Aqueous electrolytes and their influence on chemicaland biological processes have been studied for manydecades.1–6 Nonetheless, renewed interest in ion-specific ef-fects sparked considerable efforts in drawing a more realisticpicture of ionic solutions in the past few years.7 More pre-cisely, contrary to the assumptions in primitive models,where salt is modeled by oppositely charged hard spheres ina dielectric continuum, in reality ions interact with an oscil-latory pair potential due to dispersion and hydration effects.8
Furthermore, in order to study the solvation forces and theionic structure between two adjacent surfaces, detailedknowledge of the ion-surface interaction is required.9 Thesepotentials are known to be not only ion-specific but alsohighly dependent on the surface chemistry.10,11 Indeed, forcemeasurements show that solvation forces between mica3,12
and silica13,14 surfaces are nonmonotonic and ion-specific. Inthis realm, the ion-specific restabilization of dispersions ofcolloids15,16 and clays17 in dense electrolytes or the origin ofcharge reversal and the attraction between like-chargedsurfaces18,19 is not entirely understood and still a matter ofongoing research.20
The mean-field Poisson–Boltzmann �PB� theory is oftenused as a useful starting point in the examination of electro-lytes in confinement because of its simplicity and predictivepower, particularly in systems with weak surface charges andlow ion valencies.1,4 In PB theory, ions are treated as pointcharges interacting only electrostatically on a mean-fieldlevel. On such a premise, universal electrostatic ion-ion cor-relations that are important for systems with high electro-static coupling are not accounted for.5 However, numerousmethods to correct for this limitation can be found inliterature.6,21,22 In dense and very inhomogeneous electro-lytes in confinement, however, one can suspect the system tobe dominated by nonelectrostatic excluded-volume interac-tions, and thus electrostatic correlations to be of minor im-portance.
The inclusion of those nonelectrostatic �partially solvent-mediated� ion-ion and ion-surface pair potentials is a subtlematter since the solvent is treated in PB as a homogeneousbackground continuum with a uniform dielectric constant. Toovercome this obstacle, one may employ explicit-water mo-lecular dynamics �MD� simulations to compute the effectiveion-ion and ion-wall potentials of mean force �pmf� at infi-nite dilution. In this picture, the water degrees of freedom areintegrated out and dispersion, hydration, and image chargeeffects are included on a two-body level. An ion-surface pmf
a�Author to whom correspondence should be addressed. Electronic mail:[email protected].
THE JOURNAL OF CHEMICAL PHYSICS 133, 164511 �2010�
can easily be included in the Boltzmann exponent as an ex-ternal potential, whereas the nonelectrostatic ion-ion interac-tions are often treated using local or nonlocal extensions tothe PB theory.1
In this work, we pursue such a coarse-graining strategywhere effective ion-ion and ion-surface potentials for a vari-ety of monovalent salts are employed in the modified PBtheories to calculate the ionic structure and solvation forcesfor dense electrolytes in planar confinement. We will usedensity functional theory �DFT� to sketch and discuss severallocal density approximations �LDAs�—mainly followingprevious work23,24—as well as nonlocal weighted-density ap-proximations �WDAs� using the approach of Burak andAndelman25,26 and the methods that we introduced.27 In or-der to assess the validity of the DFT-derived PB �DFT-PB�theories, we perform both implicit-solvent Monte Carlo�MC� and explicit-solvent MD simulations of LiCl, NaCl,CsCl, and NaI in a one-dimensional nanoconfinement. In ourMC simulations, ions interact with pmfs derived fromMD at infinite dilution. In the first step, comparing MC- andMD-derived density profiles at molar salt concentration willilluminate whether we can correct for �solvent-mediated�many-body interactions by choosing a realistic dielectricconstant in the MC simulation, as was previously shown tobe the case for bulk thermodynamic properties.28,29 In thesecond step, comparing DFT-PB derived ion density profilesto MC, which treats nonelectrostatic and electrostatic ion-ioncorrelations exactly, enables us to evaluate the local and non-local approximations in DFT as well as the validity of ne-glecting electrostatic ion-ion correlations.
We will show that the simplest-to-implement nonlocalapproximation27 is able to correct for excluded-volume cor-relations in a wide parameter range. For this, we use aBarker–Henderson �BH� mapping procedure and treat theelectrolyte as an asymmetric and nonadditive charged hard-sphere system. The extracted effective ion diameters and thelevel of nonadditivity give a qualitative picture of the impor-tance of excluded-volume correlations in the system. Alllocal theories perform much worse than nonlocal PB �NPB�,as was already hinted earlier.30 The computational simplicityof both PB and NPB methods permits us then to investigatethe influence of excluded-volume correlations on salt expul-sion between charged plates, the Donnan effect, on solvationforces between both like-charged and uncharged surfaces,and on overcharging of a single charged plate immersed in adense electrolyte. Steric corrections, i.e., corrections due tothe inclusion of excluded-volume correlations, are found toamplify salt expulsion in all cases. Solvation forces arestrongly coupled to the ion-surface interaction and are there-fore ion-specific. The steric correction is mainly repulsive foruncharged plates for all salts but disappears or becomes at-tractive for highly like-charged surfaces at small separations.Furthermore, we observe that the charge distribution in thevicinity of a charged surface is mainly governed by ion-surface interactions but that excluded-volume correlationscan trigger overcharging in a dense, highly correlated sys-tem. Finally, we emphasize that even though the exact formof the ion-surface and ion-ion interactions depends stronglyon the studied system, NPB can be used in general for a wide
range of systems provided that ion-surface interactions andeffective diameters of the ions are accessible.
This paper is structured as follows. In Sec. II we de-scribe all methods involved in detail. In Sec. III we firstcompare the ionic structure for all salts and methods for ananoconfinement on the order of d�2 nm and effectiveconcentrations in the molar range. We show both results forlike-charged and uncharged plates. We then proceed by in-vestigating excluded-volume correlations in the case of theDonnan effect for NaI and LiCl. We then compute solvationforces for all salts for like-charged plates and compare themto the uncharged case. Finally, we inspect the charge distri-bution in the vicinity of a charged surface and will highlightwhether ion-surface interactions or ion-ion excluded-volumecorrelations drive overcharging. In Sec. IV we provide thesummary and concluding remarks.
II. METHODS
A. Explicit-water MD simulations
We perform explicit-water all-atom simulations with theMD package GROMACS 4.0.31,32 We simulate at constant par-ticle number N, constant pressure P�1 bar using an aniso-tropic Parrinello–Rahman barostat33 in the x- andy-directions, and a temperature T=300 K using a Nosé–Hoover thermostat34 �NPT ensemble�. The rectangular simu-lation box has periodically repeated edges of size of Lx
�Ly �4.15 nm and is delimited in the vertical z-directionby two walls specified in the following with a surface-to-surface distance d, which is defined as the distance on thez-axis between the centers of the respective surface C atoms�cf. Fig. 1�. We simulate a number of Nw=700–900 SPC/E�Ref. 35� water molecules and Ni=1–90 explicit ion pairs inthis one-dimensional confinement. The ions are nonpolariz-able and interact with the Coulomb and Lennard-Jones �LJ�interaction. Cross interactions are calculated with theLorentz–Berthelot mixing rules. We use the ion force-field
FIG. 1. Snapshot of a typical MD simulation with molar salt concentration.Cations �blue spheres� and anions �yellow spheres� in a one-dimensionalnanoconfinement of width d immersed in water depicted as one oxygen �redsphere� and two hydrogen �white spheres� atoms. d is the surface-surfacedistance between the centers of adjacent surface C atoms �turquoisespheres�. The picture was made using the VMD software �Ref. 76�.
164511-2 Kalcher, Schulz, and Dzubiella J. Chem. Phys. 133, 164511 �2010�
from Refs. 36–38. For completeness, the LJ interaction pa-rameters �� ,�� are summarized in Table I. Electrostatics istreated with the two-dimensional particle-mesh Ewald sum-mation method.39
The surface is modeled by a solidlike assembly of ato-mistic LJ spheres in a close-packed, harmonically restrained,hexagonal lattice arrangement. The LJ diameter is chosen sothat the atoms have the size of a methyl group �ii
=0.3905 nm,40 and the energy �ii=1.024kBT is chosen inorder to reproduce the contact angle of a simple, nonpolarorganic material, such as paraffin of �110°. This angle iscalculated by a simple mean-field integration over the inter-actions between the solid and the liquid.41 Typically,n=480 wall atoms are involved, of which ns=120 are situ-ated on the surface; the wall width is lz�1 nm, which, givenLx and Ly, corresponds to a volume number density of �wall
�28 nm−3. The positions of the wall atoms are harmonicallyrestrained in three dimensions with a force constantk=5000 kJ mol−1 nm−2, which relates the force to a one-dimensional displacement by Fi=−k�xi. The lattice con-stants are a�0.39 nm for the basal and b�0.64 nm for theheight parameter. A typical simulation snapshot of the mo-lecular slab-water-salt system is shown in Fig. 1.
In order to obtain effective ion-wall pmfs, we use um-brella sampling. A cation or an anion is placed in the waterphase near one wall and the ion-wall pmf is obtained by theweighted histogram analysis method.42,43 Runs for gatheringstatistics of 1.5 ns in time each after an equilibration periodof 500 ps are carried out in 80 distinct windows. We use aspring constant of 500 kJ mol−1 nm−2. The resulting ion-wall pmfs are shown in Fig. 2. We observe partial attractionfor the large I− and Cs+ ions, while Na+ and Cl− are repelledfrom the surface, corroborating with earlier studies.11,44 Amore astonishing feature is that the Li+ cation, even thoughhaving a very small van der Waals radius �cf. Table I�, isattracted to the hydrophobic wall instead of favoring hydra-
tion in bulk water. This could be attributed to a force-fieldproblem but is presumably due to the tightly bound first sol-vation shell that is not clearly distorted at the location of theminimum in the ion-wall pmf at a distance z=0.43 nm fromthe surface. Similar trends have already been seen at an air-water interface.10
The ion-ion pair potentials Vij�r� are taken from bulkexplicit-water MD simulations extrapolated to infinite elec-trolyte dilution.29 The ion-ion pair potential can be split intoa short-ranged part and a long-ranged Coulomb part via
�Vij�r� = �Vijsr�r� + zizj�B/r , �1�
respectively, where �B=�e2 /4��0� is the Bjerrum length,�=1 /kBT, zi is the ion valency, � is the dielectric constant,and �0 is the vacuum permittivity. Vij
sr�r� is taken from aprevious study;29 the cation-anion short-ranged pmfs areshown in Fig. 2, the cation-cation and anion-anion short-ranged pmfs are not shown. We calculated the osmotic coef-ficient in a bulk electrolyte for these particular ion-ion inter-actions for LiCl, NaCl, KCl, CsCl, and NaI bythermodynamic routes and MC simulations45 and found goodagreement with the experimental values. Due to the similar-ity between KCl and CsCl in terms of the structure and bulkosmotic coefficient,29 we compute the ion-wall pmf only forCs+ and will therefore focus on the CsCl salt.
It is well known that the dielectric constant in a bulkelectrolyte depends on the salt concentration and ision-specific.46 We obtain an estimation of the dielectric con-stant directly from MD simulations by calculating the dipolefluctuations of the water molecules and dividing by the ef-fective volume using the Kirkwood formula for three-dimensional periodic conducting boundary conditions.47 Weare aware that in our slab geometry the latter is not rigor-ously satisfied but assume that this perturbation is not criticalin relation to the magnitude of the dielectric constant of bulkwater. The volume accessible to the water molecules is lessthan the total volume inside the slab since the repulsive in-teractions with the walls prevent the water molecules to
TABLE I. Ion-water oxygen �O�, wall-water oxygen, and SPC/E water LJparameters that are used in explicit-water MD simulations. The parameters�iO and �iO are the LJ size and energy, respectively, between atom i andwater oxygen �O�, with the variable i denoting an ion, a wall carbon atom�C�, or a water oxygen �O�. Ion parameters are taken from Refs. 36–38.
FIG. 2. Effective ion-surface potentials Viext�z� �top� and the short-ranged
part of the cation-anion pair potentials Vijsr�r� �bottom�. The latter curves are
shifted vertically for a better view where the shifted x-axes are depicted bydotted lines. The potentials are obtained from explicit-water MD simulations�Ref. 29�.
164511-3 Electrolytes in a nanometer slab-confinement J. Chem. Phys. 133, 164511 �2010�
come too close. We quantify the effective distance perpen-dicular to the walls accessible to the water molecules as inprevious work,48
dwat = d −�CO + �OO
2= d − 0.335 nm, �2�
with �CO and �OO being the LJ parameters given in Table I.The accessible volume that we use in the calculation of thedielectric constant is therefore Vwat=LxLy dwat.
A previous study29 showed that the dielectric constant ina bulk electrolyte can be fitted by means of the form ����=��0� / �1+ �C��� with the salt concentration � and an ion-specific constant C given in Table II. We will use this expres-sion also for confined electrolytes introducing effective con-centrations �̄=� /d, where � is the ion area density and d isthe size of the confinement as defined before. � is easilyaccessible in a one-dimensional confinement by simply inte-grating the density profile along the z-axis �perpendicular tothe surface�, viz., �=0
d��z�dz.
B. Mapping onto hard spheres and nonadditivity
The ion-ion short-ranged pmfs Vijsr�r� exhibit oscillations
on a kBT energy scale due to hydration effects. As it is com-putationally easier and more convenient to deal with hardspheres �HSs�, we intend to map the intricate ion-ion inter-actions onto simple HS interactions. We use therefore the BHscheme that relates the short-ranged pair potentials Vij
sr�r� toan effective hard-core diameter via1
�ij = − drf ij�r� , �3�
the subscripts i , j=+,− specifying cations or anions, whichconsists of a one-dimensional integral over the Mayer func-tion of the short-ranged pair potentials
f ij�r� = �exp�− �Vijsr�r�� − 1� . �4�
Figure 3 shows both f ij�r� and �ij for all ion-ion combina-tions of the NaCl salt. The �ij are not to be confused with theLJ parameters of Table I. The Mayer functions exhibit oscil-lations and can become positive due to attractive regions inthe ion-ion short-ranged pmfs. This attraction can lead tomuch smaller than expected effective diameters. A typicalexample is Na+–Na+, which has a BH diameter of �++
=0.22 nm even though the ion-ion short-ranged potentialV++
sr �r� already diverges to infinity at a distance r�0.3 nm.
The BH diameters �ij for all studied salts are summarized inTable III. As ionic interactions and pairing affinities are gov-erned by hydration effects, we ascertain that it is not possibleto determine the effective diameters �ij by looking atvacuum van der Waals radii only. Another striking exampleis the Li+–Li+ pair, which has the smallest van der Waalsradius but exhibits the largest effective BH diameter of allcations studied.
A more detailed inspection of the values �ij in Table IIIshows that electrolytes resemble size-asymmetric and nonad-ditive hard-sphere mixtures, viz.,
�++ � �−−, �5�
�+− = 12 ��++ + �−−� �1 + �+−� , �6�
with a nonadditivity parameter �ij�0. It is well known thatnonadditivity strongly influences the fluid structure andphase behavior in binary hard-sphere mixtures.49,50 �ij �0results in an increasingly repulsive cross-correlation and canlead to stable fluid-fluid demixing transitions even at smallpositive values �ij =0.2.51,52 �ij 0 not only entails a signifi-
TABLE II. Ion-specific C parameters taken from a previous study �Ref. 29�.The salt-specific dielectric constant is then calculated with ����=��0� /�1+ �C��� with ��0� being the dielectric constant at infinite dilution and �being the concentration in mol/l �M�.
SaltC
�1 /M�
LiCl 0.31NaCl 0.27KCl 0.24CsCl 0.23NaI 0.34
FIG. 3. Mapping of ion-ion short-ranged pair potentials onto hard-sphereinteractions in the case of sodium chloride �NaCl�. Shown is the short-ranged pair potential Vij
sr �black full�, its Mayer function f ij�r�= �exp�−�Vij
sr�r��−1� �red dashed�, and the effective BH hard-sphere diam-eter �ij in the form of a delta function �violet�.
TABLE III. Barker–Henderson diameters �ij and nonadditivity �+− definedby �+−= 1
164511-4 Kalcher, Schulz, and Dzubiella J. Chem. Phys. 133, 164511 �2010�
cantly lesser correlated system but can also cause partialclustering as observed experimentally in superparamagneticcolloidal suspensions.53
The CsCl salt, for instance, has a large negative nonad-ditivity �+−=−0.49. CsCl will therefore show a tendency formixing. In strong contrast, we expect NaI, having a largepositive nonadditivity of �+−=0.36, to be strongly influencedby excluded-volume correlations and inclined to phase-separate. Indeed, at high concentrations, we will see that theNaI salt in confinement will lead to a highly asymmetricsystem and strong layering. A positive nonadditivity in gen-eral competes with electrostatics, which always favors mix-ing of cations and anions.
C. Implicit-water MC simulations
The effective ion-surface potentials Viext�z� and the short-
ranged ion-ion pair potentials Vijsr�r� are fed into standard MC
simulations in the canonical ensemble using Metropolis’algorithm.54,55 The effective potentials are linearly interpo-lated, continued to infinity on the lower bound, and set tozero on the upper bound for cut-off distances of z between0.95 and 1 nm and r=1 nm. We simulate up to N=308 saltpairs in a slab with width d=2 nm and periodic boundaryconditions in the lateral xy-directions with box lengths Lx
=Ly =4.2 nm. A typical simulation runtime is 106 MC stepswith 104 equilibration steps. To accurately account for thelong-ranged electrostatics, we use the two-dimensionalLekner–Sperb summation.56,57 The Lekner–Sperb potential istabulated using a grid size of 128 in each direction, wherethe points are quadratically distributed and linearly interpo-lated. For small separations �r 1.710−4 nm�, the poten-tial is calculated explicitly.
D. DFT approximations for electrolytes
1. Introduction
In density functional theory, equilibrium one-particledensities are obtained via minimization of a grand potentialfunctional �.1 Thus, the equilibrium ion densities �i satisfythe variational principle ����+ ,�−� /��i=0. The grand poten-tial functional of charged particles has the general form
���+,�−� = F��+,�−� − �i ��i − Vi
ext�r����i�r��dr� , �7�
where Viext�r�� is the external potential acting on ions of spe-
cies i=�, �i is the chemical potential, and F is an intrinsicfree energy functional of the one-particle densities. F is typi-cally split into ideal and excess parts via
F��+,�−� = �i
Fiid��i� + Fex��+,�−� , �8�
with the ideal contribution Fid��i�=kBT�i�r���ln��3�i�r���−1�dr�, where � is the de Broglie thermal wavelength, and anexcess contribution Fex, which we assume can be separatedinto a mean-field, purely Coulombic and a correlation part,
Fex��+,�−� =kBT
2�B ��r����r���
�r� – r���dr�dr�� + Fcorr��+,�−� ,
�9�
with
��r�� = z+�+�r�� + z−�−�r�� �10�
as the charge density.Minimizing the grand potential � with respect to �i in a
with the activity �i=exp���i� /�i3 of species i, where ��z� is
the local electrostatic potential and
ci�z� = −���Fcorr�
��i�12�
is the one-particle direct correlation function.1 For fixed�i�z�, the electrostatic potential satisfies Poisson’s equation,
e��2��z�
�2z= − 4��B��z� . �13�
Neglecting correlations �ci�z�=0�, Eqs. �10�, �11�, and �13�yield the standard mean-field PB equation. In the following,we will examine local and nonlocal approximations for thecorrelation free energy Fcorr��+ ,�−� and ci�z�.
In DFT, one can easily switch from a canonical to agrand canonical description. In the canonical ensemble �fixednumber of particles�, we fix the ion area density �. In thegrand canonical ensemble �fixed chemical potential or reser-voir density �0�, the activity is specified along with thechemical potential, see later in the text. In the case of PB, weget �i=�0.
2. Poisson–Fermi
One way to extend PB to consider steric interactions in alocal way is to derive the free energy functional from a lat-tice gas model.23 For monodisperse systems, each ion occu-pies a site using a certain excluded-volume and inhibitingother ions to occupy that site. It can easily be extended to apolydisperse system by Taylor-expanding the free spaceentropy.24 The incorporation of the effective ion-wall poten-tials finally yields for a one-dimensional confinement the so-called Poisson-Fermi �PF� distribution,
�e�2��z�
�2z= − 4��B�̄
�izi exp�zie���z� − Viext�z��
1 − �̄�i�ii3�1 + exp�zie���z���
,
�14�
with the effective density �̄=� /d, where � is the ion areadensity, d is the plate separation, and the �ii are the effectivecation-cation and anion-anion BH diameters. Note that thecation-anion BH diameters do not enter in Eq. �14�.
3. LDA for nonadditive hard spheres
In the framework of DFT, local extensions to PB can beemployed by using the local density approximation.1 In the
164511-5 Electrolytes in a nanometer slab-confinement J. Chem. Phys. 133, 164511 �2010�
LDA, the correlation excess free energy is given by an inte-gral over a local excess free energy fcorr per volume of ahomogeneous solution of density �,
Fcorr��� = dr�fcorr��� . �15�
As we showed before, after subtracting electrostatic interac-tions, ions resemble asymmetric and nonadditive HS mix-tures. Thus, fcorr should describe the free energy of a binary,asymmetric, and nonadditive HS mixture for 1:1 salts. Asthere is no closed accurate expression available, we resort toa virial expansion of fcorr up to third order given by58
�fcorr = �ij
�i� jB2ij +
1
2�ijk
�i� j�kB3ijk,
with the virial coefficients
B2ij =
2
3��ij
3 ,
B3ijk =
4
3 �
6�2
�ck;ij�ij3 + cj;ik�ik
3 + ci;jk� jk3 � ,
ck;ij = �k;ij3 +
3
2
�k;ij2
�ij�i;jk� j;ik,
�k;ij = �ik + � jk − �ij , �16�
where �k;ij is interpreted as an effective diameter of sphere kas seen from the pair i and j. In the following, we will referto the second-order virial expansion as B2 and to the thirdorder virial expansion as B3.
The direct correlation function now follows from thefirst functional derivative of the correlation term of the ex-cess free energy
���Fcorr���i
= − ci�z� = 2�j
� jB2ij +
3
2�jk
� j�kB3ijk. �17�
The correction to the PB theory is local in terms of the factthat the correlation function depends on the amplitude of thelocal density only.
4. NPB and NPB-HS
As a starting point to the nonlocal treatment of the cor-relation term of the excess free energy, we will introduce acoarse-grained density �̃�r�� defined through an appropriatelychosen, a priori unknown, normalized weight function w�r��as �̃�r��=w��r�−r������r���dr��. Equation �15� then reads as
Fcorr��� = �ex��̃���r��dr� , �18�
where �ex��̃�= fcorr��̃� / �̃ is the excess free energy per particleof the homogeneous fluid at a density �̃.1 Equation �18� rep-resents a WDA. In the low density limit, the leading term ofthe virial expansion yields �ex���=kBT�B2, with B2 beingthe second virial coefficient.1 As B2 is defined by the Mayerfunction in Eq. �4�, viz., B2=− 1
2f�r�dr�, we obtain the non-local expression1
�Fcorr��+,�−� = −1
2�ij dr�dr���i�z�� j�z��f ij��r� − r���� .
�19�
We emphasize that in our assumption Fcorr approximates theexcess free energy of a binary fluid interacting with theshort-ranged pair potentials Vij
sr�r�. Explicit Coulombic corre-lations, which become important at high electrostatic cou-pling, are therefore neglected.
The direct correlation function is now given by convo-lutions of the density profile over the Mayer functions of theshort-ranged potentials,
ci�z� = �j dr��� j�z��f ij��r� − r���� . �20�
In the case of hard spheres, the Mayer function degeneratesinto a shifted Heaviside step function, f ij�r�=���ij −r�−1.Convolutions over step functions are easier and faster tocompute than convolutions over the Mayer function of thefull short-ranged potential. In the following, we will refer toNPB-HS when using this HS expression with the effectiveBH diameters in Table III and to NPB when employing theMD-derived short-ranged potentials Vij
sr described inSec. II A.
When we refer to the grand canonical ensemble with agiven reservoir density �0, we choose the activities �i of Eq.�11� in a way that the electrolyte concentration in the middleof the slab matches �0 in the limit of infinite wall-to-walldistances d. This activity depends on the direct correlationfunctions ci.
25
We are aware that in the realm of DFT Rosenfeld’sfundamental-measure theory �FMT� �Ref. 59� is a moreelaborate way to treat hard-sphere fluids nonlocally. The rel-evant generalization of FMT to nonadditive binary HS mix-tures has been developed by Schmidt and was applied to abroad range of nonadditivities.60,61 Nonetheless, as FMT in-volves a decomposition of the Mayer function and requires avery sophisticated treatment,62 we content ourselves with ourNPB and NPB-HS theories in the following.
5. Poisson–Boltzmann solver: Numerical details
The equations of our DFT-derived approaches to the PBtheory are solved by means of a general relaxation method.The domain of interest, the z-axis in the case of our one-dimensional confinement, is approximated by a mesh of upto 2100 grid points. Each mesh point corresponds to a finitedifference equation that relates two neighboring points. Start-ing from an initial guess, the results relax to the actual solu-tion. The boundary conditions at the walls link the chargedensity � to the first derivative of the electrostatic potential
e����z�
�z= − 4��B
�
e. �21�
The convolutions in the NPB equations are treated withstandard one-dimensional fast Fourier transform techniques.We use a binning of N=8192 points and a Nyquist frequencyof fc=1 / �2�x�=250 nm−1. Further details as to the methodsinvolved can be found elsewhere.63
164511-6 Kalcher, Schulz, and Dzubiella J. Chem. Phys. 133, 164511 �2010�
E. Solvation forces
We place two infinitely large surfaces in an ionic reser-voir of concentration �0, meaning that the chemical potential�0 of the electrolyte inside the slab is equal to the chemicalpotential in the reservoir. We compute solvation �s� forces bycalculating the difference in the “internal” pressure betweenthe surfaces and the “external” bulk pressure of the reservoir.The bulk pressure is given by the internal pressure in thelimit of infinite surface separations d→�.8 Thus, we obtainfor the solvation pressure
Ps�d� = �P�d� − P���� . �22�
To calculate the internal ionic pressure, typically a contacttheorem can be derived by differentiating the free energywith respect to the surface separation.25,64 Here, we use asomewhat more general expression,65 viz.,
�p�z��z
+ �i
zi�i�z����z�
�z+ �
i
�i�z��Vi
ext�z��z
= 0, �23�
where p is the local pressure along the z-axis and � is theelectrostatic potential introduced before. By replacing thedensity in the second term by the Poisson equation �Eq. �13��and integrating from z=0 to z=d, we see that given theboundary conditions of Eq. �21�, the second expression van-ishes and the total ionic pressure on one wall inside the slabsimply is
P�d� = − �i
0
d
dz�i�z��Vi
ext�z��z
. �24�
We are then enabled to investigate the impact of stericion-ion excluded-volume correlations by comparing the re-sults of the PB and NPB-HS methods, viz.,
�Ps�d� = PsNPB-HS�d� − Ps
PB�d� , �25�
which is the steric correction to the solvation pressure. Wecan integrate again to obtain the steric correction to the in-teraction between the surfaces,
�Vs�d� = − Ad
�
dd��Ps�d�� �26�
for a unit area of A.
III. RESULTS AND DISCUSSION
This section is organized as follows. We first comparethe electrolyte structure in a slablike, one-dimensional nano-confinement of d=2 nm on an explicit-water MD, implicit-water MC, DFT-PB, and PB level, and hence validate ourcoarse-graining strategy. We next single out our nonlocalNPB-HS approach and consider salt expulsion between twolike-charged plates, the Donnan effect. We then study thedistance resolved solvation forces between neutral andcharged walls. Finally, we investigate the charge density nextto a single charged plate and discuss overcharging. We willscrutinize, in particular, the effect of excluded-volume corre-lations. During this procedure, we assume that the dielectricconstant is uniform, i.e., does not change in space. Wechoose the dielectric constant as realistic as possible, which
means concentration dependent and salt-specific according toTable II, when investigating electrolyte structure in neutralconfinement. For charged plates, however, we stick to thevalue of pure water, i.e., �=80 and �B=0.71 nm, as the sys-tem is highly asymmetric due to the counterion excess and areasonable value of the dielectric constant is hard to estimate.
A. Electrolyte structure
1. MD and MC
Figure 4 displays a comparison of MD- and MC-deriveddensity profiles for LiCl, NaCl, CsCl, and NaI. The MC pro-files are shown using, on one hand, realistic, MD-deriveddielectric constants, i.e., �=33, 38, 47, and 36 for LiCl,NaCl, CsCl, and NaI, respectively, and on the other hand,one corresponding to experimental pure bulk water. We ob-serve in general good agreement between the results for MDand MC for an area density of ��3.04 nm−2 even thoughthe MC simulations with the realistic dielectric constant per-form better. This trend substantiates even more for higherconcentrations �not shown�, i.e., larger dielectric deviationfrom bulk water. The MC simulations with MD-derived di-electric constants reproduce particularly well the ionic con-centration at the walls and in the middle of the slab in theMD simulations. However, we note that the agreement ofMC in comparison with MD is ion-specific. For LiCl, forexample, MC is able to predict the MD profiles almost per-fectly, whereas for NaCl, the cation density profile exhibitsdifferences. This is due to many-body and explicit-water ef-fects that seem to play a more important role for Na+, theeffectively smallest cation in our study, than for Li+, the ef-fectively biggest. Nonetheless, we highlight that MC simula-tions with correctly chosen ion-ion and ion-wall interactionsand a reasonable dielectric constant are able to reproduce theelectrolyte structure in an explicit-water system in a slab-confinement fairly well. The importance of a properly chosendielectric constant has been pointed out in literature so far
FIG. 4. MD- �circles� and MC-derived ion density profiles in a d=2 nmslab-confinement. The results for MC are shown for both realistic�MD-derived� dielectric constants, i.e., �=33,38,47,36 for LiCl, NaCl,CsCl, and NaI, respectively �red continuous line�, and the dielectric constantof experimental pure bulk water �=80 �blue dashed line�. The ion areadensity is ��3.04 nm−2 and the effective ion concentration is �̄�2.5M.
164511-7 Electrolytes in a nanometer slab-confinement J. Chem. Phys. 133, 164511 �2010�
only for bulk thermodynamic properties such as the osmoticpressure.28,45
2. MC and DFT-PB
Figures 5 and 6 show a comparison of ion density pro-files for MC, PB, PF, and hard-sphere LDA approaches fortwo different area densities. We observe that PF and all localcorrections to PB tend to level off regions of high and lowionic concentrations within the slab. While in the case ofNaCl the B3 and B2 LDA methods can describe well theion-ion correlations seen in MC, they fail to reproduce theion density profiles of a lower correlated system such asCsCl ��+−=−0.49 0�. Local theories in general performbadly for electrolytes with ions attracted to the interface asthe lithium cation or the iodide anion �not shown�. This isdue to the indiscriminative energy penalty imposed on highsalt concentrations. There are no systematic trends to vali-date any of the discussed local theories; on the contrary, inmost cases, the “unimproved” Poisson–Boltzmann theoryseems to be better or equally well-suited for a wide range ofconcentrations and salts such as CsCl or LiCl. For a furtheranalysis, we compute the pressure on one wall, defined as inEq. �24�, for MC and the B3 LDA for all salts at the higher
ion area density. In Table IV we see that the B3 method, eventhough performing better than PB, overestimates the pressureof MC always by 20%–30%, independently of the salt stud-ied. Furthermore, none of the tested LDAs are able to com-pute stable ion density profiles for NaI. It was noted in lit-erature before that local density approximations for hard-sphere ions lead to instabilities beyond certain ion sizes.30 Asa conclusion, we state that LDAs are able to predict neitherthe electrolyte structure nor the correct pressure for inhomo-geneous electrolytes in confinement.
In Figs. 7 and 8 we show a comparison of the same MCion density profiles with both the NPB-HS and NPB theories.In Fig. 7 we recover, on one hand, the quasi-identical perfor-mance of NPB-HS and NPB of our previous work;27 on theother hand, we observe that the nonlocal theories, eventhough neglecting electrostatic correlations, are able to re-produce the electrolyte structure of the MC simulations evenat regimes of low dielectric constants fairly well with theexception of NaI. In addition, in Fig. 8, we compare theresults for NPB-HS on an additive, i.e., �+−=0 and therefore�+−= 1
2 ��+++�−−�, and nonadditive level, assessing the im-pact of �+−. We see only a minor difference for the NaCl andLiCl salts, whereas for CsCl and NaI, a difference and betterperformance of the nonadditive model is discernible. Wequantify the latter by analyzing the relative error in pressurecompared to MC in Table IV. LiCl and NaCl perform equally
FIG. 6. The same as in Fig. 5 for an effective concentration of �̄�5M andan ion area density of ��6 nm−2.
FIG. 5. MC-derived ion density profiles compared to PB, PF, and second�B2� and third order �B3� hard-sphere LDA expansions. LiCl, NaCl, andCsCl are shown for d=2 nm, an effective concentration of �̄�3M, and anion area density of ��3.6 nm−2. The dielectric constant is chosen accordingto the formula ���̄�=80 / �1+C ��̄ /M�� in all cases with a salt-specificconstant C given in Table II.
TABLE IV. Total pressure exerted by the ions on one wall for LiCl, NaCl, CsCl, and NaI. The system is thesame as shown in Figs. 6 and 8. The pressure for PB, LDA B3, additive �NPB-HS add�, and nonadditiveNPB-HS is shown in terms of the relative error to MC pressure in percent. For NaI, the B3 method failed toyield a stable result.
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well on the additive and nonadditive levels. This is not sur-prising given their small nonadditivity �see Table III�. ForCsCl and NaI, on the contrary, the nonadditive theory is evi-dently superior. For CsCl, for example, the additive NPB-HSperforms equally bad as the B3 method in terms of pressure,while the nonadditive theory is better by a factor of 2. ForNaI, which exhibits the largest positive nonadditivity ��+−
=0.36�0� of all salts studied, even the nonadditive NPB-HSunderestimates the MC-derived pressure by roughly 50%.We argue therefore that systems with large ions in terms ofeffective size and a high positive nonadditivity are domi-nated by excluded-volume interactions and need a more so-phisticated treatment, for example, a nonadditive FMTapproach.60
Finally, in Fig. 9 we show ion density profiles for LiCland NaI for both negatively and positively like-charged sur-faces. Since the system has to be electroneutral, we observean asymmetry in the cation and anion area densities in theform of � /e= ��−−�+� /2, where � is the surface charge den-sity of one wall. This asymmetry leads to a very different ion
structure, depending on the nature of the counterions. NaI isan interesting example since Na+ is the effectively smallestand I− is the largest ion in our study. For this reason, nega-tively charged walls lead to a relatively unstructured system,whereas positively charged walls lead to accumulation ofcounterions and depletion of coions at the surface. In thelatter case, there are roughly four times as many counterionsthan coions near the positively charged surface at z�0.5 nm. For negatively charged walls, on the contrary,there are roughly as many counterions than coions for dis-tances z�0.6 nm from the wall.
B. Salt expulsion: Donnan effect
When two plates in contact with a �much larger� electro-lyte reservoir of density �0 are charged with an identicaluniform charge density, counterions will be attracted to thesurfaces in order to ensure electroneutrality, as has been an-ticipated at the end of the last section. Coions, on the otherhand, will be repelled by the latter. The salt, which has bydefinition the same concentration as the coions, is thereforeexpelled from the slab with increasing surface charge. Thisso-called Donnan effect can be treated analytically with alinearized Poisson–Boltzmann equation in the case of anideal gas and hard walls.2 Our coarse-grained PB andNPB-HS theories �in the grand canonical ensemble� permitus then to examine the impact of ion-ion excluded-volumecorrelations and ion-surface interactions on the Donnan ef-fect. We compare the results for LiCl and NaI in the follow-ing for positive and negative surface charges and a reservoirdensity of �0=3M.
1. Ideal gas: PB and analytical solution
In the case of an ideal gas, i.e., point charge particles,and �structureless� hard walls the PB equation can be linear-ized for sufficiently low surface charges. Even though hardlybeing a new result, we treat this case quickly as a starting
FIG. 7. MC- and PB-derived ion density profiles for LiCl, NaCl, CsCl, andNaI compared to nonlocal theories NPB-HS and NPB. The same distanced=2 nm, ion area density ��3.6 nm−2, and dielectric constant as in Fig. 5.
FIG. 8. MC- and PB-derived ion density profiles for LiCl, NaCl, CsCl, andNaI compared to NPB-HS in the case of additive and nonadditive HSeffective diameters �+−. The same distance d=2 nm, ion area density��6 nm−2, and dielectric constant as in Fig. 6.
FIG. 9. MC-, NPB-HS-, and PB-derived ion density profiles for LiCl �upperpanels� and NaI �lower panels� for both positively �left� and negatively�right� like-charged surfaces in a d=2 nm slab-confinement with surfacecharge densities ���=0.4e nm−2. The coion area density is ��3.6 nm−2 andthe dielectric constant is set to 80.
164511-9 Electrolytes in a nanometer slab-confinement J. Chem. Phys. 133, 164511 �2010�
point for clarity. In a one-dimensional slab geometry, theresulting electrostatic potential satisfying the boundary con-ditions of Eq. �21� is2
��z� =4��B��/e�
� sinh��d/2�cosh��z� , �27�
with � being the inverse screening length �=�8��B� and dbeing the width of the slab. In PB the salt concentration isgiven by the Boltzmann distribution ���z�=�� exp���e��z��. We can thus write
�̄s =1
d
0
d
�� exp���e��z��dz =�
d, �28�
where the � sign refers to the two distinct cases of positivelyand negatively charged coions. We will consider the first-and second-order terms in the Taylor expansion of the expo-nential in Eq. �28�. The first-order term, which can be foundin literature,2 is proportional to �; the second-order term �theexact expression is not particularly instructive� is propor-tional to �2. In the inset in Fig. 10, we show, for d=2 nm,how the salt concentration in the slab depends on the abso-lute value of the surface charge density. The sign of the latterindeed does not change the result in the case of an ideal gas.At �=0, the mean salt concentration equals the concentrationin the reservoir �0. The salt concentration decreases thenwith the surface charge. This decrease is linear in � only forsmall surface charge densities up to 0.2e nm−2. By includingthe second term in the Taylor expansion, we are able to re-produce the numerical result up to surface charge densities of0.8e nm−2.
2. NPB-HS and PB
In Figs. 10 and 11 we show the dependence of the effec-tive mean coion concentration �̄s=� /d on the surface chargedensity in the case of positive and negative surface chargesfor LiCl and NaI. As the ion-wall interaction is repulsiveclose to the surface, the ions cannot come indefinitely close.Hence, we do not recover the reservoir density �0 for �=0.
We will generally underestimate the effective mean coionconcentration in terms of the �restricted� volume accessibleto the ions. We are not bothered since we are interested in therelative decrease in the salt concentration with the surfacecharge density. Indeed, the effective concentrations for neu-tral walls will also be different for PB and NPB-HS, giventhat in the grand canonical ensemble the particle numbermay vary for the same reservoir density �0=3M.
In Fig. 10 we assess the effect of ion-ion correlations andion-wall interactions on the Donnan effect in the case ofnegatively charged walls �sodium being the counterion� andpositively charged walls �iodide being the counterion�. Wefirst turn our attention toward the results of PB and therewith“switch on” the ion-wall interactions only. We observe amore pronounced decrease in the salt concentration in thecase of negatively charged walls, indicating that sodium de-pletes the coion concentration more than iodide does. Wemake the following assertion. The attractive part of the ion-wall pmf for iodide drives the counterions to the wall andtherefore leads to a lower effective surface charge densitythan in the case of sodium being the counterion. We expectexcluded-volume effects to depend particularly on the effec-tive ion sizes of the counterions �ij and on the nonadditivityparameter �+−. As stated in Table III, NaI is a highly asym-metric and nonadditive salt. We anticipate that iodide as acounterion with an effective diameter of �−−=0.47 nm willsqueeze out more salt than sodium that is less than half asbig ��++=0.22 nm�. As we can see in Fig. 10, this is whathappens. The excluded-volume impact overcompensates theeffect of the pmf so that the Donnan effect is overall greaterfor positive surface charges.
In order to validate our assessments, we repeat in Fig. 11the same procedure for LiCl. The latter is a more symmetricand less nonadditive salt than NaI with an attractive part inthe cation-wall pmf and a purely repulsive anion-wall inter-action �see Fig. 2�. Examining the results for the PB method,we recover the same trend as for NaI. The bigger theattraction/repulsion discrepancy between cation and anionpmfs, the wider the gap in salt expulsion will be whenchanging the sign of the surface charge density. As to theexcluded-volume correlations, the small relative change ofsalt expulsion between the PB and NPB-HS methods cor-roborates with the fact that LiCl is the most symmetric salt
FIG. 10. NPB-HS- and PB-derived Donnan effect for NaI in a d=2 nmslab-confinement in the case of positively and negatively like-charging thewalls. The reservoir density is �0=3M. The difference in effective concen-trations �̄=� /d of the two methods for neutral walls ��=0� is due to differ-ent �reservoir� fugacities. Inset: numerical solution of the mean-fieldPoisson–Boltzmann theory for an ideal gas between two hard walls com-pared to the analytical expansion of Eq. �28� up to second order.
FIG. 11. The same as in Fig. 10 for LiCl. Inset: the same effect for a smallerconfinement of d=1.5 nm.
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we study and does not display a large difference in the im-pact of ion-ion excluded-volume correlations switching cat-ions for anions. In the inset in Fig. 11, we show the sameanalysis with a smaller wall-to-wall distance d=1.5 nm. Werecover the same trends, implying that our analysis is validalso for smaller confinements as long as the ion-wall pmfs ofthe two walls do not overlap.
C. Solvation forces
In an earlier study, we pointed out that for unchargedsurfaces steric corrections are mainly repulsive for all salts.27
We now broaden our scope to three different reservoir con-centrations and on positively and negatively charged walls.In addition, we compare our findings with the results ofBurak and Andelman25 and earlier work66 that predict a pos-sible net attraction due to ion-ion correlations for small plateseparations.
In Fig. 12 we compare the solvation pressure Ps�d� ofLiCl for three different reservoir densities �0=1M, 3M, and5M in an uncharged slab-confinement. As we increase thedensity, the pressure variations get more pronounced, in par-ticular, for NPB-HS. The loci of the maxima and minimastay untouched, a behavior that is expected since the ion-wallinteractions are unaffected by the increase in the concentra-
tion. The steric correction, in turn, is always repulsive andincreases with �0. We observe a change of one order of mag-nitude in the repulsive barrier going from �0=1M to �0
=3M and of roughly a factor of 2 increasing the reservoirconcentration from �0=3M to �0=5M. The repulsive barriersfor �0=1M and �0=5M are �0.4kBT and �55kBT, respec-tively, for two close colloidal surfaces with an area A=10 nm−2. This corresponds to a difference of two orders ofmagnitude.
In Figs. 13–16 we choose a reservoir density of �0
=3M and show the solvation pressure Ps�d� obtained forcharged and uncharged walls with PB and NPB-HS for LiCl,NaCl, CsCl, and NaI, respectively. A common feature of allsalts studied is the strong dependence of Ps on the shape ofthe ion-wall potentials Vi
ext. A nice example is LiCl with thecation partially attracted and the anion repelled from the sur-face. While a positive surface charge leads to a high solva-tion pressure mainly due to chloride counterions, Ps featuresregions of strong attraction for negative surface charges thatare related to the loci and magnitudes of attraction in theion-wall short-ranged potentials of the lithium counterions.We observe the same effect for NaI in Fig. 16. The anion andcation Vi
ext of NaCl being similar translate into like shapes ofPs for positively and negatively charged walls in Fig. 14.
Turning our attention now to the steric correction, wenote another trait common to all salts. The mainly repulsivebarrier induced by ion-ion correlations for uncharged sur-faces decreases with increasing surface charges, indepen-dently of the sign of the latter. We explain this behavior with
FIG. 13. Distance resolved solvation pressure Ps�d� for LiCl in the case ofneutral and charged walls with a reservoir density of �0=3M. The results forsurface charge densities of �= �0.4e nm−2 are shown. Inset: the corre-sponding steric corrections for a unit area A=1 nm2.
FIG. 12. Distance resolved solvation pressure Ps�d� for LiCl in the case ofneutral walls. The results for three different reservoir densities �0
=1M ,3M ,5M are shown. Inset: the corresponding steric corrections for aunit area A=1 nm−2.
FIG. 14. The same as in Fig. 13 for NaCl.
FIG. 15. The same as in Fig. 13 for CsCl.
164511-11 Electrolytes in a nanometer slab-confinement J. Chem. Phys. 133, 164511 �2010�
the following argument. For small confinements and highsurface charges, the pressure inside the slab is dominated bythe force exerted by the counterions on the surfaces. Weobserve that for d 1.2 nm the two pressures calculatedwith PB and NPB-HS start to converge. This is expectedsince nonlocality plays a less important role in confinementsof dimensions as small as an effective ion diameter. Notethat, in general, the ions do not come closer than �0.38 nmto the walls �cf. Fig. 2�. Since the term P��� in Eq. �22� isalways smaller for PB than for NPB-HS due to excluded-volume effects on a primitive model level, we obtain amainly negative contribution to Eq. �25� for small confine-ments and high surface charge densities. Indeed, Figs. 13–16show that for �= �0.4e nm−2 the solvation pressure for PBis always larger than for NPB-HS for wall-to-wall separa-tions of d 1.1 nm.
For LiCl in Fig. 13, only a very small repulsive barrier isstill discernible for a negative surface charge density �=−0.4e nm−2, whereas for positive surface charges, the stericcorrection becomes purely attractive. For NaCl, CsCl, andNaI in Figs. 14–16, the steric barrier disappears indepen-dently of the sign of the surface charge and gives way to apurely attractive correction.
We corroborate at this point with the aforementionedstudy25 that the steric correction due to ion-ion correlations ismainly attractive for highly charged surfaces, even thoughwe operate on a primitive model where excluded-volumeinteractions are purely repulsive, whereas Burak and Andel-man used an attractive correction. We also note that the formof the solvation interaction is strongly coupled to the ion-surface interaction and a net attraction cannot be predictedwithout knowledge of the latter.
D. Overcharging
Overcharging is in essence known as the phenomenonwhere an electric double-layer appears to attract morecharges from counterions than is needed to compensate forthe surface charge. This effect is very important to evaluatelong-ranged interactions between two charged surfaces in anelectrolyte solution. Our aim in this paper is not to provide adetailed study of the phenomenon because we use only asimple hydrophobic surface and limit ourselves to monova-lent ions. Indeed, detailed reviews as to the “physical” and“chemical” nature of overcharging can be found in
literature.20,67 Our coarse-grained methods PB and NPB-HSdo, however, allow us to compare the effect of ion-specificion-surface interactions and excluded-volume correlations onovercharging. In this simplified picture, the nonelectrostaticion-wall interactions mirror the role of chemical adsorption,whereas ion-ion correlations mimic the physical component.More explicitly, we want to inspect the crucial role assignedto excluded-volume by the authors of a previous study.68
We place one infinitely large surface in an ionic reservoirof concentration �0. The system is equivalent to that in Sec.III C except for the fact that we examine only a single sur-face, meaning that we set the intersurface distance d to in-finity. To study overcharging, we calculate the net chargearea density in the system. This charge density is given bythe surface charge density at the wall and decreases to zerofor large distances to maintain electroneutrality. We obtainthe net charge area density simply by integrating over thecharge density of Eq. �10�,
�tot�z�/e = �/e + 0
z
��z��dz�. �29�
We proceed with a similar analysis as in Sec. III B 2. Wefirst discuss results for the PB method and inspect the role ofion-surface interactions. We then compare them to the resultsfrom NPB-HS where we “switch on” ion-ion excluded-volume correlations. In Fig. 17 we show the distance re-solved net charge area density of a plate immersed in a LiClelectrolyte scaled by the surface charge density. We use sur-face charge densities of �= �0.4e nm−2 and a reservoir den-sity of �0=3M. The PB results show a distinct difference fornegative �Li+ being the counterion� and positive �Cl− beingthe counterion� surface charges. If Li+ is the counterion, thesurface charge is compensated already at a small distance z�0.5 nm from the plate. The net charge area density is thenshown oscillating around zero. In sharp contrast to the latter,in the case of a positive surface charge, the net charge areadensity becomes even slightly more positive up to distancesof z�0.6 nm and only then decays slowly to zero. Theseconsiderably different effects are obviously related to theion-surface interaction that is attractive for Li+ and repulsivefor Cl−. Figure 18 exhibits a qualitatively similar picture in
FIG. 16. The same as in Fig. 13 for NaI.FIG. 17. Distance resolved net charge area density �tot�z� divided by thesurface charge density for LiCl. The reservoir density is �0=3M. The resultsfor both negative and positive surface charge densities are shown with �= �0.4e nm−2.
164511-12 Kalcher, Schulz, and Dzubiella J. Chem. Phys. 133, 164511 �2010�
the case of NaI, keeping in mind that in this case the anion isattracted to the interface, whereas the cation is repelled. Thetrends are the same as for LiCl, even though we see nooscillations around zero of the net charge area density for apositive surface charge density, and the compensation of thelatter is more linear for a negative surface charge. At thispoint, we note that overcharging in the absence of excluded-volume correlations is in our model only possible for veryattractive interactions between the interface and the counte-rions or, analogously, very repulsive interactions between thecoions and the surface.
We now turn our attention to the influence of excluded-volume correlations and thus to the results of the NPB-HSmethod. In Fig. 17, for LiCl, the net charge area densityswitches sign for a negative surface charge. It exhibits astrong positive net charge area density of �tot�z=0.55 nm��0.13e nm−2. For a positive surface charge, on the contrary,the net charge area density even increases by almost 20% ata distance z�0.51 nm before declining slowly to zero. It isinteresting that up to distances z=0.55 nm the net chargearea density is even higher than the surface charge density.For NaI, we obtain in Fig. 18 a continuous compensation ofthe negative surface charge density with increasing distancez. The positive surface charge, on the other hand, is compen-sated already at a distance z�0.52 nm and the net chargearea density becomes negative for a distance range of 0.52 z 1.3 nm. The scale of the observed overcharging mightin this case be more of a lower limit because, reminiscent ofour structural analysis in Sec. III A 2, NPB-HS underesti-mates the excluded-volume correlations for NaI.
We can sum up our analysis stating that excluded-volume correlations can lead to overcharging for dense elec-trolytes. Nonetheless, the role of ion-specific ion-surface in-teractions is more important. Excluded-volume correlationscannot lead to overcharging if the counterions are nonelec-trostatically more repelled from the surface than the coions.In the latter case, ion-ion correlations can even increase thesurface charge and thus lead to a competing effect to chargecompensation.
IV. SUMMARY
In this work, we have investigated dense electrolytes in aone-dimensional confinement using multiscale methods. We
first calculated effective nonelectrostatic and ion-specificion-ion and ion-surface pmfs at infinite dilution usingexplicit-water MD. These potentials were then used in bothimplicit-water MC and DFT-derived local and nonlocal ex-tensions to the PB free energy functional. The performanceof the latter was analyzed in comparison with the mean-fieldPB method in which only ion-surface pmfs were included asexternal potentials.
In the first step, we were interested in how the differentmethods are able to reproduce ionic structure. We computedtherefore ion density profiles in a planar one-dimensionalnanoconfinement. We obtained good agreement betweenexplicit-water MD and implicit-water MC methods bychoosing a MD-derived dielectric constant in the latter. Wecorroborate with Hess et al.,28 who showed that a realisticdielectric constant is able to correct for many-body effects inthe case of thermodynamic bulk properties and extend thisobservation to ionic structure in confinement. Having vali-dated our MC approach, we then compared both the DFT-PBand PB methods to the latter. We achieved surprisingly goodresults with the simplest nonlocal extension to PB, namelyNPB-HS. NPB-HS, which treats the electrolyte on a primi-tive hard-sphere level, albeit asymmetric and nonadditive. Insharp contrast, all local extensions performed equally or evenworse than the mean-field PB or failed to yield stable resultsaltogether. We thus substantiate the mindset of earlierwork30,69 and maintain that local theories are unsuited for thestudy of concentrated ionic systems. They should give wayto nonlocal methods. The mean-field PB theory yields rea-sonable results at low concentrations ��1M� and for nearlyuncorrelated systems in terms of effective ion diameters andnonadditivity, but is always outplayed by NPB-HS for highsalt concentrations.
On the basis of NPB-HS and PB, we proceeded in in-vestigating steric corrections to the Donnan effect, solvationforces, and overcharging. Excluded-volume corrections werealways found to enhance salt expulsion between two like-charged plates upon charging them and thus increase theDonnan effect. Ion-specificity is dominated by the effectivesize of the counterions. Intuitively, the bigger the latter, themore salt �i.e., coions� is expelled. In contrast, solvationforces are dominated by the ion-surface pmf. Steric correc-tions are mainly repulsive for uncharged plates but decreasewith increasing charge upon like-charging the two surfaces.For high surface charges, the steric correction not only dis-appears but can even become attractive. Recent experimentaldata show indeed that restabilization patterns of charged col-loids at high salt concentration depend strongly on the vary-ing degree of hydrophobicity and hydrophilicity of the sur-face and thus on the ion-specific �counter�ion-surfaceinteraction.70 Finally, the net charge area distribution aroundone charged plate was found to be mainly influenced by ion-surface interactions. More precisely, if counterions arestrongly attracted nonelectrostatically to the plate, the sur-face charge is compensated already at roughly half a nano-meter distance from the wall �at high salt concentrations� andcan reverse its sign. This overcharging, though, can also betriggered and amplified by excluded-volume interactions, aswas noted by Messina et al.68 Iodide as a counterion will, for
FIG. 18. The same as in Fig. 17 for NaI.
164511-13 Electrolytes in a nanometer slab-confinement J. Chem. Phys. 133, 164511 �2010�
example, lead to substantial overcharging since the anion isboth attracted to the hydrophobic surface and is big in termsof effective diameter—suggesting a highly correlated sys-tem.
The main point of this paper is that both ion-specificion-surface and ion-ion excluded-volume correlations play aprominent part in highly concentrated electrolytes. On acoarse-grained level it is therefore essential to includeexcluded-volume correlations nonlocally with ion-specificeffective ion sizes, as given in Table III, which are asymmet-ric and nonadditive. This can be achieved by our simpleNPB-HS method.
An obvious important issue is the extension of themethod to three-dimensional geometries, where curvatureplays a significant role, as, for instance, in protein �ion�channels71,72 or charged protein binding pockets,73 where lo-cally the ion concentration can be enormous. A recent paperhighlights that this is indeed feasible.74 Those methods couldthen be implemented in existing continuum Poisson–Boltzmann solvers such as the Adaptive Poisson-BoltzmannSolver �APBS�.75 Furthermore, ongoing work at our labora-tory aims to include more realistic, biologically relevant sur-faces.
ACKNOWLEDGMENTS
The authors thank Nadine Schwierz for useful discus-sions and for reading and improving the manuscript, theDeutsche Forschungsgemeinschaft �DFG� for funding withinthe Emmy-Noether Program, the Bavaria California Technol-ogy �BaCaTec� Center for travel support, and the Leibniz-Rechenzentrum �LRZ� for computing time on the Höchstleis-tungsrechner Bayern II �HLRB II�.
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