AD-A~iR 997 PUERTO RICO UNIV RIO PIEDRAS DEPT OF PHYSICS F/6 7/4THE APPLICATION OP THE MODIFIED GOUY-CHAPMAN THEORY TO AN ELECT--ETCIU)SEP 82 L B SHUIYAN, L BLUM, D HENDERSON NOOOIIIBI C 0776

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The Application of the Modified Gouy-Chapman Theoryto an Electrical Double Layer Containing Asymmetric Ions

by

L. B. Bhuiyan*, L. Blum*, and D. Henderson**

Prepared for Publication inThe Journal of Chemical Physics

* University of Puerto RicoDepartment of Physics

Rio Piedras, Puerto Rico

** IBM Research LaboratorySan Jose, California

September 21, 1982

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Prepared for publ-ication in Journal of Chemical Physics

19. KEY' WORDS (Continus on ,o..eo *ide dt noco...yan Id.,ify *y btoCk ALM0b.,)

Gouy-Chapman, tiectrode, interface, asymmetric ions.

.Zo. ASS'- nAC7. 'nnU. ori fve teo I! t.co-AV Mria Iawnt:?. by blockt numone)

IThe modified Gouy-Cbapman Theory is applied to 1:1, 2:1 and 1:2 electrolytesof unequal sizes near a flat electrode. This modejofeatures a potentialof zero charge in the absence of specific adsorption. t

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INTRODUCTION

-The accumulation of ions near a charged electrode, resulting in a so-called electrical

double layer, has been the subject of considerable recent interest. Both integral equation

methods 1 5 and Monte Carlo simulations6 have been applied to this problem.

The classical theory of the electric double layer is the Poisson-Boltzmann theory of

.ODUY and Chapman+ inwhich the solvent is assumed to be a uniform dielectric medium

.. . + h -.o s.e+ + .. . . + : . . - ..... ._/.~)Sdelectric.-..constant is--- - -.

In the Gouy-Chapman (GC) theory, the ion. are neglected. It is a theory for •

, -point ions. However, if the ionic profiles are displaced by a/2, where a is the ionic

diameter, so that the distance of closest approach is o/2 rather than zero, the GC profiles

for monovalent ions are reasonably good. We refer to this modification as the modified

-: - Gouy-Chapman (MGC) theory. In the MGC theory, the hard-core interactions among the

ions are neglected, but the hard-core interaction of an ion with the electrode is not

-- neglectd..TheMGC is less satisfactory for multivalent ions but is still of interest because .

it is a useful starting point for more accurate interactive procedures based on integral

S. equations.

.. .... The GC theory was originally applied to symmetric ions.7,8 "ihe extension to 2-1 and

1-2 electrolytes was obtained by Grahame. 9 Recently, Valleau and Tonie10 have applied

the MGC theory ions which are symmetric in charge and asymmetric in diameter. In this

note, we apply the MGC theory to ions which are asymmetric in both diameter and charge.

C.14 0' c13.... . ..

Is 130

4-j.. . . . -- . ... . . . . . ..- +i" ... . . .-

2

THEORY

The starting point of the GC is the one-dimensional Poison-Boltzmann equation

- Zip eXp JPZ e4.(X)1 1- ~dx2 -

where O(x) is the mean electrostatic potential at a distance x from the electrode, e is the

oi he lecronc carg, z an p~arethevalence (including the sg)~n

' *which Eq. (1) is being applied. If mmO, the RHS of Eq. (1) is zero and Eq. (1) bcmes

Laplace's equation for charge free space. . - -

Independently of the value of m2! 1, Eq. (1) can be integrated to give

S[ i. PiCXP JPZ~eo(X)1 + Aj 1/ 2)

w*here A is a constant of integration.-- -

-To obtain -0(x), Eq. (2) must be-integrated. This can always be done numerically. -

T- _-However, results in closed form have been obtained ony for m=O, 1 and 2 with either all

the z1 equal (symmetric charge) or with z, I and z2 =2, or equivalently, z, 2 and Z2 =-

Therefore, we restrict ourselves to n-n, 1-2 and 2-1 binary solutions.

The geometry of a general binary system is shown in Fig. 1. There are three regions

to consider: O:5x:5c 1 /2, a1 /2:5x:S 2 /2 and x=9 2 /2, where 01" and cF2 are the

*diameters of the two species of ions. In these three regions, there are, respectively, no ions

present, only ions of one species, or both species of ions.

3

In the first region (mmO), O:Sx:Saj/2 and integrating Eq. (2) we get

OWx 0 (O) + X*O'(O) .(3

-where Ss'(O) is the derivative of O(x) at the electrode. In the second region (mml1)),

01 /2Sx~v2 /2,

Im .p MW

whm u~Swkfa./e'1ntegrating gives .A

An~[ 1-B exP{-zie-O(x)J ~In - + A>O

Pzleo(x) - (5)

ffA.[, + tan 2(Bxpze~i2]y1

where B is a constanit of integration.

~jr 'The solutio or. (x) in- the third region (z-2). x a/, is well known. ]For

CjYMMetjj'dharge,-3 I=z. the result s'

tanh [1zeO(x;] - an P[Ze(2/2)] (6)

For the unsymmetric 2-1 and 1-2 case, the result is9 - . . . .

exp [pe#(x)]-

1 + C-,Y + 1(7b)

4

where

O + V/73 (8)

with

(2 eXp [e0,(O2/2)] + Il/2 (9a)

-" eXP -P . .(C-2 /2)] + 11 -..7-

In Eqs. (6) and (7), y-x-- 2 /2, and .b- -

~2 4wPe 2 2-b '-- . (10)

- In obtaining Eqs. (6) and (7), we have employed the condition O(x), .'(x).O as .-...

~ A~-~Equations (7a) and (9a) apply to the case where the electrode has a positive charge and

Eqs. (7b) and (9b) apply to the case where the electrode has a negative charge.

* ~.... ~. . " ". . .- " " .1

e constants of integration are determined by requiring that #(x) and (x) be

-. continuous at x-al/2 and al/ 2 and #(0) be equal to the applied potential. In principle,

analytical expressions could be developed for these constantsof "integration. However, in

. practice, we found it convenient to satisfy the boundary conditions by an iterative

procedure.

RESULTS

The potential drop across the double layer (measured from x-0) is plotted inFigs. 2-5 as a function of the charge density on the electrode. As would be expected at

large charge densities, the potential drop is largely determined by the counterions since the

,.. co-ions are excluded from the vicinity of the electrode. Consequently, the results for

+. +, +-L, .

- . o. •- , .

5

positive charge densities are insensitive to the value of 01/02 since, for positive charge

densities, the counterion always has the diameter 92=4.25,A.

For negative charge densities on the electrode, there is a wider variation in the.i. .. ..... potential since the ,ounterions now have a variable diameter 01 <02. At large negative

-charge densities, the potential, as expected, approaches that of a symmetric electrolyte of

- as

7_NI. .,Uh sm e c ag n diameter asthe cou tei.. . ."... ....

1 M .'At small charge densities on the electrode, the smaller ions can approach the wall

moreclosely This results in a separation of charge for ions of unequal diameter even at

zero charge on the electrode and a resulting nonzero potential difference across the double

-,-layer. This has already been noted by Valleau and Torriel' for ions which have a

,... ..symmetric charge. *_._..ec.-. -..... . _ ..-

The charge profile and concentration profiles for a 0.5M solution with 0l/02O.%0 are

plotted in Fis. 6 and 7 for the case where the potential drop across the double layer is, --.. -.'-". ".- '-- Z . .' -'.' - - - .-. '- .- ". -- - - - - . . .

. -zero. The potential profile is not monotonic, as is the case when a,-=2 or when the .

potential drop is appreciably different from zero. The concentrations profile of the smaller

.. . .ions is also nonmonotonic at V-0. - --

.. .: ..." .. .. ..

-' - As is seen from Figs. 8 and 9, even for the relatively small potential, lV 1-0.026 -

volts, the concentration profiles are rather similar to those for the equal diameter case. At

large potential drops, the concentration profiles near the electrode will approach those of

the symmetric electrolyte whose charge and diameter are equal to those of the counterions.

6

. . CONCLUSIONS

The MGC theory has been applied to electrolyte which are asymmetric in charge and

size. Within the MGC theory, the most interesting results are obtained in the region of

very small potential drop and very small charge density on the electrode because at large

- potentials or charge densities, the co-ions are excluded from the vicinity of the electrode.

Consequently, the counterions dominate and the double layer properties approach these of

- -a symmetric electrolyte whose charge and diameter are equal to those of the counterion.

If the ions differ in size, then the smaller ions can penetrate closer to the electrode

than the larger ions. At large charge densities, this effect would be negligible cmpared to .

the coulombic effects. However, this effect is significant at small charge densities. In

-:!* :". . particular, charge separation and a nonzero potential can occur even when there is not

charge on the electrode.

It is conventional to interpret experimental results in terms of the GC theory in which - ,.., .-.. - . . . . .-s u-.---..-.,.---.---, -.-. .- . ..

ion diameters do not appear. As a result, noncoulombic effects uch as specific adsorption,

are required to explain such phenomena as a nonzero potential at zero charge on the

. . electrode. It is not necessary to invoke such effects since unequal ion sizes can also give

rise to such phenomena. This is not to say that specific adsorption is not an important -

phenomena in double layer studies. However, estimates of the importance of specific

adsorption obtained by subtracting equal diameter Gouy-Chapman results from

experimental studies may well be misleading. \ t'-t, b M W0-0 ' , 4i LWA,

. .. . . . . . . . . . - .7 .. . " ' - ---* \ . .. . . .. - - -

7

.1 L Blum, J. Phys. Chem. 81, 136 (1977).

2. D. Henderson and L. Blum, J. Chem. Phys. 69, 5441 (1978).

3. D. Henderson, L. Blum and W. R. Smith, Chem. Phys. Letters 63, 381 (1979);

(1980).

5. T. L. Croxton and D. A. McQuarrie, Mol. Phys. 42, 141 (198 1). -

6. G. M. Tomre and J. P. Valieau, J. Chem. Phys. 73, 5807 (1980).

~ G. Gony, J. Phys. 9, 457 (1910).

~ ~ ~,8. D. L. Chapman, Phil. Mag. 25, 475 (1913). -

9. D. C. Grahame, J. Chem. Phys. 21, 1054 (1953).

10. J. P. Vallean and G.. M. Torre, J. Chem. Phys. -1 (P +eZ~ I (cl a.)

... ~.1IL

FIGURE CAPTIONS

Figure 1. Double layer geometry.

Figure 2. Potential drop across a double layer as a function of the charge density on the.298K, .254 ,1 ,0.

electrode for the case where zl-1, z2 -- 1, e-78.5, T-298K, .25 and l/ 1, 0.5

"and 0.1. The solid and broken curves give the MGC results for concentratiops of A, and

0.05M, respectively.

S . Figure 3. Potential drop across a double layer as a function of the charge density on the

electrode for the case where zl -I and z2 --2.. The other parameters have the same values

as in Fig. 2 and the curves have the same meaning as in this figure.

Figure 4. Potential drop across a double layer as a function of the charge density on the - - -

electrode for the case where z1=2 and z2 =-l. The other parameters have the same values as - "

in Fig. 2 and the curves have the same meaning as in this figure.

Figure 5. Potential drop across a double layer as a function of the charge density on the

electrode for the case where zj-2 and z2 m-2. The other parameters have the same values as

in Fig. 2 and the curves have the same meaning as in this figure.

Figure 6. Potential profiles for the case where V-0 and where 02/02-0.1 and zj-2, z2 -- l

and z l1, z2 -- 2. The concentration is 0.5M. The other parameters have the same values

as in Fig. 2.

Figure 7. Concentration profiles for the case where V=O and where 01/0 2 =-1 and zl=2,Z2a.-O. 1 and z1 , z2 -- 2. The concentration is 0.5M. The other parameters have the same

values as in Fig. 2.

Figure 8. Concentration profiles for the case where V=0.026 volts and where a /0r2-0.1and zj-2, z2 =- and zj=1, z2 =-2. The concentration is 0.05M.- The other parameters-

vthe same values as in Fig. 2. .- --

-Figure 9. Concentration profiles for the case where V=-0.026 volts and where 01/02=0.5and zj-1, z2 =-2 and zl=2, z%.=-1. The concentration is 0-5M. The other parameters have

,.. *~ the same values asin Fig. 2.

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