A new correlation effect in the Helmholtz and surface potentials of the electrical double layer

A new correlation effect in the Helmholtz and surface potentialsof the electrical double layer

Enrique Gonzalez-Tovara)

Instituto de Fı´sica, Universidad Autońoma de San Luis Potosı´, Alvaro Obregon 64, 78000 San Luis Potosı´,S.L.P., Me´xico and Programa de Ingenierıá Molecular, Instituto Mexicano del Petro´leo, LazaroCardenas 152, 07730 Me´xico, D.F., Mexico

Felipe Jimenez-Angelesb)

Programa de Ingenierıá Molecular, Instituto Mexicano del Petro´leo, Lazaro Cardenas 152,07730 Mexico, D.F., Mexico

Rene Messinac)

Institut fur Theoretische Physik II, Heinrich-Heine-Universita¨t Dusseldorf, Universita¨tsstrasse 1,D-40225 Dusseldorf, Germany

Marcelo Lozada-Cassoud)

Programa de Ingenierıá Molecular, Instituto Mexicano del Petro´leo, Lazaro Cardenas 152,07730 Mexico, D.F., Mexico

~Received 18 December 2003; accepted 25 February 2004!

The restricted primitive model of an electrical double layer around a spherical macroparticle isstudied by using integral equation theories and Monte Carlo simulations. The resulting theoreticalcurves for the Helmholtz and surface potentials versus the macroparticle charge show an unexpectedpositive curvature when the ionic size of uni- and divalent electrolyte species is increased. This isa novel effect that is confirmed here by computer experiments. An explanation of this phenomenonis advanced in terms of the adsorption and layering of the electrolytic species and of thecompactnessof the diffuse double layer. It is claimed that the interplay between electrostatic andionic size correlation effects, absent in the classical Poisson–Boltzmann view, is responsible for thissingularity. © 2004 American Institute of Physics.@DOI: 10.1063/1.1710861#

I. INTRODUCTION

The electrical double layer~EDL! is a key concept inphysical chemistry referring to the ionic distribution formedby an electrolyte around a charged colloid or electrode. As ageneral rule, an EDL is developed whenever a particle isimmersed in a polar solvent and, in addition to this ubiquity,most of the equilibrium and transport properties of colloidalsystems are determined by the specific structure of the EDL.Thus, a precise account of the charge arrangement close todissolved macroparticles is highly desirable and representsthe basis of an adequate understanding of colloidal disper-sions and its concomitant practical applications. The study ofthe EDL is a long-standing problem that has been addressedexhaustively via experiments, theory, and simulations.1–6

Notwithstanding, at present, the detailed information of thecharge density profiles is inaccessible to direct measure-ments. On the other hand, the role of the most fundamental~nonphenomenological! theories of double layer is to estab-lish a framework to interpret experimental data and to evolveinto a predictive tool, all of that sustained on microscopichypothesis. By construction, statistical mechanics theoriesand simulations do provide the prime quantities in the EDL,

viz., the ionic distribution functions, at the cost of beingmodel dependent and of an approximative nature.

For a given model, simulations are the ultimate test fortheoretical formalisms of the double layer, nonetheless, anddue to the necessary approximations involved in computerexperiments, those conclusions drawn from comparisons be-tween theories and simulations are not totally exempt of un-certainties. To alleviate this, in the past, diverse exact condi-tions to be followed by the ionic distribution functions andsome thermodynamic quantities have been derived startingfrom basic principles of statistical mechanics. The electro-neutrality condition, the Stillinger–Lovett and other sumrules, and the contact theorem are examples of such exactresults in the EDL.1–3 Hence, alternatively to the corrobora-tion by simulations, these formally exact restrictions allow usto examine the consistency and validity of approximate theo-ries. During many years the positivity of the inverse differ-ential capacity, (Cd)21, defined as

~Cd!215dc0

ds0, ~1!

wherec0 is the total potential drop~from the macroparticlesurface to infinity! and s0 is the macroparticle charge den-sity, was believed an exact theorem7–9 and was indeed usedto argue the inadequacy of some integral equations treat-

a!Electronic mail: [email protected]!Electronic mail: [email protected]!Electronic mail: [email protected]!Corresponding author. Electronic mail: [email protected]

JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 20 22 MAY 2004

97820021-9606/2004/120(20)/9782/11/$22.00 © 2004 American Institute of Physics

Downloaded 07 May 2004 to 193.6.32.106. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

http://dx.doi.org/10.1063/1.1710861

ment, i.e., the hypernetted-chain approximation~HNC!. Nev-ertheless, Attardet al.10 showed later that there was no validproof that the relationshipc0(s0) has necessarily a positivederivative, and, even more, Monte Carlo simulation datahave been already published11 showing instances of negativedifferential capacities in model EDLs. Subsequently, Parten-skii and Jordan12 have revisited the issue and ascertained thecircumstances under whichCd,0 is admissible. In this re-spect, further computational efforts are desirable in order togive a definitive answer to the possibility of a nonmonotonicbehavior in thec0(s0) curves. A clearer situation is ob-served in the case of the Helmholtz or diffuse layer potential,cH , which is the potential drop between the distance of clos-est approach of the ions to the macroparticle and the bulkelectrolyte. Accordingly, a great deal of theoretical and simu-lation evidences1,13–25is known about EDL systems with di-verse geometries for which

dcH

ds0<0. ~2!

In fact, the inequality expressed in Eq.~2! is so well estab-lished for most high-coupled EDL systems~for example,with large macroparticle charges and/or multivalent electro-lytes! that could be even employed as aninversetest of the-oretical approaches, i.e., under certain conditions, the pre-dictedcH(s0) curves, to be correct, should exhibit maximaand sections with a negative derivative. Notably, some treat-ments such as that of the classic Poisson–Boltzmann~PB!equation fail to satisfy Eq.~2!.

From all the above, it is clear that most of the attentionhas been directed to examine the variation of thefirst deriva-tive of the mean electrostatic potential~MEP! as a functionof the macroion charge. However, as it has been foreseen byTorrie in his investigation on negative differentialcapacities:11 ‘‘...many other instances of nonmonotone be-havior of c0(s0) in more complex systems...remain to bediscovered... .’’ In particular, no explicit pronouncement isknown related to higher-order derivatives of theMEP.1,12,13,22–25Thus, the main purpose of the present con-tribution is to report theoretical and simulation studies of thespherical EDL, where the occurrence of an uncommonposi-tive second derivativein the curves ofcH(s0) andc0(s0),for arbitrary surface charge density, is observed. We furtheroffer a possible explanation for this effect by relating theMEP to ionic adsorption and layering, and to the compact-ness of the diffuse layer, which leads us to an examination ofthe role of such microscopical mechanisms. The paper isorganized as follows: In Sec. II we describe the model,theory, and simulation methods. Section III contains the re-sults and their discussion, and, finally, some conclusions aregiven in the closing part.

II. MODEL AND METHODS

A. Model

An isolatedEDL has been most studied recurring to theprimitive model of an electrolyte. In this concise representa-tion, a dispersed colloid, or an electrode–electrolyte system,consists of a hard, uniformly charged, object~the macropar-

ticle or electrode! surrounded by various species of sphericalions, all immersed in a structureless solvent~see Fig. 1!.Most commonly, it is supposed that the same dielectric con-stant permeates all the system in order to avoid image chargecontributions. Despite its apparent simplicity, the resultingprimitive model of a double layer~PM-EDL! has two mainadvantages, namely, it is computationally tractable and takesinto account the essential interactions operating in a Coulom-bic fluid: electrostatic and hard-core potentials. Althoughmore sophisticated models could be employed,2,25–27 thePM-EDL has largely proved to be sufficient to exhibit themain qualitative phenomenology present in double layer sys-tems, for instance, ionic layering, charge inversion, chargereversal, and overcharging are features already seen in thePM-EDL.16,17,20,28–34We remind the reader that charge inver-sion occurs when counterions and coions change their rolesat some point in the electrolyte, which results in a largercoion density compared to that of the counterions there. Onthe other side, charge reversal is the excessive presence ofcounterions next to a macroparticle causing an overcompen-sation of its bare charge and inclusive of the reversion of thesign of the enclosed total charge~native1adsorbed charges!.Overchargingis the situation in which the adsorbed chargeon the colloid’s surface is of the same sign of the residentcharge in the colloid.32–34Such a situation has been observedonly in electrolyte solutions made up of unsymmetric par-ticles, and requires a high steric contribution. The PM-EDLwill be employed here in its particular modality of sphericalgeometry and for an equally sized binary electrolyte, i.e., therestricted primitive model of the spherical electrical doublelayer ~RPM-SEDL!. To be more specific, the ‘‘colloid’’ is asphere of radiusR, bearing a uniform surface charge densitys0 , the electrolytic ions of speciesi (51,2) are deter-mined by their diametera and chargezie, wherezi is thevalence ande is the protonic charge, and the dielectric con-stant of the system ise. For clarity, hereinafter, we will as-sume thats0>0. We have chosen the simplest RPM-SEDLsince, as it is evidenced below, the new effects in the meanelectrostatic potential to be exposed, due to excluded volume

FIG. 1. Schematic representation of the model.

9783J. Chem. Phys., Vol. 120, No. 20, 22 May 2004 Potentials in the electrical double layer


correlations, do not require any type of asymmetry in theelectrolyte~in size or charge! or a particular macroion shapeto be manifest.

B. Integral equations for the spherical electricaldouble layer

To describe theoretically the RPM-SEDL, we havemade use of the integral equation approach in the formof the hypernetted-chain/mean-spherical approximation~HNC/MSA!.18 The HNC/MSA theory is a hybrid closureto the Ornstein–Zernike equation with a long row of thriv-ing applications modeling simple35,36 and confinedelectrolytes,37–40 double layer systems of manygeometries,17,18,20 including the spherical one, and other in-homogeneous fluids.41 In all these instances this approxima-tion has proven its adequacy by showing a very good quali-tative agreement with computer simulations1,14,16,19,36 andother theories,1,42–44depending on the quantities that are ana-lyzed. Such an agreement confirms that HNC/MSA reason-ably well takes into account, both electrostatic and size cor-relations. The HNC/MSA equation produces the ionicdistribution profiles through the solution of the followingintegral equations:

gi~r !5expH 24pR2s0

bzie

er

1 (j 51,2

r jE hj~ t !ci jMSA~s!d3tJ , ~3!

for i 51,2, and r>R1(a/2). Therein,gi(r ) is the radialdistribution function~RDF! of an ion of speciesi at a dis-tancer from the center of a macroparticle, which is located atthe origin,hj (t)5gj (t)21 is the total correlation function ofan ion of speciesj at t, r j corresponds to the bulk numberdensity of that ionic component, andci j

MSA(s) is the bulkMSA direct correlation function between two electrolyte ionsof the ith and jth types separated bys5ur2tu. The explicitform of ci j

MSA(s) for the RPM is a well-known rational func-tion of s containing the hard-sphere and Coulombic parts ofthe interionic correlations.45 As usual,b51/(kBT), wherekB

is the Boltzmann constant andT is the absolute temperature.After some lengthy algebra, Eq.~3! can be recast as

gi~r !5expH 24pR2s0

bzie

er1ziE

R1~a/2!

`

rd~ t ! f 1~r ,t !dt

1ziER1~a/2!

`

rd~ t !Kd~r ,t !dt

1ER1~a/2!

`

rs~ t !Ks~r ,t !dt1rTA~r !J , ~4!

with rT[( j 51,2r j ,

rs~ t !5 (j 51,2

r j@gj~ t !21#, ~5!

rd~ t !5 (j 51,2

zjr j@gj~ t !21#, ~6!

f 1~r ,t !522pbe2

e

t

r@r 1t2ur 2tu#, ~7!

and

A~r !52E0

R1~a/2!

Ks~r ,t !dt. ~8!

Equation~4! represents a pair of nonlinear integral equationswhich, due to its complexity, has to be solved numerically bymeans of finite element techniques. The detailed expressionof the analytical kernels,Kd(r ,t) and Ks(r ,t), which em-body the short-range interionic terms, and a complete de-scription of the numerical methodology can be consulted inRefs. 18 and 46.

C. Monte Carlo simulations

In parallel to HNC/MSA calculations, we also performedMonte Carlo~MC! simulations. These computer experimentswere implemented using an efficient procedure previouslyemployed in recent surveys of the phenomenon of chargereversal in spherical geometry,31 image potentials,47 and ofpolyelectrolyte multilayering on charged substrates.48

Two types of charged hard spheres compose the system:~i! a macroion of radiusR with a positive bare chargeQ0 ;and~ii ! small microions~counterions and coions! of diametera57.14 Å with chargeqi5zie ~with i 51,2) ensuring theelectroneutrality of the assembly. All these ions are confinedin an impermeable cell of radiusRcell and the macroion isheld fixed at the center of the cell. Typically, the total numberof mobile microions is about 1000, so that at a salt concen-tration of 1 M we haveRcell58.17l B , with the Bjerrumlength, l B[e2/(ekBT), corresponding to the distance atwhich two monovalent ions interact withkBT. This lattervalue has to be compared with the Debye–Hu¨ckel screeninglength ~'3 Å for a 1:1, 1 M electrolyte!, which proves thatsize effects are negligible~here withR51.5l B). To avoid theappearance of image forces we suppose that the dielectricconstants of the macroion and the solvent~i.e., water withe578.5! are identical.

The total potential energy of interactionU tot can be ex-pressed as

U tot5(i

FUCoul~m! ~r i !1(

j . iUCoul

~c! ~r i j !G , ~9!

where the first~single! sum stems from the interaction be-tween a microioni ~located at a radial distancer 5r i fromthe macroion center! and the macroion, and the second~double! sum stems from the pair interaction between ionsiand j with r i j 5ur i2r j u, where r i5 xiex1 yiey1 ziez . Moreexplicitly, the macroion–microion interaction is given by

UCoul~m! ~r i !

kBT5H b

Q0zie

r i, for R1a/2<r i<Rcell ,

`, for r i,R1a/2,

`, for r i.Rcell .~10!

The microion–microion pair interaction is given by

9784 J. Chem. Phys., Vol. 120, No. 20, 22 May 2004 Gonzalez-Tovar et al.


UCoul~c! ~r i j !

kBT5H zizj

l B

r i j, for r i j >a,

`, for r i j ,a,

~11!

and to link our simulation parameters to experimentalunits and room temperature (T5298.15 K) we choose theBjerrum length of water,l B57.14 Å.

The equilibrium properties of our model system wereobtained by using standard canonical MC simulations fol-lowing the Metropolis scheme.49,50 Single-particle moveswere considered with an acceptance ratio of about 50%.Typically, a simulation run consists of 106 MC steps perparticle, and about 3000–10 000 uncorrelated~and equili-brated! configurations were used to perform measurements.

III. RESULTS AND DISCUSSION

We have obtained results for the RPM-SEDL in 1:1 and2:2 electrolyte solutions. To construct the correspondingcH(s0) andc0(s0) curves for different systems, we variedthe radius and charge density of the macroparticle, and theconcentration and diameter of the ionic species. In all thecalculations,e578.5 ands0>0. Once the solution of theHNC/MSA equation and the simulations have been carriedout, most of the thermodynamic and structural properties canbe deduced via simple integration of the RDFs. Among them,the mean electrostatic potential,c(r ), is defined by

c~r !524p

e Er

`

rel~ t !F t2

r2t Gdt, ~12!

with rel(t)[( j 51,2zjer jgj (t). c(r ) is a radially symmetricfunction and measures the effect atr of all the charges in thesystem, averaged statistical mechanically over all the pos-sible configurations.

In the first section, the two central quantities for thisstudy, namely, the surface and Helmholtz MEPs,c0 andcH ,respectively, were already introduced.c0 andcH are impor-tant and particular cases of the prior expression, i.e.,

c0524p

e ER

`

rel~ t !F t2

R2t Gdt ~13!

and

cH524p

e ER1~a/2!

`

rel~ t !F t2

R1~a/2!2t Gdt. ~14!

On the other hand, since the complete system is electroneu-tral, the RDFs must obey the exact condition:

s0521

R2 ER1~a/2!

`

rel~ t !t2 dt, ~15!

and, analogously to Eq.~12!, the accumulated charge, up tothe distancer, is

Q~r !54pR2s01Q~r !, ~16!

where

Q~r !54pER1~a/2!

r

rel~ t !t2 dt, ~17!

or, by electroneutrality,

Q~r !524pEr

`

rel~ t !t2 dt. ~18!

Notice that overcharging is the situation in which, at somepoint r 0 , the integrated charge of adsorbed ions from thesurface up tor 0 , Q(r 0), has thesame sign of Q0 anduQ(r 0)u.uQ0u.33,34

Then, the local cumulative chargedensityis

s~r !5R2

r 2 s011

r 2 ER1a/2

r

rel~ t !t2 dt

521

r 2 Er

`

rel~ t !t2 dt. ~19!

Combining Eqs.~13!–~15! we have the electrostatic relation-ship

c05cH14pR2s0

e F 1

R2

1

R1~a/2!G . ~20!

At this point, it is pertinent to comment that the integralversion of the Poisson–Boltzmann theory with the Stern cor-rection is an excerpt of the HNC/MSA integral Eq.~4! whena50 in Kd(r ,t) andKs(r ,t). Since these kernels constitutethe excluded volume contributions, the former vanish in thepoint-ion limit and, therefore, the ensuing steric-dependentcorrelations are lost. As a sequel of it, we arrive to the fa-miliar result of monotonic PB ionic distributions for uni-formly charged macroparticles and, eventually, to the stan-dard results in the PB mean electrostatic potential-surfacecharge curves and their derivatives, which are always mono-tonic and one-signed.

In Fig. 2 we present succinctly our main finding, namely,the occurrence of Helmholtz potential versus macroparticlecharge curveswith an upward concavity for all values ofs0 ,i.e., cH9 (s0).0. The data correspond to a colloid of radiusR510.71 Å bathed in 1:1 and 2:2 electrolytes, such that theionic diameter isa57.14 Å and thetotal bulk concentrationrT5r11r252 M. The temperature isT5298.15 K.Clearly, both the HNC/MSA predictions and the MC simu-

FIG. 2. Mean electrostatic Helmholtz potential,cH[c(R1a/2), as a func-tion of s0 . This plot contains Monte Carlo~MC! and HNC/MSA results fora 1:1, 1.0 M electrolyte and for a 2:2, 0.5 M electrolyte witha57.14 Å,T5298.15 K,e578.5, andR510.71 Å.



lations~see also Table I! display apositive second derivativefor both the 1:1 and 2:2 cases, confirming that this feature isa genuine comportment of the model and not an artifice of anapproximate theory. Notice, however, that in both cases, the1:1 and 2:2 electrolytes, some quantitative differences be-tween the HNC/MSA and MC curves can be observed. How-ever, we reiterate their coincidence in the sign of the concav-ity, and, even more importantly, it is remarkable that thecomputer experiments reveal a more pronounced bending ofcH(s0), which is symptomatic of the strong ionic size cor-relations present in these highly coupled divalent systems. Aprominent characteristic of the phenomenon in question is itshappening for thewhole range ofs0 . Such an upward con-cavity, i.e., for the whole range ofs0 , had not been reportedbefore.52 In electrical double layer studies thecH versuss0

curves are typically monotonically increasing, in PB or forlow ionic concentration in the RPM, or with a down-ward concavity for the RPM at higher concentra-tions.13–15,18,20,21,51,53In some other preceding studies of theplanar electrical double layer,22–25,54,55wherein the RPM anda molecular solvent model were used, a slight to moderatechange to an upward curvature ofcH(s0) has been found,but, differently from the current RPM-SEDL investigation,those episodes come about only for very high surfacecharges (s0.0.4 C/m2), so that the phenomenon receivedscant attention and the source of the effect was not sought. Inthose previous studies of the planar EDL the ionic diameterwas always taken of the order ofa'4 Å. It is important tomention that we have taken care of keeping the values of allthe parameters of our systems in the liquid range where theHNC/MSA closure is reliable.

Once the behavior of the charge second derivative ofcH

has been authenticated by simulations, we now explore themorphology of thecH(s0) curves for uni- and divalent so-lutions when the ionic diameter is varied, in preparation foran explication of the change in curvature. Therefrom, in Figs.3 and 4 we have plotted the Helmholtz potential as a functionof the surface charge density for a macrosphere ofR580 Åin 1:1 and 2:2 electrolytes withrT52 M and 1 M, respec-tively, and variable ionic size.T is fixed at 298 K. In Figs. 3and 4 the change in the concavity ofcH(s0) from negativeto positive, whena is increased, emerges as the main con-clusion and clearly unveils the possible origin of the uncom-mon tendency ofcH9 (s0): an exacerbation of the entropiccorrelations arising from the interplay between the Coulom-

bic and hard-core interactions in highly coupled systems,viz., with large ions, densities, and/or valences. On the otherhand, the specific evolution of the curves for 1:1 and 2:2 hasparticularities worth to be mentioned. First, comparing thecurves corresponding toa54.25 and 9 Å for 1:1 with thosefor a54.25 and 10 Å for 2:2, it is perceived that, regardlessof the valence, the initial and final stages in the developmentof the cH(s0) function whena is modulated have, in gen-eral, the same shapes~downward concave and upward con-cave, respectively!. Nevertheless, for intermediate ionic di-ameters thecH-s0 relationship bends differently dependingon z1 :z2 , e.g., notice the undulating curves for 2:2 anda58 or 9 Å. Besides, and contrasting with the 1:1 case, fordivalent ionscH becomes negative before the sign of thesecond derivative ofcH(s0) experiences a [email protected].,beforecH9 (s0).0]; see, for instance, data for 7 Å in Figs. 3and 4. This last observation also points up the relevance ofanalyzing the second derivative of the MEP, i.e., the positiv-ity of cH9 (s0) since, for example, for 1:1 anda57 Å thisphenomenon evidences the existence of major entropic ef-fectsthat otherwise should be overlooked as the mean elec-trostatic potential and its first derivative behave ‘‘normally,’’

FIG. 3. Mean electrostatic Helmholtz potential,cH[c(R1a/2), as a func-tion of s0 , from HNC/MSA. The different curves correspond to 1:1, 1 Melectrolyte solutions with successively increasing values of the ionic diam-eter. In all casesT5298 K, e578.5, andR580 Å.

FIG. 4. The same as Fig. 3, but for 2:2, 0.5 M electrolytes.

TABLE I. Monte Carlo data for the mean electrostatic Helmholtz potential,c(R1a/2), computed for a set of values of the surface charge densitys0

~left-hand side column!. The central column contains the data for 1:1, 1 Melectrolytes, whereas the third column those for 2:2, 0.5 M electrolytes. Inall casesT5298.15 K,e578.5,R510.71 Å, anda57.14 Å.

s@C/m2#

cH @mV]

1:1 electrolytes 2:2 electrolytes

0.111 9.51 27.330.222 22.4 212.40.333 40.3 214.60.444 64.0 213.20.555 92.0 28.1



i.e., they stay non-negative. As another example of the im-portance of analyzing the second derivative of the MEP, inFigs. 5 and 6 selected graphs of the surface potential–chargerelationship,c0(s0), are included to illustrate how the cur-vature ofc0(s0) can also turn upward, whereas the differ-ential capacity,Cd , never gets abnormally negative. All thediscussed features of the Helmholtz and surface potentialsversus charge curves can be understood on the basis of thestructural arrangement of the electrolyte next to the colloid,i.e., in terms of the ionic distribution functions, as we willsee below.

In order to discuss the origin of the conduct ofc09(s0)andcH9 (s0) we proceed to recast the relations~13! and~14!in a more convenient form by introducing new variables. Acombination of Eqs.~12! and ~18! leads to

c~r !5Q~r !

er1

1

e Er

` rel~ t !

ut20u4pt2 dt. ~21!

Letting r 5R and r 5R1(a/2) in Eq. ~21!, we can rewriteEqs.~13! and ~14! as

c05Q0

eR1

1

e ER1~a/2!

` rel~ t !

ut20u4pt2 dt ~22!

and

cH5Q0

e@R1~a/2!#1

1

e ER1~a/2!

` rel~ t !

ut20u4pt2 dt. ~23!

It should be noticed that, due to the hard-sphere condition,Q(R1a/2) andQ(R) are equal to the colloidal chargeQ0

54pR2s0 and the lower limit in the integral in Eq.~22! isR1(a/2) instead ofR. Let us now define the charge averageof the inverse distance,^1/t&c :

K 1

t Lc

5

*R1~a/2!` rel~ t !

ut20u4pt2 dt

*R1~a/2!` rel~ t !4pt2 dt

, ~24!

or, using Eq.~15!,

K 1

t Lc

5

*R1~a/2!` rel~ t !

ut20u4pt2 dt

2Q0. ~25!

Hence

c054pR2s0

e F 1

R2

1

tcG ~26!

and

cH54pR2s0

e F 1

R1~a/2!2

1

tcG , ~27!

where we have defined

tc[ K 1

t Lc

21

. ~28!

Then, for givene, a, R, ands0 , the sign and, in general, thebehavior of the functionsc0(s0) and cH(s0) depend en-tirely on tc . The expression forc0 is equivalent to that forthe potential of a spherical capacitor made up of two concen-

FIG. 5. Mean electrostatic surface potential,c0[c(R), as a function ofs0 ,from HNC/MSA. The different curves correspond to 1:1, 1 M electrolytesolutions with successively increasing values of the ionic diameter. In allcasesT5298 K, e578.5, andR580 Å.


FIG. 7. An equivalent capacitor made up of two concentric spheres of radiiR andtc .



tric spherical shells with radiiR and tc and with surfacecharge densitiess0 and2s0 , respectively~see Fig. 7!, i.e.,of capacitance,C, given by

1

C5

1

e F 1

R2

1

tcG . ~29!

Hence,tc can be seen as the distance to the center of themacroparticle, where the induced charge on the fluid can belocated. Let us refer totc as the induced chargecentroid.The tc value can also give us a measure of thecompactnessof the EDL, i.e., the smaller the value oftc , the more com-pact the EDL. A widely used interpretation for the surfaceand Helmholtz potentials in EDL studies is that of two ca-pacitors connected in series. From Eqs.~26!, ~27!, and~29!,it is easy to show that

1

C5

1

CH1

1

CD, ~30!

where

1

CH5

1

e F 1

R2

1

R1a/2G , ~31!

and

1

CD5

1

e F 1

R1a/22

1

tcG . ~32!

We will come back to this point later.Since Eqs.~26! and ~27! differ just by a geometrical

factor, we can study the relationshipcH(s0) in terms oftc

and, later, translate the analysis toc0(s0). To illustrate theconvenience of this alternative venue to interpretcH , as afirst example, let us consider an electrical double layer with acharge distribution function,rel(r ), constrained to beone-signed. This condition is typically meet in low-concentratedmonovalent solutions and, otherwise, is the unfailing mark ofall the Poisson–Boltzmann~PB! charge distributions. Grant-ing such a hypothesis, a straightforward use of the FirstMean Value Theorem for integrals56 yields

ER1~a/2!

` rel~ t !

ut20u4pt2 dt5

1

g ER1~a/2!

`

rel~ t !4pt2 dt, ~33!

with g>R1(a/2). Thereby, from Eq.~24!,

K 1

t Lc

51

tc5

1

gor tc5g, ~34!

i.e., the one-sign condition ofrel(r ) warrants that gP@R1(a/2),`). Thus, Eq.~34! together with Eqs.~26! and~27!, assures the conditionsc0>0 andcH>0 ~for s0.0). Ifwe had chosens0,0, the opposite can be shown to be true,i.e., c0<0 and cH<0. Thus, in general,if rel(r ) is one-signed, thenc0s0>0 andcHs0>0. In the case of the modi-fied PB equation including the Stern correction, these condi-tions are always corroborated and supplemented by theknown results c08(s0).0, cH8 (s0).0, c09(s0),0, andcH9 (s0),0. Hence, for monotonic charge distributions thereis a total absence of ‘‘anomalies’’ in the first and secondderivatives of the MEP.

Returning to our main point, ifs0.0, the observedchange in the curvature ofcH(s0) when the size of the ionicspecies is increased comes from the propensity ofcH , atlarge ionic diameters, to become negative for lows0 , and itsposterior return to ‘‘regular’’ or positive values for highs0

@see thecH(s0) curves for 9 Å and 1:1 in Fig. 3 and that for10 Å and 2:2 in Fig. 4#. Therefore, a plausible explanation ofthe physical mechanisms behind the crossover in sign ofcH9 (s0) should be able to justify satisfactorily its limitingbehavior for largea. Such a rationale can be indeed con-structed relying upon the definition oftc and the expression~27!. With this purpose in mind, in Figs. 8 and 9 the reducedlocal charge densityrel* (r )[( i 51,2zigi(r ) for 1:1 and 2:2has been portrayed for several ionic diameters at fixed andvery low s050.0005 C/m2. Initially, for these barelycharged situations, we observe that the absolute value ofrel* (r ) at contact,urel* (R1a/2)u, grows whena augments, asit is expected from the knowledge of systems of pure hardspheres next to walls.30–34,57However, in our case, the exis-tence of electrostatic interactions impose to the ionic distri-butions the severe restriction of electroneutrality, which, inturn, obligates to every set ofs0-fixed curves in Figs. 8 and9 to keep constant the integral in Eq.~15!. Therefore, tocomply with both the requirements of an increasingurel* (r )uat contact, driven by hard-core effects, and a constant area,dictated by electroneutrality, the functionrel* (r ) has to be

FIG. 8. Reduced charge density profiles,rel* (r )[( i 51,2zigi(r ), fromHNC/MSA. The different curves correspond to distinct 1:1, 1 M electrolytesolutions with increasing ionic diameters. In all casesT5298 K, e578.5,R580 Å, ands050.0005 C/m2. In the inset the corresponding cumulativecharge density profiles,s(r ), for each diameter is shown.




oscillatory for growinga. A pair of important consequencesthen arise, coming from the accentuated short-range correla-tions: ~i! as it is visible at the graphs ofs(r ) in the insets ofFigs. 8 and 9, in the proximities of the macroparticle, thehigh values of2rel* (r ) indicate an enhancement of thescreening capacity of the EDL asa increases. This meansthat the native colloid charge,Q054pR2s0 , is rapidly neu-tralized by the excess counterions and, for large values ofa,even charge reversal occurs~e.g., note the wide regions ofnegatives(r ) therein!, and ~ii ! the layered and dampingionic structure in Figs. 8 and 9 implies a diffuse chargemainly concentrated in the neighborhood of the surface or,else, signifies a very compact electrical double layer withtc,R1a/2 for large a and lows0 . This result implies anegativecH and a negativeCD @see Eq.~32!#. While a nega-tive cH is perfectly meaningful, anisolatedcapacitor with anegative capacity to store charge has no physical meaning. Anegative capacitance or negative differential capacitance im-plies not energy storage but an energy source. This concepthas been long recognized in the field of electronics, whereI 5Cd(dV/dt), with Cd negative, refers to an energy source.Here I is the circuit current,t the time, andV the appliedvoltage. In this respect, we fail to see any possible energysource in an equilibrium EDL. Hence, a negativeCD prob-ably implies that the EDL picture of two capacitors in seriesis not correct. However, notice that the total capacitance ofthe EDL, C, is positive and has a physical meaning. This issomewhat in agreement with the Partenskii and Jordan12 pic-ture, where the capacitance of an isolated electric cell, whichis part of a more complex system, can be negative, but thecapacitance of the whole system must be positive.

In Figs. 10 and 11 we depict 1:1 and 2:2 data ofrel* (r ),respectively, for a fixed and largea and varyings0 , to makeevident the evolution of the ionic distribution as the macro-particle’s charge is enlarged to reach values at which theHelmholtz potential turns back to be positive. Ass0 in-creases, the portrayed charge profiles clearly show the devel-opment of a second layer of counterions, promoted by thecooperative action of Coulombic and steric causes. The ex-cess counterions are increasingly attracted to the colloidalsurface in response to the escalating electrostatic forces com-ing from the growth ofs0 , but, by excluded volume re-

straints, only a maximum number of counterions can be ac-commodated in a first stack contiguous to the macroparticle’ssurface so thata secondary layer is built up, prompted by thepersistent adsorption of neutralizing charges whens0 aug-ments ~see, for instance, the hump in the curves fors0

.0.25 C/m2 at Figs. 10 and 11!. Consequently, for largeaands0 , the ionic distributions regain a larger spatial exten-sion, the distancetc returns to the range@R1(a/2),`) andcH becomes positive, or ‘‘normal,’’ again. Thus, contradic-tory as it can appear, the behavior ofcH(s0) and its secondderivative for largea is just the reflection of the peculiar wayin which the subjacent ionic structure solves the compromisebetween the two dominating interactions included in theRPM-SEDL model, i.e., the hard sphere and the electrostaticforces.

In Figs. 12 and 13 we plot (tc2R)/(a/2), as a functionof s0 , for a 1:1, 1 M and a 2:2, 0.5 M electrolytes for severalvalues ofa. Notice that (tc2R)/(a/2).1(,1) implies tc

.R1a/2 (tc,R1a/2). As we pointed out in the discussionof Eq. ~33!, tc.R1a/2 is a direct consequence of the one-sign property of therel(r ) function. Therefore, the eventual-

FIG. 10. HNC/MSA reduced charge density profiles,rel* (r )[( i 51,2zigi(r ) for 1:1, 1 M electrolyte solutions witha59 Å, T5298 K, R580 Å, ande578.5. The different curves correspond to increas-ing values of the surface charge density,s0 .

FIG. 11. The same as Fig. 10, but for 2:2, 0.5 M electrolytes witha510 Å.

FIG. 12. The computed value oftc2R @in units of ionic radius (a/2)] as afunction of the surface charge density,s0 , for distinct 1:1, 1 M electrolytesolutions, with varying ionic diameters. In all casesR580 Å, T5298 K,e578.5.



ity of tc,R1a/2 has to do with the occurrence of oscilla-tions in the charge distribution profiles,rel(r ). In terms ofthe capacitor representation@see Fig. 7 and Eq.~29!# thenarrower electrolytic charge distribution is equivalent tohave a spherical shell with charge2Q0 of radius tc,R1a/2, i.e., asa increases,tc decreases, and the capacitorelectrostatic free energy, given by

U05Q0

2

2e F 1

R2

1

tcG , ~35!

decreases. This less participation of the electrostatic energyis, in some way, compensated by the steric contribution. It isthis energy requirement that forcestc to become lower thanR1a/2, lying in a zone where no charge is present.

In Fig. 13 we see that for lows0 , larger ionic diametersimplies lower values of the induced charge centroid,tc . Thisbehavior is reversed for highs0 . On the other hand, forsmall a, the derivative oftc , as a function ofs0 , is nega-tive, whereas for largea, tc8(s0).0. In other words, for lows0 , as a increases, the EDL becomes more compact, and, ass0 increases, the EDL becomes more compact, for smallvalues of a, and less compact for large ionic diameters. Aspointed out above, and from Figs. 8–11, larger ionic diam-eters increase the system excluded volume, which, in turn,promotes the particles adsorption,30–33 as an entropic effect,thus, decreasing the system electrostatic energy [see Fig. 4and Eqs. (27) and (35)] and making the EDL more compact.As the macroparticle electric field increases, i.e., ass0 in-creases, ionic adsorption increases. However, whereas forsmall a this implies a more compact EDL, i.e., lowertc , forlarge a, layering is needed since there is not enough room toaccommodate more ions next to the macroparticle surface,hence increasingtc , making the EDL less compact, and in-creasing the system electrostatic energy [see again Fig. 4and Eqs. (27) and (35)]. This analysis explains whycH(s0),in Fig. 4, becomes negative as a increases and whycH9 (s0),0 for small a, andcH9 (s0).0 for large a. A simi-lar analysis can be made for Figs. 12 and 3, for the 1:1

electrolyte, where these effects are somewhat less accentu-ated due to the less electrostatic energy due to their less ioniccharge.

Now, regarding the comportment of the surface poten-tial, c0(s0), it is notable in Figs. 5 and 6 that, even if theRPM-SEDL can exhibit curves withc09(s0).0 for posi-tively charged colloids, the functionc0(s0) itself never getsnegative, signaling the dominance of the term 4pR2s0 /eR.From Eq. ~26!, one can thus infer that, for the restrictedprimitive model,tc.R, ;s0 .

From the discussion of Figs. 2–13, it is clear, we hope,that the decrease of the system accessible volume promotesthe decrease oftc , at low s0 and the increase oftc , ass0

augments, due to the counterion layering. Hence, a fair ques-tion is as follows: How does the concentration affectscH9 (s0)? Because an increase of the electrolyte concentra-tion implies a decrease of the system accessible volume,from the above argument it follows that for a sufficientlylarge concentrationcH9 (s0) will change from negative topositive, just as it happens when we increase the ionic size.To show this, in Fig. 14 we have plottedcH(s0), for a 1:1,2 M electrolyte, for two different ionic diameters,a54.25 Å anda56 Å. Also in Fig. 14, we show our resultsfor a 2:2, 1 M electrolyte fora54.25 Å anda58 Å. OurFig. 14 results for the 1:1, 2 M electrolyte should be com-pared with their counterparts in Fig. 3, for a 1:1, 1 M elec-trolyte. Notice that in Fig. 3 thea54.25 Å and a56 Åcurves show negative or nearly negative value ofcH9 (s0). InFig. 14, while for a54.25 Å, cH9 (s0),0, for a56 Å cH9 (s0) is clearly positive. A similar result is found ifwe compare our Fig. 14 results with those for a 2:2, 0.5 M,shown in Fig. 4. Fora54.25 Å, even though we increasedthe electrolyte concentration, in Fig. 14,cH9 (s0),0. For alarger diameter, however, while in Fig. 3, fora56 Å, andFig. 4, for a58 Å, cH(s0)&0, in Fig. 14cH9 (s0) is clearlypositive. If the solvent is considered in the model, one of itsconsequences would be to decrease the accessible volume,and hence aCH9 (s0).0 can be expected.

Concerning Eqs.~33! and ~34!, the essential ingredient

FIG. 13. Computed value oftc2R @in units of ionic radius (a/2)] as afunction of the surface charge density,s0 , for distinct 2:2, 0.5 M electrolytesolutions, with varying ionic diameters. In all casesR580 Å, T5298 K,e578.5.

FIG. 14. Mean electrostatic Helmholtz potential,cH[c(R1a/2), as afunction of s0 , from HNC/MSA. The different curves correspond to 1:1,2 M (a54.25, 6.0 Å! and 2:2, 1 M (a54.25,8.0 Å) electrolyte solutions. Inall casesT5298 K, e578.5, andR580.



to proves0c0>0 ands0cH>0 was supposingrel(r ) of onesign. However, independently of the monotonicity and/or theone-sign property ofrel(r ), and based on the second meanvalue or Bonnet’s theorem for integrals,56 the expression of^1/t&c can be rewritten to produce a corollary. Since 1/t is apositive, monotone decreasing function in@R1(a/2),`), byBonnet’s Theorem, there exists ajP@R1a/2, ), such that

ER1~a/2!

` rel~ t !

t4pt2 dt

51

R1~a/2!E

R1~a/2!

j

rel~ t !4pt2 dt, ~36!

and, from Eqs.~17! and ~25!,

K 1

t Lc

51

R1~a/2!F Q~j!

2Q0G . ~37!

Hence, by Eq. ~37!, and since Q0Q(r ),0 for r .R1(a/2), if no overcharging is present,32–34 we have^1/t&c

.0, and, from Eqs.~26! and ~27!, c0,4pR2s0 /eR andcH,4pR2s0 /e„R1(a/2)… if s0.0. In other words, themean electrostatic potentialsc0 and cH are always lowerthan the corresponding unscreened electrostatic potentials. Ifovercharging is present, for example, for a size-unsymmetricsalt, there are values ofr P@R,`) such thatQ(r )Q0.0.32–34

This result would, in principle, allowtc to be negative,which is unphysical. Hence, even if overcharging is present,Q(j)Q0,0.

IV. CONCLUSIONS

In this paper we have studied the restricted primitivemodel of a spherical electrical double layer, via the HNC/MSA integral equation, and reported the occurrence, forlarge ionic diameters and all charges, of an unexpected posi-tive second derivative in the Helmholtz and surface poten-tials as functions of the macroparticle charge. AccompanyingMonte Carlo simulations were also performed that corrobo-rate the novel phenomenon and, moreover, these computerexperiments indicate that the effect could be still more in-tense for divalent electrolytes than that initially shown by theHNC/MSA formalism. To gain a qualitative insight into theunderlying physical mechanisms, the behavior ofc0(s0),cH(s0) and their derivatives has been explained thoroughly,for the first time, on the grounds of a magnification of theionic adsorption and layering, driven by the increasing elec-trostatic and size correlations for big ions, i.e., the decreaseof the accessible volume versus the wall-ion electrostatic en-ergy. Accordingly, we introduced the related concept of com-pactness of the electrical double layer, and a way to measureit by dint of the induced charge centroidtc , to prove that thetendency ofcH(s0) to bend upwardly stems from the changeof value of this centroid, meaning that the EDL passes froma very compressed and highly localized charge distributionaround the colloid, for lows0 , to a more extended ioniccloud, at highs0 , in which a first interfacial layer of coun-terions is saturated, giving rise to a second maximum in thecounterion density. Some important characteristics of the en-ticing curvature ofcH(s0) deserve a final remark. First, dif-

ferently from a previous and collateral reference,22,23,25 inour study there is a manifest incidence of a positivecH9 (s0)for the whole range of charge densities, attesting the preva-lence of enhanced entropic effects for highly coupled EDLsystems~for large ions, in this case!, and, second, the exis-tence of anomalous upward-concavec0(s0) andcH(s0) isindependent of an specific double layer geometry, as it hasbeen confirmed by the authors,58 and of the electrolyte sizeor charge asymmetry, which is evident from the present useof the RPM. In other words, the phenomenon in questionseems to be an essential property of the most basic represen-tation of an ionic solution in contact with a surface, i.e., oneincluding, at least, the intertwined correlations alreadypresent in a system of equally-sized hard-spheres with em-bedded point charges of equal magnitude. In this last respect,even if the asymmetry of the ionic species is not a prerequi-site for the conditionss0c09(s0).0 ands0cH9 (s0).0 to befulfilled for the whole interval ofs0 , a significant incrementof the effects on the mean electrostatic potentials discussedhere and an unexplored phenomenology, including over-charging, are expected to emerge in the case of EDLs inasymmetric salts. We will explore these issues in a forthcom-ing investigation.

ACKNOWLEDGMENTS

E.G.T. thanks CONACYT~NC0072! and PROMEP.R.M. thanks IMP for its hospitality.

1S. L. Carnie and G. M. Torrie, Adv. Chem. Phys.56, 141 ~1984!.2L. Blum, in Fluid Interfacial Phenomena, edited by C. Croxton~Wiley,New York, 1986!.

3P. Attard, Adv. Chem. Phys.92, 1 ~1996!.4K. S. Schmitz,Macroions in Solution and Colloidal Suspension~VCH,New York, 1993!.

5F. Fenell-Evans and H. Wennerstro¨m, The Colloidal Domain: Where Phys-ics, Chemistry, Biology and Technology Meet~Wiley-VCH, New York,1994!.

6P. C. Hiemenz and R. Rajagopalan,Principles of Colloid and SurfaceChemistry~Dekker, New York, 1997!.

7L. Blum, J. L. Lebowitz, and D. Henderson, J. Chem. Phys.72, 4249~1980!.

8L. Blum and D. Henderson, J. Chem. Phys.74, 1902~1981!.9L. D. Landau and E. M. Lifshitz,Electrodynamics of Continuous Media~Pergamon, Oxford, 1960!, Chap. 3.

10P. Attard, D. Wei, and G. N. Patey, J. Chem. Phys.96, 3767~1992!.11G. M. Torrie, J. Chem. Phys.96, 3772~1992!.12M. B. Partenskii and P. C. Jordan, J. Chem. Phys.99, 2992~1993!.13G. M. Torrie and J. P. Valleau, J. Phys. Chem.86, 3251~1982!.14L. Degreve, M. Lozada-Cassou, E. Sańchez, and E. Gonza´lez-Tovar, J.

Chem. Phys.98, 8905~1993!.15L. Degreve and M. Lozada-Cassou, Mol. Phys.86, 759 ~1995!.16M. Deserno, F. Jimeńez-Angeles, C. Holm, and M. Lozada-Cassou, J.

Phys. Chem. B105, 10983~2001!.17E. Gonzales-Tovar, M. Lozada-Cassou, and D. Henderson, J. Chem. Phys.

83, 361 ~1981!.18E. Gonza´lez-Tovar and M. Lozada-Cassou, J. Phys. Chem.93, 3761

~1989!.19M. Lozada-Cassou, W. Olivares, and B. Sulbarań, Phys. Rev. E57, 2978

~1998!.20 Lozada-Cassou, R. Saavedra-Barrera, and D. Henderson, J. Chem. Phys.

77, 5150~1982!.21M. Lozada-Cassou and D. Henderson, J. Phys. Chem.87, 2821~1983!.22Z. Tanget al., Mol. Phys.71, 369 ~1990!.23L. Mier-y-Teran, S. H. Suh, H. S. White, and T. Davis, J. Chem. Phys.92,

5087 ~1990!.24L. Mier-y-Teranet al., Mol. Phys.72, 817 ~1991!.



25Z. Tang, L. E. Scriven, and H. T. Davis, J. Chem. Phys.97, 494 ~1992!.26W. Schmickler and D. Henderson, Prog. Surf. Sci.22, 323 ~1986!.27G. M. Torrie, P. G. Kusalik, and G. N. Patey, J. Chem. Phys.90, 4513

~1989!; 89, 3285~1988!; 91, 6367~1989!.28T. Das, D. Bratko, L. B. Bhuiyan, and C. W. Outhwaite, J. Chem. Phys.

107, 9197~1997!.29H. Greberg and R. Kjellander, J. Chem. Phys.108, 2940~1998!.30S. Lamperski and C. W. Outhwaite, Langmuir18, 3423~2002!.31R. Messina, E. Gonza´lez-Tovar, M. Lozada-Cassou, and C. Holm, Euro-

phys. Lett.60, 383 ~2002!.32F. Jimenez-Angeles and M. Lozada-Cassou, J. Phys. Chem. B108, 1719

~2004!.33M. Lozada-Cassou and F. Jimeńez-Angeles, arXiv:physics/0105043.34F. Jimenez-Angeles and M. Lozada-Cassou, arXiv:cond-mat/0303519; J.

Phys. Chem. B~in press!.35E. Gonza´lez-Tovar, M. Lozada-Cassou, L. Mier-y-Teran, and M. Medina-

Noyola, J. Chem. Phys.95, 6784~1991!.36D. Henderson, M. Lozada-Cassou, and L. Blum, J. Chem. Phys.79, 3055

~1983!.37M. Lozada-Cassou and E. Dıáz-Herrera, J. Chem. Phys.92, 1194~1990!;

93, 1386~1990!.38M. Lozada-Cassou and J. Yu, Phys. Rev. Lett.77, 4019~1996!.39L. Yeomans, S. E. Feller, E. Sańchez, and M. Lozada-Cassou, J. Chem.

Phys.98, 1436~1993!.40J. Yu, L. Degre`ve, and M. Lozada-Cassou, Phys. Rev. Lett.79, 3656

~1997!.41J. Alejandre, M. Lozada-Cassou, E. Gonza´lez-Tovar, and G. Chapela,

Chem. Phys. Lett.175, 111 ~1990!.42L. B. Bhuiyan and C. W. Outhwaite, inCondensed Matter Theories, edited

by L. Blum and F. Malik~Plenum, New York, 1993!, Vol. 8; Philos. Mag.B 69, 1051~1994!.

43C. W. Outhwaite and L. B. Bhuiyan, J. Chem. Soc., Faraday Trans.79,707 ~1983!.

44C. W. Outhwaite and L. B. Bhuiyan, Electrochim. Acta36, 1747~1991!;Mol. Phys.74, 367 ~1991!.

45K. Hiroike, Mol. Phys.33, 1195~1977!.46L. Mier-y-Teran, E. Dıáz-Herrera, and M. Lozada-Cassou, J. Comput.

Phys.84, 326 ~1989!.47R. Messina, J. Chem. Phys.117, 11062~2002!.48R. Messina, C. Holm, and K. Kremer, Langmuir19, 4473~2003!.49N. Metropolis, J. Chem. Phys.21, 1087~1953!.50M. P. Allen and D. J. Tildesley,Computer Simulations of Liquids~Claren-

don, Oxford, 1987!.51G. M. Torrie and J. P. Valleau, J. Chem. Phys.73, 5807~1980!.52After this paper was completed we became aware~private communication

of D. Henderson to MLC! of a preprint by D. Bodaet al., where a densityfunctional theory calculation for a very high concentrated, 2:1 planar EDLalso shows an upward concavity. However, the result is simply reportedand no explanation is offered.

53S. L. Carnie, Mol. Phys.54, 509 ~1985!.54P. Nielaba and F. Forstmann, Chem. Phys. Lett.117, 46 ~1985!.55M. Plischke and D. Henderson, Electrochim. Acta34, 1863~1989!.56Encyclopedic Dictionary of Mathematics, edited by S. Iyanaga and Y.

Kawada~MIT Press, Cambridge, MA, 1977!, Vol. I, p. Article 218~X.10!;S. M. Nikolsky, A Course of Mathematical Analysis~MIR, Moscow,1977!.

57I. Snook and D. Henderson, J. Chem. Phys.68, 2134~1978!.58F. Jimenez-Angeles, E. Gonza´lez-Tovar, M. Lozada-Cassou, and R.

Messina~unpublished!.



A new correlation effect in the Helmholtz and surface potentials of the electrical double layer

Documents

Transcript of A new correlation effect in the Helmholtz and surface potentials of the electrical double layer