unfortunately, they lead to poor agreement with experiment. Agreement with experiment can be obtained by means of semi-empirical modifications of this “primitive” model. A compact layer next to the electrode is postulated. This compact layer is presumed to be a monolayer of solvent molecules whose properties are obtained in some ad hoc manner. No attempt is made to require that this monolayer has the same free energy as the rest of the interfacial layer which is referred to as the diffuse layer. Experimentally, it appears that at high concentrations the capacitance of the interface is independent of the electrolyte concentration. This seems to lend support to the idea of a compact layer of solvent molecules. However, conceptually at least, the idea of a special layer near the electrode is unsatisfactory. A theory in which all the ions and solvent molecules are treated in a consistent manner would be much more satisfactory. 0039-6028/90/S 03.50 0 Elsevier Science Publishers B.V. (North-Holland)

Such a conceptually satisfactory theory was developed a decade ago by Camie and Chan (CC) [ 1 ] and Blum and Henderson ( BH ) [ 2 ] who solved the mean spherical approximation (MSA) integral equations for a model double layer consisting of a mixture of charged hard spheres (the ions) and dipolar hard spheres (the solvent molecules). Even though a dipolar hard sphere is an overly simplified model of a solvent molecule (especially water), it is an improvement over a continuum. More importantly, no special compact layer was introduced in the MSA. CC and BH obtained a general expression for the double layer potential which is implicit and useful only for numerical calculations. However, an explicit expression, useful at low to intermediate concentrations, was obtained by a two-term expansion in powers of (concentration) ‘I2 . The first term in this expansion is concentration dependent and is the usual diffuse layer term obtained for the “primitive” model theories. The second term is concentration independent and plays the role of the compact layer and has properties which are of the correct order of magnitude (without semi-empirical adjustment ). Even more importantly, some numerical values of the solvent profiles were given in these papers. These profiles make it clear that there is no special compact layer for the solvent molecules. The interfacial region for the solvent molecules is as thick as that of the ions

80

D. Henderson /Ion and solvent profiles in the electric double layer

(i.e. the diffuse layer). The notion of a compact layer is artificial. In addition, BH give analytical results for the Laplace transforms of the ion and solvent profiles. Again the results are implicit. Some expressions for the low concentration expansion of these formulae have been reported [ 31 but without proof. In this paper we outline a derivation of these formulae and find that the previously stated result for the dipole profile at low concentrations is correct but that the previously stated result for the ion profile at low concentrations is in error. Further, we give explicit formulae for the inversion of these Laplace transforms in the form of a series and give some numerical results. These formulae permit rapid calculation of the ion and solvent profiles which are consistent with the potentials and capacitances obtained earlier by CC and BH.

For the mode1 of the electric double layer used in this paper, these equations are applied to a mixture of charged hard spheres (ions) of diameter di and dipolar hard spheres (solvent molecules) of diameter d, in the presence of a single charged hard sphere of exceedingly large diameter d,,. The density p. of this large sphere is such that both p. and podi vanish. For this system

(da) where z,< is the charge of a sphere of species i, e is the elementary charge and i and j are the charged hard sphere subscripts, R12
39

COS0,s,

RI2

>dis

2

Cab)

R:2

2. Theory The MSA consists of the Ornstein-Zernike equation

•t

1 Pk k

hik(RI3)Ckj(R23)

dr3

3

(OZ)

(1)

s

~s(R12)

wheregi,(R) =/Q(R) + 1 is the pair correlation function and cij(R) is the direct correlation function, together with the ansatz h,(R)=-1

where ~1is the dipole moment of the solvent molecule, the index s is the solvent subscript and cos 0,s gives the orientation of the dipole relative to the vector between centers of the ion and solvent molecule. Finally

)

R
R12

=a~

,

=-

$0(12)

) R,z >ds,

3 R>dij.

where

(2a) (4d)

(2b)

In eqs. ( 1) and (2), pk= Nk/ V, where Nk is the number of particles of species k and I/is the volume of the system, and /3= 1/kBT, where kB is Boltzmann’s constant and T is the temperature of the system. The parameter du is defined by dij=(di+d,)/2)

(4c)

12

0(12)=3~~.R12)(li2.R12)-~il.~2r C,(R)=-puu(R)

cd,,

(3)

where di is the diameter of a particle of species i. Finally, Uij(R) is the pair energy between particles of species i and j. Eq. (1) is just the definition of the direct correlation function and eq. (2a) is an exact condition reflecting the fact that the particles have a hard core. Eq. (2b) is the approximation.

withA and d,, being unit vectors. In the mode1 double layer specified by eqs. (4a)-(4d) there are no image forces. The solutions of CC and BH are both based on the factorized form of the OZ equation due to Baxter [ 41. Both CC and BH found that goi

=go(x)

go,(x)

=go(x)+ff2(x)

+ziHi (x)

T

9

(5)

(6)

where x=R -do/2 and go(x) is the profile for hard spheres near a hard wall. For simplicity, we take the ion and solvent diameters to be equal, di = d, = d. The subscript 0 denotes the large charged sphere which,

81

D. Henderson /Ion andsolvent profiresin the electricdouble layer

in the infinite limit &+cc considered here, becomes a flat charged hard wall. The Laplace transform of g, ( X) is known from the work of Lebowitz [ 51. Henderson and Smith [ 61 have inverted this transform in a series of the form

RI0)

~cA21 + -y

-PclE,(S)

+H2(s)

(12) go(x)= f, &s(x) u(x--nd+d) ,

(7)

where U(X) is the unit step function,

x
u(x)=O,

=l,

s-0.

(8)

Henderson and smith obtained expressions for the gg (x) for n G 5. Subsequently, Henderson [ 71 has given expressions for the g; (x) for all n. CC and BH give general expressions for the double layer potential, which results from the sum of the first moment of H, and the zeroth moment of Hz. This general result is implicit. However, an explicit result can be obtained by expansion to first order in powers of the Debye parameter, K, given by (9)

where the Q:(s) are the Laplace transforms of the Baxter Q functions [4]. Expressions for the Q,(r) have been given by CC. The parameter E= 4x0 is the electric field at the surface of the large sphere and u is the uniform charge density on the surface of the large sphere. The parameter A, and the parameters appearing in the Q,(r) are from the simultaneous solution of eqs. (4.26a) to (4.26k) of CC. Solving eqs. ( 11) and ( 12) for E?, (s) and fi2 (s) gives R,(S)=-

se

x AI, [I-

f~s&2(~)

I+ fA2,~s&2W

(13)

D(s)

r7,(s)=-$e

x

-A,,P,@,

(~1

+A21

[ 1

-P&,

(~11

D(s)

2

(14)

>I

(15)

where where Eis the dielectric constant and D(s) =s PC= c zfp’pi.

(10)

I

x

BH also give general expressions for the Laplace transforms of H, and Hz. These too are implicit and not readily usable. Explicit expressions, based on an expansion in powers of K, have been given without proof. Taking the Laplace transforms of eqs. (4.43) and (4.44) of CC gives

HI(S)

1-Pc&,w+

+H2(s)

PEA,, ---y

,,(z>$,

(11)

(

K

I-

;~s&z(s)

l-p&,(s)+p+

)

> -P&,(S)+

P~AZI --y

.

Eqs. (13)-(15)arethesameaseqs. (52) and (53) of BH, apart from slight differences in notation. Thus, it has been established that the methods of CC and BH both yield the same Laplace transforms. In eqs. ( 13) and ( 14) and in what follows, the wall is assumed to be at X= -d/2 so that the plane of the closest approach of the ions and dipoles is at x= 0. Eqs. ( 13)-( 15 ) are correct but not too useful for computation since A,, and A2, and the parameters determining the Q, are defined implicitly. Explicit results for R, (s) and E?,(s) can be obtained by ex-

D. Henderson /Ion and solvent proJles in the electric double layer

82

pansion in power of K. This procedure has already been used by CC and BH to obtain explicit results for the double layer potential. Expansion in power of ic gives

and @2(s)

=

1 -s+s*/2-e-” s3

Expanding

(16)

(27)

.

D (s ) gives

D(s)=A(s)(s+K)+....

(28)

Using these results, we have to order K (17) and &(.s)=

++-1)-- 8:

27rj3e2 - &

b/3,2

(18)

x ( 1+ 4@P2P4,2)@, (s) +... ,

1 +

x

e;&(s) = - 2K;y

s (S+K12

@,(s) +... )

28

(19)

b2@2(s)+M,(S)

8;

[

(29)

A(s)

(

and f~sQl2b)

= 2scpepBi 3838,2

Ps[b2@20)

+a$,

(s) 1

!!p

g*(S)=-

3

(20)

+ ... , hQ52@)=

s

x

[B~2(~)+8,2~,(~)1+...

>I 1

=1-A(s)+..., where/I=

(21)

1lkBTas before but

jj=1-(-1)“/3bz

and E is the dielectric constant. and E are related by 4MP2PS _ E-l 3 e

(22)

n’

The parameters

/I:

b2, ,u,

’ (s+K)A(s)

(30)

.

Eq. (30) is the same as that stated previously [ 3 ] but eq. ( 29 ) is not. Since the results in ref. [ 3 ] were stated without proof, the results given here are to be preferred. Expansion of fi, (s) and Hz(s) in power of s gives, to order K,

(23)

Pd

(31)

and Hz(s)= l3:/3:2

9

(24)

E=Bt’

Eqs. (23) and (24) can be combined

x

to give

(25)

(

1+,-l)

>

+0(s))

where (33)

J.= 83lP6 . a familiar form which is due originally [ 8 1. Finally

to Wertheim

In obtaining

eq. (32) the following identity

/%--/%=28,&--83) 9 I (s)- -

1--s-e-S s2

(26)

(32)

and the expansions

(34)

D. Henderson /ion and solvent projiles in the electric double layer

@I(S)=--f+w)

(35)

9

&(s)=;+W), and 5

A(s)=I%IBa+o(s)

(37)

were used. Eqs. (3 1) and (32) were obtained earlier by CC and BH. The contact values of HI (x) and Z-I2(x) can be obtained from lim sE?Js) . s-co

Thus, to order H

(38)

(44)

(K+S)[M(S)+N(S)eS].

Oncef(x)

is obtained,

(45) and

Y-‘(

(K!:;(S))

=--f’((x-d))+f’(x)-f”(x).

K

(()),_@

I

es

(36) =

H,(O)=

83

(39)

EK

(46)

Using eq. ( 38 ) , the boundary its derivatives are

conditions

f(O)=f’(O)=f”(O) pfl

H2(0)=-83B;

-

.

Eqs. (39) and (40) were obtained

(40) earlier by BH.

Eqs. (29) and (30) are too inversion. Numerical inversion venient at low concentrations H2 (x) are long ranged. Thus, a nique of Smith and Henderson To illustrate, we invert

complex for analytic is possible but inconwhere H,(x) and variation of the tech[ 6,7,9] is useful.

1

and f”(O)=l.

(48)

s3eS ’

m= && (--l)“-’ p”-‘(S)

N”(s)

(41)

e-(n-‘)s.

“3+Br’3

,

s, =

- $a+wA

s2 =

-ja+co2A’/3+coB1/3

(50)

1/3+02~1/3 (51)

:

(52)

and

where

S3=-K,

M(s)=

(49)

The inverse ofr( s) is determined by the zeros of the denominator which are located at So= - fa+A

(K+S)A(S)

= (K+S)[hf(S)+N(S)e']

(47)

Note, we define f (x), f’ (x), and f )) (x) to be zero for negative arguments. Expanding

3. Inversion of H,(s) and l&(s)

F(s)=Y[F(x)l=

on f (x) and

4

(S/%2+83)

(42)

6

where o=e2ni/3

and (43)

(53)

02=1/w,

(54) (55)

i being ( - 1) ‘j2, Since multiplying to differentiation

by s in Laplace space corresponds in real space, we may consider

a= &//?a ,

(56)

D. Henderson /Ion and solvent profiles in the electric double layer

84

A

=

b,( 108+ 1862-b:) 1OS/I2

‘+

26: (108+18bz-b;)*

>

112 1 ’

(57)

m
(62)

and

and

dk B

=

b2( 108+ 18b,-b;) 108/?:

‘+

2b: (108+18b2-b:)2

>

l/2 1 .

The above formulae come from the standard tion of a cubic equation. Thus f(x)=

F L(x)

u(x-nd+d)

>

Cn,k(ti)

(58)

-K)

N”(-K)

’

exp

[ -K(X-tZnd+d)

(- 1)+-l i lir,, d”-’ (n- 1 )! i=,, t+li dt”-’ M”-‘(t) (t+K)N"(t)

(64)

,io t-

’ )’

can be obtained

&- . :$;(‘?, )

from

(65)

where $[($$)“I,

mtn.

(66)

The B,,,,(t) can be obtained from eq. (27) of ref. [7] byreplacingS(t),S,(t),S2(t),and&(t)byN(t) and

1

2

N,(t)=N’(t)

(f-t,)”

exp[t(x-nd+d)]

(63)

k
from eq. (28 ) of ref. [ 7 ] byM(t) and

Using the chain rule B&(t)

B,,,(t)=

f,(x)=(-l)“-’

+

The C,,k( t ) can be obtained byreplacingL(t) andL(t)

&,k( t ) =

where

x W-‘(

3

M,(t)=M’(t). solu-

(59)

n=l

~k[““-‘(‘)]tdi

&(t)=2N;(t)=N”(t))

>

.

(60)

iv3(t)=gv;(t)=N”‘(t).

To invert fi, (S ) , we invert first

All that remains are the differentiations in eq. (60). These can be done with the standard chain rule formulae. Hence

I=

J(s) =Y[J(x)

1

(68)

(K+s)2A(s)

s3eS =

.

(K+s)2[M(s)+N(s)e~]

f,(x)=(--1)“-’ Again we consider x Mn-l( N”(

-K)

exp{-K[X-(n-l)d])

-K)

+ (-l)“_’ (n_l)!

j’(s)=Ylj(x)]

Once j( x) is obtained, J(x)=Y-’

I)! (x--nd+d)“-“-’

and

m! k=~k!(m-k)!

where

(69)

2

i&exP[ti(x-nd+d)l

n-1 (n-l)! xrn;, m!(n_m_ Xf

es

=

Bn,k(ti)Cn,m-k(fi)

9

(61)

1 (K+S)2A(S)

(

>

=Y”(x)

(70)

D. Henderson /Ion and solvent profiles in the electric double layer

91(s) (K+syA(s)

y-’ (

>

=j'(x-d)+j'(x)--j"(x),

(

y-’

(71)

85

negative of the derivative of eqs. (6 1) and (65 ) with respect to K. We have developed a program to calculatef(x) and j(x) and its derivatives for OGX< 5d. Some numerical results are give in section 5.

@2(s)

( K+s)2A(s)

>

=-j(x-d)+j(x)-j'(x)+sj(x).

(72)

4. Some numerical results

The boundary conditions on j( x) and its derivatives are

In fig. 1 results for F(x) are plotted. For comparison, the approximation

j(O)=j’(O)=j”(O)=j’“(O)=O

F(x)=Wexp(-=)/B3

(73)

and j”” ( 0 ) = 1 .

(74)

From this point, things proceed in exact analogy to the inversion off(x). Hence j(x)=

f jn(x)U(x-nnd+d)

n=l

,

(75)

where (n- 1Pf,

j (x) = M”-’ ”

N”

is also plotted. Eq. (77) has the same area as F(x) and is a good approximation to F(x) at large x. At higher concentrations, F(x) is sufficiently short ranged that the use of eq. (79) is of marginal value. However, at exceedingly low concentrations, where Kd is small, F(x) is of such a long range that eq. (79) is needed to supplement the values of F(x) obtained from the formulae reported here for 0 < XQ 5d. Results for H2(x) are plotted in fig. 2. Again for comparison, values obtained from Hz(x)

-- nN, N

A4

(

j?;[l+fKd(l-A-‘)]

H,(o)= /%(1- td%/&B,z) xexp( --Kx)

-x+nd-d

exp[

-K(X-d+d)]

n-1

(n-l)! l)!

(80)

are also plotted. Again eq. (80) is a good approximation at large x. Both figs. 1 and 2 show oscillations indicating layers of strongly and less strongly oriented dipoles near the surface. However, the most notable feature of

>

XC m=lJm!(n-m-

(79)

(x-nnd+d)“-“-I

1.0

Dn,k(&)Cn,m-k(ti) 7

(76)

and z

m
0.5

(77)

Using the chain rule Difl=

c i=O

(-I)‘-

(i+ 1)k! Bn,k__i( t) (k-i)!

(t+K)‘+’

0.0

’

(78)

Eqs. (76)-( 78) can also be obtained by taking the

Fig. 1. Results for F(x) for d=0.276

nm,e=78.4, and T=25”C.

D. Henderson /Ion and solvent projZes in the electric double layer

86

5. Conclusions An explicit set of formula are given for the inversion, to order K, of A, (s) and fiZ (s), the Laplace transforms of the electrode-ion and electrode-solvent profiles, as obtained from the mean spherical approximation. These formulae are valid for all x. A program for XG 5d has been developed. Supplementary approximations for larger x can be used.

1M

0

1

2

3

x/d

Fig. 2. Results for H*(x) for d=0.276 nm, ~~78.4, and T=25”C.

these figures is the fact that HZ(x) decays asymptotically as exp ( - KX). As a result, the interfacial region of the solvent molecules is as thick as that of the ions. This feature has been pointed out earlier [ 1,2] by CC and BH and is physically reasonable but is lacking in the conventional picture of the compact inner layer against the wall followed by a diffuse layer in which the solvent molecules are unaffected by the electrode. We have tested our results by comparison with a numerical inversion of Hi (s) and HZ(s). Except for exceedingly low concentrations, the agreement is excellent. The problem at low concentrations lies with the numerical inversion which is hard to implement because Hi (x) and HZ(x) are slowly varying. No doubt this can be overcome by experimenting with different truncations and numbers of integration points and interval widths. The advantage of the formulae given here is that they can be applied at all concentrations.

Acknowledgement This work was commenced in 1988 while the author was the Manuel Sandoval Vallarta Visiting Professor of Physics at the Universidad Autonoma Metropolitana/Iztapalapa, Mexico. The hospitality of UAM, the Sandoval Vallarta Foundation, Dr. M. Lozada-Cassou and Mrs. Manuel Sandoval Vallarta are gratefully acknowledged.

References [I] S.L. Camie and D.Y.C. Chen, J. Chem. Phys. 73 (1980) 2949. [ 2 ] L. Blum and D. Henderson, J. Chem. Phys. 74 ( 1980) 1902. [ 31 D. Henderson and L. Blum, J. Electroanal. Chem. 132 ( 1982) 1; D. Henderson, L. Blum and M. Lozada-Cassou, J. Electroanal. Chem. 150 (1983) 291. [4] R.J. Baxter, J. Chem. Phys. 52 (1970) 4559. [S] J.L. Lebowitz, Phys. Rev. A 133 (1964) 895. [6] D.Hendersonand W.R. Smith, J. Stat. Phys. 19 (1978) 191. [ 7 ] D. Henderson, J. Colloid Interface Sci. 121 ( 1988) 486. [8] M. Wertheim, J. Chem. Phys. 55 (1971) 4291. [ 91 W.R. Smith and D. Henderson, Mol. Phys. 19 ( 1970) 4 I 1.