AD-RI63 452 RN IMPROVED CLOSURE OF THE BORN-GREEN-YYON EQUATION FOR L/1THE ELECTRIC DOUBLE LAYER(U) PUERTO RICO UNIV RIO

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An Improved Closure of the Born-Green-YvonEquation for the Electric Double Layer Interim Technical Report

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-" Primitive Model of Charged Interfaces Integral Equations,,,

' 20. ABSTRACT C . .oin.e on rew o. @#do It nog.. e ay mid Idenltify OF .... l. nu"m b )An extensive series of calculations are made to compare an improved

closure of the Born-Green-Yvon (BGY) equation for the electric double layerof primitive electrolytes to existing Monte Carlo simulation and othertheories, such as the modified Poisson Boltzmann (version: 5) (MPB5) and theHypernetted chain/mean spherical approximation (HNC,MSA), and its recentimprovements. In contrast to these theories the BGY equation satisfies thecontact theorem always. Furthermore the bulk pair correlations function used

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StC.JJI i' CLAJ&WF1PCATICW COF ? IS PA0jz,%.r Dw ,

-'are the most accurate available, namely the HNC bulk results. The resultswhow very good general agreement to the computer work for high densities andsurface charges. -

In particular .o.re .-ah1to go *o, higher densities~than those- hitherto published in the literature. -For low density 2-2 and 2-1 elec-" trolytes the agreement is not as good as for the other theories. . % 1

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AN RIPROVED a)SUR& OFME ONGREJVO ATIQN4 ER THlEELECMIC DOUSILE LAYER

C. Gaccanmo G. Pizzimenti and L. dlu:30'Departent of PhysicsLiversity of Puerto Rico.

Rio Piedras, PR 00931

Istituto Ji fisica teoricaUniversitg & L.ssina

A-ssina, Italy ..ior .. ':

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-. JOY AL 0& Ct4ICAL PHYSICS

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855

. '.'.

, 85 12 .

.I ' ,_' .. ' . -: ,. .. ../ .'"" .. """- " .-' -.-.-. ."' ". ..._i ." -"'" " "" "" . .

.. :-,: tatE .-A:'.

u.Caccamo~~% an %*ziet

Istt .Cadccamo . eoic and Giment

Unitrersiti di Messina, Messina, Italy,

and

L.Blum

Department of Physics, University of Puerto Rico

Rio Piedras, Puerto Rico 00931.

§ALso supported by NATO grant 816/83

Supported by the Office of Naval Research and NATO grant 816 8-1

Abstract.

An extensive series of calculations are made to compare

an improved closure of the Born-Green-Yvon (BGY) equation for the

electric double layer of primitive electrolytes to existing Monte

Carlo simulation and other theories, such as the Modified Poisson-

Boltzmann (version 5) (MPB5) and the Hypernetted Chain/Mean Sphe-

rical Approxirmation (HNC/MSA),and its recent improvements.

In contrast to these theories the BGY equation satisfies

the contact theorem always. Furthermore the bulk pair correlation

functions used are the most accurate available, namely the HNC

bulk results.

The results show very good general agreement to the compu-

ter work for high densities and s4rface charges. In particular

we are able to go to higher densities than those hitherto publi-

shed in the literature. For low density 2-2 and 2-1 electrolytes

the agreement-is not as good as for the other theories.

-.. ..- -S * S

:-. -:.... :

or

Introduction.

The primitive model of the electrode interface consists of

a perfectly smooth surface, which represents the metal, and a col-

lection of charged hard spheres representing the ions.All of it 4s

immersed in a dielectric continuum.

Clearly this model is not an accurate representation of the

real electrode- electrolyte interface, but it can be simulated In

computers, and therefore it is possible to awss the accuracy of

approximation schemes for it.

The early experimental work has used the theories of Gouy

(2) (3)Chapman and Stern with reasonable success, in spite of the

fact that these theories neglect completely the excluded size effects

and correlation effects between the ions. Recently, various theories . .

have been proposed to include these correlations in the treatmen 4 'O )

One of the primary goals of this exercise is to develop an e-

quation ca;%le of handling very dense and higly coupled systems, such

as the molten salts and the non-primitive electrolyte (mixture of

hard ions and dipoles) against a metal electrode.

In this regime it is most important that the contact relati-(11,12)

on

2k= E/ 8 ()B (? (0 B o

be exactly satisfied. In (1) k T Is the Boltzmann thermal factor,

P is the bulk pressure, e is the dielectric constant (8 SO forB

water in the primitive model, and 1 =1 for the non-primitive case),

Qt(O) is the contact density of species I ( which could be eithera * -

an ion or solvent molecule in the non-primitive case) and t.he sum-

mation is over all the species; E is the applied electric field.0

o .:- -...

PLO .*-_

All species are assumed to be hard spheres.

The BGY equation and any of its closures will always sati-

sfy the contact relation. For this reason we have chosen this e- .(9 , O ) .

quation in this work In this paper we study and extend an

improved ansatz for the closure of the BGY which we proposed earlier".

Here the pair correlation function is forced to satisfy the local(9,14,15)

electroneutrality condition and to be positive definite.

We compare the results of this theory to the extensive work of ','al-(16) (17)

leau and coworkers and Snook et al. . We find that the agreemen-

is excellent for the values of the contact potentials for the more

concentrated solutions . The density and potential profiles are in

general good agreement also.

With our method we have been able to get results for very con-

centrated solutions (up to 4 M of a 1-1 electrolyte). In these ca-

ses we observe a marked layering of solute against the electrode

wall. For the dilute ionic solutions ( 0.05 M for the 2-2 and 2-1

electrolytes, 0.1 M for the 1-1 electrolytes) the results are in

general inferior to the MPB5.

In Section 2 we describe our. basic equations. In Section 3

we give the results for the 1-1, 2-2 and 2-1,1-2 electrolytes.

A brief discussion of the general conclusions is given at the

end. -

Sect. 2: Basic equations.

We consider a model in which the ions are hard spheres of

diameter o. and number density The solvent is a continuum

of dielectric constant a , and the electrode is flat, smooth and

has the same dielectric constant s

...............................

. • -... .

For this model we define the one and the two particle di-

stribution functions

where r 1 (x y,Yiz 1 is the position of particle 1, zbeingits distance from the wall

The pair correlation function is defined by

i(1,2) Q1 (1) e~ (2) g.(1.2) (4)

and we also need'

h (1,2) g (1,2) -1i ii

These functions must satisfy the asymptotic conditions

lim: Q(1.) Q(6),z -o

9j(1.2) g gulk(r) (7),

bulkh (1,2) =h (r )(8),ij ij 12

where r12 jrl-7r2 This means that all the correlations far

away from the electrode must be equal to those of the homogeneous

bulk.

Our basic approach is to write the Born-Green-Yvon equation

as a modification of the Poisson -Boltzmann equation(18)

Consider the first BGY equation

-kTV n (1) =-e Eo 45dr2 ~ (2)& (1,2) v ti CrB1 2o i ij 12

for z> a /2

* --- - -. -<-. - -. -p -.-.

where the pair potential U (r) has two ccntributions .J

U (r ) U° (r ) ee/r ( srij 12 ij 12 i j 12

0where U (r, ) is the short ranged part cf Wiie ion-ion interacticn. kvij 12

We introduce the local Glec-rostatic potential

S(1) = -EoZl Sr (2) 1 r

Adding and subtracting the gradient of the second term of (Ii.

into (9), we ekinate the divergence of the right hand side of (II).

Using also the relation

V Ui (r1 2 ) =-kBT 6(r 1 2 - a.) (z Ir (12),I ij 1 3 1212 12

where a ( aj)/2 (13)

z =z-z12 1 2

and 6(x) is the Dirac delta function, we get after some simple

transformations:

-k T V in Q (lJ

e (i) kT 2d1(2)gi (1,2) ar- a )(z /r

B j12 ij 12 12

d? qj( 2) h1 J(1,2) V, (e~e, / sBr 2 )5)

'de assume first that the inhomogeneous pair correlation function

is known. Then equations (11) and (15) form a closed pair of e-

quations for the unknown *(I) and Q (1) A computationaily con- r

venient form of these is obtained by noting that (1i) is really the

3% -

7-, -- ::::y

Poisson equation

2 (1) = - a /s e Q (I) (16),

and that integrating (15) once, we obtain a modified Boltzmann e-

quation

F FIn( Q (1)/ 1/) (IkBT)( e, 1(1) JF(1) e. (1)) ,

where the excluded volume term is

z Q ij +i .°

J (z) = -2XkT 2 ,|dzjdz2z1 2 J(Z 2 ) gJ (0 .z z2 ) iB Z21212 ij 122

,Z- a] '

.here gj( I 2, is the contact inhomogeneous pair correlation fun.

.J 12ction and the fluctuation potential is

e (z) -2a ej dz I dz2 j2)z12 12h (1,2)/rj

z 0(2 (z2) z2 J0h.

where R is a vector in a plane perpendicular to z12 .

in our work we use the arisatz of ref. 13. For the restricted

case of equal diameters a= v j--i. Gj we have:

g. ') A(Zlz) h (1 2) f.(1)f (2)h bulk( ) (20),- 2 1j ' iJ r12

w4he re.-A (z ,z ) = A for z z < 20

ij1 2 ij 1 Z 2 < 2

= 0 otherwise (21) . 44

Pyand h (r ) is Lhe bulk Percus-Yevick pair correlation functicn.

1.1 12In general, however, we have used Aj = A

p .

.7-7

Tho functions f.(z) are found by requiring that the local(

electroneutrality is satisfied(910)

-e = d 2 (2) h. (1.2)

The effect of the first term in eq. (20) is to. eliminate the reg:n

of negative g (1,2) which otherwise could occur in the neighb,-'ir-

hcod of the electrode surface

Sect. III: Results.

The caldulations were performed by the technique described(10)

elsewhere , integrating the Poisson differential equation using

a more refined version of the predictor-corrector integration rcu-

tine than in ref. (10), which enhances significantly the convergen-

ce rate

Typically 3 to 5 iterations are needed for the lower den-

sities. For higher molarities we need at most 10 iterations to reach

convergence.

In our calculations we have used the same parameters as thc-

se in the Monte Carlo simulations of Torrie and Valleau(16)0

a =4.25 A, 8= 78.4, T=298 K The adimensional surface cnarze2is U = q 0 /e where q is the surface charge density (usually

5 5

in units P&C/cm ) The plasma parameter r = Pe2/Ea = 1.6809 and

= 1,k T. Similarly, the adimensional potential drop isB

:-.: = eO .

where * is the potential drop across the interface (usually in r

millivolts). We have explicitily, for the above parameters,

2qs = 88.76* ( AC/ ) *= 25.7 * (mV)

r

-. - .......... 7777i

a) 1-i electrolytes. . ...

We have computed the potential and density profiles for so- VS

lutions ranging from 0.1M to 4M, and surface charge a ranging

from 0.1 to 0.7. The diffuse layer potential drop for all the re-

ported cases is given in Table I where we also display the Monte

'arlo results and those of competing theories. The most extensi-ve results are those of the MPB5 and the HNC, MSA

It is noteworthy to say chat the MPB5 ,ses a Debye-Huckel

like bulk correlation functions, while the good agreement shown by

the HNC/MSA is obtained through the use of bulk correlation func-

tions which are poorer than those used in the HNCIHNC which is not

a very good theory ( ref. 6, 1979).

In our work we use most accurate bulk correlation functions ( 9

which are in good agreement with earlier reported work of Rasaiah

(20)and Friedman for those cases that these last authors have stu-

died.

Furthermore, our work will always satisfy the contact rela-

tion (I) within the numerical accuracy of the calculation,which is

generally less than 1% in all cases investigated and typically clo-

se to 0.3%. This is particularly important for high densities and

Low couplings Again, neither the MFB5 nor the HNCtMSA will sati-

sfy this relation exactly. For the 0.1M the results are given for• • as

a 0 1,0 2 and 0.3 : the potential drop W are not accurate as

those of the MPB5.

In figs.1-2 we show the potential and density profiles for

the case 0 =0.3. The agreement of the BGY is better than that of

the MGC, and in general quite good. We do not have data for this

case from the MPB5. In the low density regime we experience nume-

rical difficulties with the electroneutrality condition (22), which

could be of technical nature.

" ,- . . .. . .

The results are ,iuch better for .0 '-1 scl:2tizrs, . - -

se that the overall agreement with the Monte Carlo is quite g4.,d. V...'

:he BGY is slightly better than the HNC,MSA and slightly worse :nan

t'h,,e MPB5 for surface charges below a =0.5.For 0 =0.55,0.6

the agreement to iiC is good ( see fig.3)

Recently Nielaba and Forstmann(2 1.) have reported a new appr:-

ximation, the HNC/LMSA, where the correlation functions are cop ----

ted using the MSA at a local average concentration. The results :':*r .

the potential as a function of the reduced charge are siown In f;g.

3, where it is also visible the failure of the HNC,MSA for high

charge densities.

In fig. 4 we show a comparison of the case with 0 =0.141

(1-1 case at l.OM) with MC results. The agreement is very good and(16)of comparable "accuracy than MPB4

For highest surface charge we show both the potential prf:-:

le (fig. 5) and the charge profiles(fig.6). It is interesting to0

note that the BGY reproduces quite well the qualitative features

of the counterion density profile, in particular the density -scll-

Lations near z=l. The agreement is however not as good as that of* (21)Nielaba and Forstmann ( . but we do not have any adjustable para-

meter in our calculation.

This feature of density oscillation due to a layering near

the electrode-6urface is also present in the higher density reglme. 1-or the 2.OM case, already investigated in ref .(13), the agree'en -

in the potential drop is quantitative, while the density profile.-

are in remarkably good agreement with the AC results.

:n fig. 7 we show the vr, ntial drop as a function of the s'.r-

face charge for 2.OM and 3.0A On the basis of these results we

also are able to compute the differential capacitance of the dif-

fuse layera quantity of interest to the electrochemists.

In fig. 8 we show the potential and density profiles for

3.0 and 4.0 M solutions These show very pronounced layering.

The potential profile also shows a minimum which is deeper f-r

the higher density case.

b) 2-2 electrolytes.

We have computed results for two concentrations: 0.05 M

and 0.5 M. The results for the potential drop are displayed in

Table II. Here again, the results for the more dilute case are not

as good as those of the MPB5, while the 0.5M case is in excellent

agreement. We experience also here severe problems with the elec-

troneutrality relation, which could again be ofatechnical nature.

The results for the potential drop are shown in fig. 9, whe-

re we also display the differential capacitance.

In fig.l0 we show the charge distribution of the 0.05M 2-2.

case. In spite of the poor agreement of the potential drop, the

density profile are in qualitative agreement with the MC results.

The counterion profile is displaced to the right, and the coion .-:

profile has a peak at the correct place.

For the-O.5M case the results for the potential and iensity

profiles are shown in figs. 11 and 12. The agreement here '-s gcCd:n particular.

both in the potential drop and in the density profiles, the poten-

tial profile is in good agreement near the wall, but the BGY is

higher than the MC far from the wall rhe MPB5 is always lower than

the MC result.

c) Asymmetric case.

Two concentrations were investigated:0.05M and 0.5 M for the• ,%'p

2-1 and 1-2 cases. The results for tne potential drop are dispiuye

in table III. Againhere the agreement for the more dilute case is

poorer, presumably due to technical reasos.

ig. 13 shows the results of the potential drop,as well as

those of the differential capacitance,for the asymmetric case. As

can be seen the agreement is better for singly charged counterio.ns.

Figs. 14 and 15 show the potential and density profiles for the 0."-

(1-2, i.e. doubly charged counterion) at surface charge 0 =0.1704.

While the potential profile is very similar to that of the MP35,

the density profiles are slightly better and in excellent agreement

with the MC profile.

We have also performed the calculation of the 0 %0, 0.5 M(16)

case, recentl! reported by Torrie, Valleau and Quthwaite Fig.

16 shows the potential profile, which is slightly inferior to the

MPB5 result, and fig. 17 the density profiles , which are slightly

"etter than the =95

Concluding remarks. '

In the present work we have presented an extensive discussion

of the use of the BGY for charged interfaces. The BGY equation is

an exact relation but requires a closure, which is an ansatz for the

inhomogeneous pair distribution function g..(1,2).( 13)

In this work we have used a construct that removes some

of the unphysical regions with negative gi (1,2) that occurred when(9,10) J '

the modified superposition ansatz was used. The results are

in very good agreement for high salt concentrations but definitely

poorer for dilute solutions. The addition of a repulsive part to

the pair interaction near the wall takes into account higher repul-

sive interactions due to a crowding effect. We rmust remark that the

:....................-.. . . .. . . .. . . .. . . .

'..-.. ,' . L 2-- . : ~ . . & K * \l > § ~ . .. - ,A.'.,-

" ,t_,'.." " #' " .~" • -'.'.' '.'.' '.'..' '..." ,../, . ",='%, ",".. . . . . . .. . . . . . . . .-.. . . . . . . . . . . . . .".. .-.. .. ".'..'.''4-'.'. ' - - - ° "*,.'-

S- -;" .~ ."

effect of this term is in general not very large. However......-

creases very significantly the stability of the iteration prccedu- I-r e .

..

The poor performance for dilute solutions is attributed t.

difficulties with the electroneutrality rule, which could asic be

of a technical nature. We will investigate this point in the f.'..eJ -"

The main objective of our work is to find an equation t.a:t-i--

will perform well in the high density-high coupling regime whi:-

is that of the molecular solutions or molten salts near a meta!

electrode. Our equation is indeed iell behaved in this regime.and

we have been able to obtain physically reasonable sclutions at 3._ M

and 4.OM,for fairly high surface charges.

LAi

4 o.

......................................... ... .... .... ...

~-.. ; - .

1.G. Gouy, J.Phys.Radiun 3.3

2. D.L..Thapman,Phil.Mag. 25,475 (1913).

3. 0. Stern, Z.Elektrochem. 30, 508 (1924).

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

D.H-enderscn and tL.Blum, :an.J.Chemistry 50, 1906 (1981).

5. K.R.Painter, P.Ballone,M.P.Tosi,P.Grout and N.H.March,

Phys.Chem.liq. 13, 269 (1984).

6. L Blum and G.Stell, J.Stat.Phys. 15, 439 (1976);

D.FHenderson and L.Blum, J.Chem.Phys. 69,5441 (1978);

D.Henderson, L..Blum and W.R.Smith, Chem.Phys.Lett. 63, 381 37I?

7. S.L.Jarnie,D.Y.C-.Z han,D.J .Mitchell,B.W.'Uinham, J .Chemn.Phys.

74, 1472 (1981);

M.Lozada-CassouR.Saavedra-Berrera and D Henderson, J .Chem.

Phys. 77, 5150 (1982) ;r.Lozada-Cassou and D.Henderson,J.Fhys.Then.871282T-(1983);U-Marini Hettolo, Marconi,J.'diechen and F.Forstmann, Chemn. -

Phys.Lett. 107, 609 (1984).

8. S.Levine and C.W.Outhwaite, J.Chem.Soc.Faraday Trans. I'i 74,

1670 (1978);

S.Levine,C.W.Outhwaite and L.B.Bhuiyan, J.Electrcanalh.2he..

123, 105 (1981);

L.B.Bhuiyan,C.W.Outhwaite and L.Levine, Mol.Phys. 42,1271 (1981 ;

C.W.Outhwaite and L.B.Bhuiyan, J.Chem.Soc.Faraday Trans. II

79, 707 (1-983).

9. T.L..Croxton and D.A.McQuarrie, Mol.Phys. 42. 141 1981.).

10. L.Blum,J.Hernando and J.L..Lebowitz, J.Phys.Chem. 87,2825 (3 83).

C~

102,315 (1979).

12. L.Blum and D.Henderson, j.:hem.Phys. 74,1902 (1981).

13.C.Caccamo,G.Pizzimenti and L.Blum, Phys.Chem.Liq. 14,311 3i.

*14. R.W.,Pastor and J.Goodisman,J.Chem.Phys., 68. 3654 (1978>.

15. L.Blum,C.Gruber,D.HendersonJ.L.Lebcwitz and P.A2.lartin,

J.Chem.Phys. 78,3195 (1982).

16. G.M.Torrie and J.P.Valleau, J.Chem.Phys. 73,5807 (1980);

G.24.Torrie,J.P.Valleau and G.N.Patey, ibid. 78, 4615 (1982:>

5 J.P.Valleau and G.M.Torrie, ibid. 76, 4623 (1982);

G.M.Torrie and J.P.Valleau, J.Phys.Chem. 86, 3251 (1982):

G.M.Torrie,J.P.Valleau and C.W.Outhwaite, J.Chem.Phys. 81,

g 9606 (1984). .

17. I.Snook and W.Van Megen, J.Chem.Phys. 75,4104 (1981).

18. M.Born and G.H.Green, Proc. R.Soc. L ndon 188 10 (1946);

J.Yvon, "La Theorie Statistique des Fluldes", Herman

Pai (13)19. C.Caccamo and G.Pizzimenti ,uipublished.

20. J.C.Rasaiah and H.L.Friedman, J.Chem.Phys. 48, 2742 (1968).

21. P.Nielaba and F.Forstmann, Chem.Phys.Lett. (in press).

~~fl 0l N C , -

'\j m~ 0 c l- '

cn C). Mf ) ~ -

C C) O 0 -1 IV -4j v 0'

- c~j r\ (j q Iq L') C~

- 00

0) OD On Iq0

C.r- t 0 V ('

4

C r-

C' M IC T - -N Z

- ~ ~ L -MX f

r: I ( * q r

- - - rs ~ 'C

C! 0

fco

C) 0 O 0

41-

N 4 Cl)C)

cn m Cu)

cc ~

0) -4

Ci) if) Ci r \ -t- 0 r) CD r- r-

0 Ci ) r ) .

- 4 C*~j

4 .- 4 0 4) OrI 0) OD M

0l CY) m) 4zJ m

cc

0 0 ) 4l)

0 0) 0 (n- C 0 4). 0 ru 0) Iq r

11 L - f ) r- tC- 0. ) O (0 (L ) 4) L L 4f)

' -- 4 '-4 '-4 -4 :0 0 -4 0 000 0

-4 -4 r -

0. Q4) IV.C Q.) m - w 0

u w 44- 0 OD w i~~C- 0 0 0 0

C\j

-' .4 C\

4r) 0 r) t- t%- t- 0o %r)

W-V4 0 C 00C0 0

Ci C

c c-l C', 0, 0It:

±~~ - - - - -000 r

C

C-.

c c c C

.a b e 2a c

able 1: (a) Diffuse layer potential drop V (0) for 1.-1 elec-

trolytes. MC results ;G.M.Torrie and J.P.Valleau (1980),

(ref. 16);MPB5:C.W.Outhwaite and L.B.Bhuiyan (1983),(ref.8);HNC,MNSA:M.ozada-Cassou et al. (1982),Cref. '

(b) As calculated from the r.h.s. of eq. (1).

T1able II:Diffuse layer-potential drop V (0) for 2-2 electrolytts.

MC results:G M.Torrie and J.P.Valleau (1982),(ref. 16):

MPB5 and HNC/MSA same captions as Table I. The HNC,:4SA

results at 0.05 M have been taken from a graph.

Table III: Diffuse layer potential drop f# (0) for 1-2 and 2-1

electrolytes. Same captions as Table II.

Potential -" .

F~g. , ?: Potential prof~le for a I-I e.ectrcl.yte at :--.>: .

a =0.3. Dots, MC results(ref.16): full line, this Aork.

Fig. 2 Density profiles for the case of fig. 1. Dots and crosses.

Ic results;full line, this work.IFig. 3 Diffuse layer potential drop V (0) ( 0/2 is taken as o:cr-

dinate origin) as a function of the surface charge densi-

ty e • Dots, MC;dashed line, HNCLMSA;dashed-double dot

line,MP85;dash-dot line,HNCMSA; full line, this -work.

Also shown is the differential capacitance in the BGY(_his -"I2

work), whose scale is on the right(In 8.854 pF/cm Units;.

Fig. 4 Density profiles for the 1-1 electrolyte at c1..,i and a =

=0.141. Dots and crcssesMC results;full line. dashed Ii- -,

ne, this work.

Fig. 5 Same captions as fig. 1 except that C=1.OM and a =0.7.

Fig 6 Density profilesfor fig. 5 case. Dots and triangles, >--

results;open circleR and crosses,HNC/LMSA; full line, da-

shed line, this work.

Fig. 7 W(0) vs. a for the 1-1. case at c=2.OM and 3.0M.BGY re-

sults:fuli line, 2.OM;dashed line, 3.0 M. Also shcwn

are the differential capacitances(same units as fig 3).

Fig. 8 Potential and density profiles for 3.0 aid 4.0i con-

centration of the 1-1 electrolyte. Frop top to bottcm:

potential, counterion, coion profile. Dash-dot line,3.0,.

at 0 =0.485.Full line,3.OM at 0 =0.396;dashed line, 4.0*

LM at a =0.56.

Fig. 9 Wp (0) and diff. capac. vs. a for the 2-2 electrolyte

at c=0.5M.(in the same units of fig.3)

S... .... . . . . . . . .. '.. "

-% -. -. _. ..- '-.-o. .. ._ 1. - -o.ri 1 %. .-. .%- --.-. --o--- . .o. - -_v -y---- -

=0.284. Dots qnd :~s~. ~ ~ ~ j~ ~

dashed line, Medified Gcuy-:hapm"an.

Fig. 11 Potential profile for the 2-2 case at c=0.5M and a =O.'-.

Dots, MC results; full line, this work ; dashed ii.e,

Fig. 12 Density profiles for the case of fig. Ii. Dots and crcs-

ses, MC results; the rest, same captions.

Fig. 13 vp (0) vs. a for 1-2,2-1 electrolytes at different m; a-

rities. Counterion single charged for a > 0.dcubly nargi

for r< O.Concentration c=0.5M:dots, .,iC results:f . 'I e,

this work. c=0.05M: crosses, MC results;dashed Line, this• .

work. Dash-dot line: differential capacitance for trhe ..

c=0.05M case (same units of fig 3).

Fig. 14 Potential profile for the 1-2(doubly charged countericn)

case at c=0.5M and o =0.1704. Dots,MC results; dashed

MPB5; full line, this work.

Fig. 15 Density profiles for the case of fig. 14. Dots and crosses,.

MC results. For the rest same captions.

Fig. 16 Potential profile -or the 1-2 case at a =0 and c=0.5M.

Dots, MC results; crosses, MPB5; full line, this work.

Fig. 17 Density profiles for the case of fig.16.Single charged

ionF crosses, MC results;dash-double dot line, MP23;

dashed line, this work.Doubly charged ion: dots, 2 res..:

dash-dot line, APB5;full line, this work.

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