Surface tension of electrolyte interfaces: Ionic ...rudi/reprints/tomer-long.pdfSurface tension of...

THE JOURNAL OF CHEMICAL PHYSICS 142, 044702 (2015)

Surface tension of electrolyte interfaces: Ionic specificitywithin a field-theory approach

Tomer Markovich,

1

David Andelman,

1

and Rudi Podgornik

2

1Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Ramat Aviv,Tel Aviv 69978, Israel2Department of Theoretical Physics, J. Stefan Institute and Department of Physics, Faculty of Mathematicsand Physics, University of Ljubljana, 1000 Ljubljana, Slovenia

(Received 18 November 2014; accepted 4 January 2015; published online 22 January 2015)

We study the surface tension of ionic solutions at air/water and oil/water interfaces by using field-theoretical methods and including a finite proximal surface-region with ionic-specific interactions.The free energy is expanded to first-order in a loop expansion beyond the mean-field result. We calcu-late the excess surface tension and obtain analytical predictions that reunite the Onsager-Samaraspioneering result (which does not agree with experimental data), with the ionic specificity of theHofmeister series. We derive analytically the surface-tension dependence on the ionic strength, ionicsize, and ion-surface interaction, and show consequently that the Onsager-Samaras result is consistentwith the one-loop correction beyond the mean-field result. Our theory fits well a wide range of saltconcentrations for di↵erent monovalent ions using one fit parameter per electrolyte and reproduces thereverse Hofmeister series for anions at the air/water and oil/water interfaces. C 2015 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4905954]

I. INTRODUCTION

Surface tension of ionic solutions strongly depends ontheir salt composition and, in general, increases with ionicstrength for low salt concentrations.1,2 Wagner3 was the firstto connect this finding with the dielectric discontinuity at theair/water interface, suggesting that dielectric image chargeinteractions could explain the increase in surface tension.This suggestion was implemented in the pioneering work ofOnsager and Samaras (OS), who used the Debye–Hückel (DH)theory for electrolytes.4 The OS result presents a universallimiting law for the excess surface tension. It depends onthe dielectric mismatch at the interface and on the bulk saltconcentration5 and implies that the increase in the surfacetension would be independent of the ion type. However, thissimplified observation turned out to be violated in manyexperimental situations6 and led over the years to numerousinvestigations of non-electrostatic, ion-specific interactionsbetween ions and surfaces,6,7 and their role in modifying thesurface tension of ionic solutions.2

In fact, ion-specific e↵ects date back to the late 19thcentury when Hofmeister8 measured the amount of proteinprecipitation from solution in presence of various salts andfound a universal series of ionic activity. The same Hofmeisterseries emerges in a large variety of experiments in chemicaland biological systems.9–11 Among others, they include forcesbetween mica and silica surfaces,12–14 osmotic pressure in thepresence of (bio)macromolecules,15,16 and, more specifically,measurements of surface tension at the air/water and oil/waterinterfaces.17,18 For simple monovalent salts, the surface tension(in particular at the air/water interface) was experimentallyfound19 to depend strongly on the type of anion, while thedependence on the cation type is much weaker. This finding

is consistent with the fact that anion concentration at theair/water interface exceeds that of cations. Furthermore, forhalides,6 the lighter ions lead to a larger excess in surfacetension in a sequence that precisely corresponds to the reverseof the Hofmeister series.

Discrepancies between the OS predictions and theobserved ion-specific surface tension motivated numerousattempts to modify the original OS model. Here, we limitourselves to briefly review some of the more recent workson surface tension of ionic solutions,20–26 which are directlyrelated to the present study.

Dean and Horgan23 calculated the surface tension of ionicsolutions to first order in a systematic cumulant expansion,where the zeroth order is equivalent to the DH linear theory.4The specific ion-surface interactions are modeled via anionic exclusion (Stern) layer of finite thickness. Thus, theinteraction of ions with the interface contains only a lengthscale without any energy scale. This model is solved via aformal field-theoretic representation of the partition function.The OS result is reproduced exactly, and ion-specific e↵ectsare described in terms of the thickness of the salt-exclusionlayer. In, yet, a separate study by the same authors,22 a systemof two interacting surfaces including a cation-specific short-range surface interaction was addressed. The consequencesof this cation-specific interaction on determining the e↵ectivecharge, as well as the disjoining pressure as a function of theseparation between the surfaces, were studied in detail.

In a series of papers, Levin and coworkers20,21,27

calculated the solvation free-energy of polarizable ions atair/water and oil/water interfaces. Their mean-field theory(MFT) modifies the Poisson-Boltzmann (PB) theory by addingan ion-surface interaction potential. The modified potential isbased on several ion-surface interaction terms that are added

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044702-2 Markovich, Andelman, and Podgornik J. Chem. Phys. 142, 044702 (2015)

in an ad hoc way to the PB equation. These interactionsinclude the image charge interaction, Stern exclusion layer,ionic cavitation energy, and ionic polarizability. While theadditional interaction terms may represent some physicalmechanisms for ion-specific interaction with the surface, onecannot, in general, simply add such terms to the MFT potentialin a self-consistent way. These terms, which are sometimesmutually exclusive, are neither completely independent norcan they be obtained from a MFT formulation.6,8

Computing numerically the surface tension for ahomologous series of sodium salts, Levin and co-workersfitted their predictions to the Hofmeister series. They usedthe hydrated size of the sodium ion as a single fit parameter.Furthermore, the surface tension at the oil/water interface wasfitted for another series of potassium salts.27 In order to applytheir theory to the di↵erent interfaces, a second fit parameterwas used to account for the dispersion forces at the oil/waterinterface.

A di↵erent line of reasoning was initiated by Netzand coworkers,24–26 who calculated the surface tensionfor both charged and neutral surfaces using a two-scale(atomistic and continuum) modeling approach. Explicitsolvent-atomistic molecular-dynamics (MD) simulationsfurnished non-electrostatic, ion-specific potentials of meanforce. These interaction potentials were then added to the PBtheory that provides the electrostatic part of the potential ofmean force. Within this framework, Netz and coworkers wereable to show that the polarity of the surface may reverse theorder of the Hofmeister series. The fitted results agree wellwith experiments performed on hydrophobic and hydrophilicsurfaces.

Although many works20,21,23–30 tried to generalize theseminal OS theory, it nevertheless remains largely misun-derstood what is the accurate theoretical framework of theOS theory. The OS theory makes use of the (linearized) PBequation in the presence of a planar dielectric boundary toobtain the one-dimensional image charge potential of meanforce [see Eq. (4) in Ref. 5]. It should be stated that thelatter is not a solution of the one-dimensional PB equation.In fact, the PB solution cannot describe any image chargee↵ects on its own. This subtle, yet essential point, gets oftenirreparably lost when generalizations of the OS theory areattempted based on elaborate decorations of the PB equationitself.

If the OS theory is not simply a redressed version of thePB theory, then what exactly is the relation between them?While the OS theory cannot be obtained from the mean-fieldtheory, it is deduced from the thermodynamic fluctuations ofthe instantaneous electric fields around the PB solution.31 Thefree energy is expanded in a loop expansion and only theone-loop correction to the MFT result is retained.

While the detailed formal derivation (as shown below)is complex, we believe that its physical basis is quite simpleand straightforward. The OS result does not generalize thePB equation to include image charge e↵ects at an interfacebetween two dielectric media, but rather it solves the problemon a higher level of approximation by going beyond the MFTlevel. We consider this conceptual clarification to have largeimportance on generalizations of the OS theory itself.

In this paper, we introduce two important modifications,relevant to the calculations of the surface tension of elec-trolytes. First, we demonstrate that the OS contribution ise↵ectively fluctuational in nature and follows from the one-loop expansion of the Coulomb partition function around themean-field solution. Second, we propose a phenomenologicalapproach that consistently describes ion-interface interactionsin the form of a coupling term in the free energy. This newformulation allows us to obtain a simple analytical theory thatreunites the OS pioneering result, which does not agree withexperimental data, with the ionic specificity of the Hofmeisterseries.

We take ionic specificity into account through theionic size and an ion-surface interaction. Each ionicspecies is characterized by a phenomenological adhesivityparameter.22,29,32 Specifically, short-range, non-electrostatice↵ects such as the ion chemical nature, size, and polarizability,as well as the preferential ion-solvent interaction,7,33–36 areintroduced by adding one phenomenological parameter to thefree energy. This allows us to obtain a modified PB mean-field theory and to evaluate the contribution of fluctuations(beyond mean-field) to the surface tension. The latter includesthe dielectric image charge e↵ects (OS), as well as thecouplings between image charge e↵ects and surface-specificinteractions. Our analytical surface tension prediction fitswell a variety of interfacial tension data at the air/water andoil/water interfaces. Using one fit parameter per electrolyte, itreproduces the reversed Hofmeister series for several types ofmonovalent anions.

The outline of this paper is as follows. In Sec. II, wepresent the model and a general derivation of the grand-potential to the one-loop order. In Sec. III, we find the freeenergy and treat the spurious divergencies of our model,while in Sec. IV, we derive an analytical expression for thesurface tension. Finally, a comparison of our results withexperimental data (Sec. V) and some concluding remarks(Sec. VI) are presented.

II. THE MODEL

We consider an ionic solution as in Fig. 1, which containsmonovalent symmetric (1:1) salt of charge ±e and of bulkconcentration nb. The aqueous phase (water) volume V = ALhas a cross-section A and an arbitrary large length, L!1.The surface between the aqueous and air phases is chosenat z = 0. The two phases are taken as two continuum mediawith uniform dielectric constants "w and "a, respectively, suchthat

"(r)=8><>:"a z < 0"w z � 0

. (1)

Note that we can equally model the water/oil interface. Byassuming no ions penetrate the oil phase, we model theoil/water system by taking "a as the dielectric constant of theoil phase (Fig. 1).

The ion self-energy will not be considered explicitlybecause it is well known that this self-energy is extremelylarge (⇠100kBT) in the air phase. It will only be taken into



FIG. 1. Schematic image of the system. The water and air phases have thesame length L, where L ! 1. In the water phase, there is a proximal region,0 z d where the anions and cations interact with a surface interaction,modeled by an external potential V±(z).

account implicitly by confining the ions into the water phase.We also assume a proximal region inside the water phasewhere there are ion-specific interactions of the ions with theinterface. The width of this region is denoted by d, and theinteractions are modeled by a potential V±(z) for anions andcations, respectively. We also assume that these interactionsdepend only on the z coordinate, which is a reasonableassumption if the surface is rather uniform and flat.

The model Hamiltonian is

H =12

X

i, j

qiqju(ri, r j)+X

i ✏ anions

V�(zi)[✓(zi)�✓(zi�d)]

+X

i ✏ cations

V+(zi)[✓(zi)�✓(zi�d)], (2)

where qi =±e is the charge of monovalent anions and cations,respectively, and we use the Heaviside function

✓(z)=8><>:

0 z < 01 z � 0

. (3)

The first term is the usual Coulombic interaction, whichsatisfies r · ["(r)ru(r, r0)]=�4⇡�(r�r0), where the divergingself-energy of point-like ions is subtracted. The second andthird terms are the ion-surface specific interaction for anionsand cations, respectively.

The thermodynamical grand-partition function can bewritten as

⌅ =1X

N�=0

1X

N+=0

�N��N�!�N++N+!

⌅ N�Y

i=1

d3riN+Y

j=1

d3r j

⇥ exp *.,� �

2

X

i, j

qiqju(ri, r j)

� �X

i

V�(zi)[✓(zi)�✓(zi�d)]

� �X

j

V+(z j)⇥✓(z j)�✓(z j�d)

⇤+/-, (4)

where � = 1/kBT , kB is the Boltzmann constant, T is thetemperature, and N± are, respectively, the number of cationsand anions. Note that the sum is over all i and j. The fugacities

�± are defined via the (intrinsic) chemical potentials, µ±, as

�± = a�3 exp[�µtot± ],

µtot± = µ±(r)+

12

e2uself(r, r),(5)

where the (diverging) self-energy, uself(r, r), of the 1:1monovalent ions is subtracted, and µtot

± is the total chemicalpotential. The length scale a is a microscopic lengthcorresponding to the ionic size and is assumed to be equal foranions and cations.

We can then write the charge density operator, ⇢̂, as

⇢̂(r)=X

j

qj �(r�r j), (6)

and in order to proceed, we introduce the functional Diracdelta function,

�[⇢(r)� ⇢̂(r)]= �

2⇡

!N⌅D�(r)

⇥ exp"i �

⌅d3r �(r)(⇢(r)� ⇢̂(r))

#, (7)

with � being an auxiliary field and N = N++N�. The functionalintegral representation is a functional integral over all valuesof {�(r)} at all space points r. It can be thought of as thecontinuum limit of multiple integrals over values of �(r)at di↵erent points in space,

⇤ QNi=1d�(ri)!

⇤D�(r), for

N!1. Rewriting the grand-partition function using Eqs. (6)and (7) and the Hubbard-Stratonovich transformation31 yields

⌅=(2⇡)�N/2

pdet[��1u(r, r0)]

⇥⌅D� exp

"� �

2

⌅d3rd3r 0�(r)u�1(r, r0)�(r0)

+

⌅d2r

⌅ d

0dz

⇣�+e�i�e�(r)��V+(z)

+ ��ei�e�(r)��V�(z)⌘

+

⌅d2r

⌅ L

ddz

⇣�+e�i�e�(r)+��ei�e�(r)

⌘#. (8)

The grand-partition function can be written in the form

⌅⌘ (2⇡)�N/2

pdet[��1u(r, r0)]

⌅D� e�S[�(r)], (9)

where S[�(r)] plays the role of a field action,

S[�(r)]=⌅

d3r�"(r)8⇡

[r�(r)]2

�⌅

d2r�

⌅ d

0dz

⇣e�i�e�(r)��V+(z)

+ ei�e�(r)��V�(z)⌘

+ 2�⌅ L

ddz cos[�e�(r)]

�, (10)

with "(r) is the dielectric function defined in Eq. (1).In the above equation, we have also used the in-verse Coulomb potential u�1(r, r0) = � 1

4⇡r · ["(r)r�(r� r0)]that obeys

⇤d3r 00u(r, r00)u�1(r00, r0) = �(r� r0). The electro-

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For slowly varying potentials, V± within the proximalregion, one can write,

⌅ d

0dz f (z) e��V±(z)'

De��V±(z)

Ez

⌅ d

0dz f (z), (11)

where hOiz = 1d

⇤ d0 dz O(z) denotes the spatial average in the

[0, d] interval. Indeed, if V± vary slowly in the 0 z d region,we can write the field action using Eq. (11) as

S[�(r)]=⌅

d3r�"(r)8⇡

[r�(r)]2

�⌅

d2r �

⌅ d

0dz

⇣e�i�e�(r)��↵+

+ ei�e�(r)��↵�⌘

+ 2�⌅ L

ddz cos[�e�(r)]

!, (12)

where e��↵± ⌘⌦e��V±(z)

↵z, or to first-order in a cumulant

expansion, ↵±= hV±iz. This approximation is also valid for athin layer, in which one does not consider the layer details,but only its average e↵ect.

The above equation represents the full grand-partitionfunction of our electrolyte system as represented by a fieldtheory. Because of the existence of specific ion-surfaceinteractions, it contains additional length scales besides theelectrostatic ones. These additional length scales depend on↵± and do not allow to introduce a single electrostatic couplingparameter, which leads to the weak- and strong-couplinglimits.37 Nevertheless, we can still introduce the weak-coupling limit as it corresponds to a saddle-point configurationwith Gaussian fluctuations. This is consistent not only withweak electrostatic interactions but also with weak specificsurface-ion interactions. Up to first-order in a loop expansion,the field action, Eq. (10), can be written as

S[�(r)]'S[�0(r)]

+12

⌅�2S[�(r)]��(r)��(r0)

��0��(r)��(r0) d3rd3r 0. (13)

It is useful to define the Hessian of S as

H2(r, r0)=�2S

��(r0)��(r00)��0. (14)

The subscript “0” stands for the value of the field action atits stationary point defined as �S/��(r)|0= 0, and �0(r) is thevalue of �(r) at this point. The functional integral in Eq. (13)is Gaussian and can be evaluated explicitly giving the grandpotential, ⌦=�kBT ln ⌅, in the form

⌦'⌦0+⌦1

= kBT S0+12

kBT Tr ln (H2(r, r0)), (15)

where we have dropped irrelevant constant terms and used thematrix identity, ln det(A)=Tr ln (A).

A. Mean field theory

The MFT equation corresponds to the saddle-point of Sand can be written for the mean-field electrostatic potential

(r) by identifying (r) = i�0(r). The stationary point ofEq. (12) then implies

r2 1= 0 z < 0,

r2 2=4⇡enb

"w

�e��↵� e�e 2�e��↵+ e��e 2

�0 z d, (16)

r2 3=8⇡enb

"wsinh(�e 3) z > d

and is equal to the standard PB equation in the z > d region.The three spatial regions are denoted by “1” for z < 0 (air),“2” for 0 z d (proximal layer), and “3” for z > d (distalregion). Note that on the mean-field level, the fugacity � can bereplaced by the bulk salt concentration, nb (see Appendix C).

Because of the (x, y) in-plane translation symmetry, theMFT potential varies only in the perpendicular z-direction, (r)= (z). We simplify the treatment by linearizing the MFTequations (the DH limit), and obtain

001 =0,

002 =1�e

2D

2�e��↵��e��↵+

�+ ⇠2 2, (17)

003 = 2D 3,

with

⇠2⌘2

D

2�e��↵�+e��↵+

�, (18)

and the inverse Debye length defined as

D=

q8⇡ �e2nb/"w. (19)

The linearization is valid for a surface potential that is rathersmall, |�e |⌧ 1, and corresponds to ↵� ' ↵+.

Using the continuity of the electrostatic potential and itsderivative at z = 0 and z = d, with vanishing electrostatic fieldin the bulk (z! L) and in the air phase (z < 0), we obtain thelinear MFT solution

1=D(cosh ⇠d�1)+ ⇠ sinh ⇠dD cosh ⇠d+ ⇠ sinh ⇠d

�,

2=D(cosh ⇠d�cosh ⇠z)+ ⇠ sinh ⇠d

D cosh ⇠d+ ⇠ sinh ⇠d�, (20)

3=⇠ sinh ⇠d

D cosh ⇠d+ ⇠ sinh ⇠de�D(z�d)�,

with the parameter � defined as

�e� ⌘ e��↵+�e��↵�

e��↵++e��↵�. (21)

Notice that for ↵± = 0, the parameter � = 0, the surface atz = d is a phantom surface, and (z) vanishes everywhere.Another limit where = 0 everywhere is obtained for d! 0(no proximal region).

Keeping only linear terms in d, the electrostatic potentialis

�e 1= Dd�e��↵+�e��↵�

�,

2= 1, (22) 3= 1e�Dz.

By taking ↵+ = 0, while keeping ↵�, 0 in the above equation,the result of Ref. 38 is recovered for a single type of adsorbing



ion subjected to a surface interaction. This is equivalent to thelimit of a proximal layer of zero width.

The approximation for the field action is then

⌦0=�⌅

d3r�"(r)8⇡

[r (r)]2

� kBTnb

⌅d2r

⌅ d

0dz

�e��e e��↵++e�e e��↵�

�

� 2kBTnb

⌅d2r

⌅ L

ddzcosh(�e ), (23)

where the MFT potential is obtained from Eq. (20) and (r)= i�0(r).

B. One-loop correction

In order to obtain the one-loop correction, one shouldfirst calculate the trace of the logarithm of the Hessian (seeEq. (15)). To do so, we start by considering the eigenvalueequation of the Hessian

⌅d3r 0H2(r, r0)u⌫(r0)= ⌫u⌫(r). (24)

Because of the planar symmetry of our system, r = (⇢, z),where ⇢ is the inplane vector, it is convenient to write downthe eigenvalue problem in the transverse Fourier space, wherek is coupled to ⇢ and U⌫(k, z) is the transverse Fouriertransform of u⌫(⇢, z),

u⌫(⇢, z)= A4⇡2

⌅d2k U⌫(k, z)eik·⇢. (25)

In Fourier space, the eigenvalue equation is �"w2

D cosh(�e )[✓(z�d)�✓(z�L)]

� 12"w

2D�e��↵�e�e +e��↵+e��e

�[✓(z)�✓(z�d)]

+ @z"(z)@z�"(z)k2!U⌫(k, z)= ⌫U⌫(k, z), (26)

where "(r) = "(z) as in Eq. (1), and the correspondingboundary condition at z = 0 is

"w@zU(2)⌫

��0+�"a@zU (1)⌫

��0�= 0, (27)

where is the MFT solution for the electrostatic potentialobtained in Eq. (20), and the notations u(i)

⌫ (⇢, z) andU (i)⌫ (k, z), i = 1, 2, 3, correspond, respectively, to the solutions

in the three regions: z < 0 (air, i = 1), 0 z d (proximallayer, i = 2), and z > d (distal region, i = 3). For simplicity, theexplicit dependence on k is suppressed hereafter. A secondboundary condition is given in terms of the macroscopicsystem size L and is written as @zu⌫(±L)! 0. Note that inthe air, the solution can be obtained by simply taking D= 0in Eq. (26). This leads to an exponentially decaying solutionin the air ⇠exp(

pk2+ ⌫z), @zU (1)

⌫ =p

k2+ ⌫U (1)⌫ at the z! 0�

boundary, and k ⌘ |k|.The boundary conditions can then be written in a matrix

form

AV��z=0+BV��z=d+CV��z=L = 0, (28)

where the four vector V=⇣U (2)⌫ , @zU

(2)⌫ ,U (3)

⌫ , @zU(3)⌫

⌘and the

4 ⇥ 4 coe�cient matrices A, B, and C are detailed inAppendix A.

The boundary conditions are satisfied when thedeterminant of the coe�cient matrix of Eq. (28) vanishes.This determinant, D⌫(k), is called the secular determinantand can be written as39

D⌫(k)= det[A+B �⌫(a)+C �⌫(L)], (29)

where the 4⇥4 matrix �⌫(z) is also detailed in Appendix Ain terms of the two independent solutions of the eigenvalueequation (Eq. (26)). These two independent solutions aredenoted by h(i)

⌫ (z) and g(i)⌫ (z) with i = 2, 3 for regions “2”(proximal) and “3” (distal), respectively.

As was mentioned above, the one-loop fluctuationcontribution requires to calculate Tr ln H2. We rely on previousresults28,39 where it has been shown that only the ⌫ = 0 valueof the secular determinant, D ⌘ D⌫=0, needs to be evaluated.This represents an enormous simplification as there is no needto find the entire eigenvalues spectrum of the Hessian. Let usstress that ⌫ = 0 is not an eigenvalue of the Hessian.

The fluctuation contribution ⌦1 around the MFT can thenbe written as28,40

⌦1=12

kBT Trln(H2(r, r0))

=AkBT8⇡2

⌅d2k ln

D(k)

Dfree(k)

!, (30)

where the integrand depends on the ratio D(k)/Dfree(k), andDfree is the reference secular determinant for a “free” systemwithout ions.

The next step is to calculate D(k). We return to Eq. (26)to find the solution for ⌫ = 0, U0(z)⌘U⌫=0(k, z), dropping forconvenience the k-dependence. Using �e ⌧ 1 and keepingonly terms of order O

�e��↵��e��↵+

�, Eq. (26) yields

@

@z"(z) @

@z�"(z)k2�"w2

D[✓(z�d)�✓(z�L)]

� 12"w⇠

2[✓(z)�✓(z�d)]!U0(z)= 0. (31)

The four independent solutions of Eq. (31), hi(z)⌘ h(i)⌫=0(z)

and gi(z)⌘ g(i)⌫=0(z), i = 2, 3, can be written as

h2= cosh qz; g2=sinh qz

q,

h3= cosh pz; g3=sinh pz

p,

(32)

where p2= k2+2D, q2= k2+⇠2, and U (i)

0 is a linear combinationof hi and gi.

By substituting Eq. (32) into Eq. (29), it is straightforwardto compute the secular determinant which gives, in thethermodynamical limit, L� d,

D(k)'""ak

⇣ pq

sinh qd+cosh qd⌘

+ "wp⇣cosh qd+

qp

sinh qd⌘#

ep (L�d). (33)



The secular determinant can be written in a more familiarway as41

D(k)'epLe(q�p)d "wq+"ak

q

!

⇥"q⇣1��(q, k)e�2qd

⌘+ p

⇣1+�(q, k)e�2qd

⌘#,

(34)

with �(q, k) defined as

�(q, k)⌘ "wq�"ak"wq+"ak

. (35)

Keeping linear terms in d, the secular determinant fromEq. (34) yields

D(k)'⇥"ak+"wp+"wd

�⇠2� 2

D�⇤

epL. (36)

Note that the DH result, as obtained in Ref. 38, is recoveredin the above equation for d = a.

III. FREE ENERGIES

The grand potential ⌦ can be calculated to the one-looporder by inserting D obtained in Eq. (33) into Eq. (30) andexpressing D of Eq. (19) in terms of the fugacities instead ofthe bulk densities

⌦=⌦0+V kBT12⇡

�⇤2+ 2

D�3/2� 3

D�⇤3�

+AdkBT

12⇡

�⇤2+ ⇠2�3/2�

�⇤2+ 2

D�3/2� ⇠3+ 3

D

�

+AkBT

4⇡

⌅ ⇤

0dk k

ln""wq+"ak

2("w+"a)kq

#

+ lnfq�1��(q, k)e�2qd�+ �1+�(q, k)e�2qd�g! , (37)

where q2 = k2+ ⇠2 and ⇤ is the ultraviolet (UV) cuto↵.As shown in Appendix C, for symmetric electrolytes,�± = n(±)

b = nb. This simplification is exact for the one-looporder of the free energy, F, but not for the grand potential, ⌦.In order to find the one-loop grand potential, one has to findconsistently the one-loop correction to the fugacities.

Note that the integral in Eq. (37) has an UV divergencyfrom the upper bound of the k-integral, as ⇤!1. AlthoughCoulombic interactions between point-like ions diverge atzero distance, such a divergence is avoided because of stericrepulsion for ions of finite size. A common way in field theoryto avoid this issue without introducing explicitly yet anothersteric repulsive interactions is to employ a short length cuto↵.For isotropic two-dimensional integrals, as in Eq. (37), theUV cuto↵ is taken to be ⇤= 2

p⇡/a, where a is the average

minimal distance of approach between ions. This distance canbe approximated by a ' 2r , with r being the ionic radius.An alternative way of avoiding the divergence is to use theself-energy regulating technique as in Ref. 42. However, forsimplicity we will employ only to the UV cuto↵ hereafter.

In order to calculate the surface tension, we now calculatethe free-energy,43 which is related to the grand-potential by,

F =⌦+X

i = ±

⌅d3r µi(r)ni(r). (38)

It is instructive for the surface tension calculation to separatethe volume and surface contributions of the free energy,F = FV +FA. Taking the ⇤!1 limit in the volume term ofEq. (37), and using Eq. (5) for µi, yields an expression for FVto the one-loop order

FVV'⌦0

V+2kBTnb ln(nba3)� kBT

12⇡3

D

� dL

kBT12⇡

�⇠3� 3

D�+

kBT8⇡⇤

"2

D+dL�⇠2� 2

D�#

� e2nbuself(r, r)*,1+dL⇠2� 2

D

2D

+-. (39)

Note that in the above equation, we neglect all terms of orderO(⇤�1). The first term is the MFT grand potential, ⌦0, thesecond one is the usual entropy contribution, and the thirdone is the well-known volume fluctuation term, as in the DHtheory.4 The fourth and fifth terms are the bulk self-energiesof the ions (diverging with the UV cuto↵) and will be shownto cancel each other exactly.

The surface free-energy FA to one-loop order is

FAA=

kBT4⇡

⌅ ⇤

0dk k

ln""wq+"ak

2("w+"a)kq

#

+ lnfq�1��(q, k)e�2qd�+ p

�1+�(q, k)e�2qd�g! . (40)

We now treat the spurious divergencies that originatefrom the bulk self-energies of the ions. The Coulomb potentialobeys

r2u(r, r0)=�4⇡"w�(r�r0) for z � 0

r2u(r, r0)= 0 for z < 0.(41)

As our system exhibits translational invariance in thetransverse (x,y) directions, we can simplify the transverseFourier transform as in Eq. (25) into the Fourier-Besselrepresentation

u(r, r0)= 14⇡2

⌅d2k U0(k;z, z0)eik·⇢

=

⌅dk k U0(k;z, z0)J0(k |⇢�⇢0|), (42)

where J0(z) is the zeroth-order Bessel function of the 1st kind,and ⇢ = (x,y) is the in-plane radial vector. The solution ofEq. (41) is

U0(k; z, z0)= 2⇡"wk

⇣e�k |z�z

0|+"w�"a"w+"a

e�k(z+z0)⌘, (43)

for z � 0, and

U0(k; z, z0)= 2⇡"wk

⇣1+

"w�"a"w+"a

⌘ek(z�z

0) (44)

for z < 0. The self-energy of an ion in the bulk is obtained bysetting r= r0 and z!1,

uself(r,r)=1"w

⌅ ⇤

0dk

1+

"w�"a"w+"a

e�2kz!'⇤/"w. (45)

The integral over k for the self-energy, Eq. (45), has an UVcuto↵ that was replaced by its most divergent term, which is



linear in ⇤. Finally, by substituting Eq. (45) into Eq. (39), thetwo diverging terms in FV cancels each other.

To summarize, we have shown that within our approach,the self-energy diverging terms cancel out and, as anticipated,they do not a↵ect the free energy.

IV. SURFACE TENSION

We are interested in calculating the surface tension of anaqueous ionic solution. The excess surface tension, ��, can beobtained from the Gibbs adsorption isotherm or, equivalently,by taking the di↵erence between the free-energy of the two-phase system and the sum of the free energies of the twosemi-infinite systems of the air and the bulk ionic solution

��=fF (2L)�F (air)(L)�F (B)(L)

g/A. (46)

Here, �� = ��A/W is the excess ionic contribution to thesurface tension with respect to the surface tension betweenpure water and air, �A/W.

The bulk electrolyte free energy, F (B)(2L), is obtainedfrom Eqs. (39) and (40) by considering the entire [�L, L]interval as a uniform dielectric medium with "w, ↵± = 0,and mobile ions. This implies "a! "w and q! p. Then,D(k)/Dfree(k) = p/k, while ⌦0 = �2nbV is obtained for theMFT solution of = 0 (bulk electrolyte phase). Applyingthese limiting values gives the free-energy of a system ofwidth 2L. This system includes two boundaries at the twoextremities, ±L. Thus, one needs to divide the free energy bytwo in order to get the bulk free-energy of a slab of width, L,with one interface at infinity,

F (B)(L)= kBTV2666642nb ln

�nba3��2nb�

3D

12⇡

377775+

AkBT8⇡

⌅ ⇤

0dk k ln

✓ pk

◆, (47)

where p2= k2+ 2D as defined before. The free-energy of the

air phase is equal to zero, F (air)(L)= 0, as there are no ions inthe air phase. For calculating the interfacial tension betweenwater and oil, for typical oil with dielectric constant "a ' 2,the assumption that no ions penetrate into the oil phase willbe maintained, and the free energy of the oil phase will betaken as zero.

We calculate in Subsections IV A and IV B the excesssurface tension to one-loop order, ��=��0+��1, where ��0is the MFT contribution to the surface tension, while ��1 isits one-loop correction.

A. Mean-field surface tension

Using the mean-field (z) from Eq. (20), the MFT surfacetension is

��0=�⌅ 1

�1dz"(z)8⇡

d dz

!2

+ kBTnb

⌅ d

0dz

�2�e��e e��↵++e�e e��↵�

�

+ 2kBTnb

⌅ L

ddz (1�cosh �e ). (48)

As the electrostatic potential and its derivative are of the orderO�e��↵��e��↵+

�, the MFT surface tension to that order is

simply

��0'�kBTnbd⇥�

e��↵��1�+�e��↵+�1

�⇤. (49)

On the MFT level, the equation above is exact for ↵�= ↵+.To show this, we first substitute ↵�= ↵+ in Eq. (20), leadingto vanishing electric field and potential. Further substitutionof = 0 and @z = 0 in Eq. (48) leads to Eq. (49) without anyfurther approximations.

In order to make contact with the PB surface tension ofa charged surface with adhesivity, we write Eq. (48) to linearorder in d. Substituting the first integration of Eq. (16) intoEq. (48), one gets

��0=�nbdf�

e��↵��1�e�e +

�e��↵+�1

�e��e

g� 8nb

�1D

⇣cosh(�e s/2)�1

⌘. (50)

The second term in the above equation is the PB surfacetension. For our purposes, the electrostatic potential is smalland the same second term is negligible. In this case, the aboveequation exactly coincides with Eq. (49).

B. One-loop correction to the surface tension

The one-loop correction to the surface tension, ��1, isobtained from Eqs. (40) and (47)

8⇡kBT��1=

⌅ ⇤

0dk k

ln266664

kp

"wq+"ak

2("w+"a)kq

!2377775+ ln

fq�1��(q, k)e�2qd�

+ p�1+�(q, k)e�2qd�g2

!

� 2d3�⇠3� 3

D�, (51)

recalling that q2= k2+⇠2. Taking the limit of d! 0 or ↵±! 0(which implies q! p) gives the OS result with a correctiondue to the finite ion size.31 By keeping terms linear in d, thelinearized fluctuation contribution yields44

8⇡kBT��1'

⌅ ⇤

0dk k ln

266664kp

"ak+"wpk ("w+"a)

!2377775+ 2d

"⌅ ⇤

0dk k (q� p) "wq�"ak

"wp+"ak

�⇠3� 3

D

3

377775 . (52)

The above result contains the OS result5,23,31 and an ionic-specific correction, as will be discussed later. It is clear fromthe above results that as long as the adhesivity parameters↵± are small, the MFT contribution to the surface tension,��0, is small and the dominant contribution comes from thefluctuation term, ��1. This observation goes hand in hand withthe fact that the OS result by itself originates from fluctuationsbeyond MFT.

The leading asymptotic behavior of the integral of Eq. (51)reveals the OS result and its correction terms. Writing downonly the remaining ⇤-dependent terms, the one-loop surface



FIG. 2. Comparison of the fitted excess surface tension, ��, at the air/water interface with experimental data from Ref. 45 (for the K+ series) and Ref. 46 (for theNa+ series), as function of ionic concentration, nb. For (a) KX and for (b) NaX, where X stands for one of the Halogen anions F�, Cl�, Br�, and I�. The bottomdashed line represents the OS surface tension.5 The fitted adhesivity values, ↵⇤ in units of kBT , are shown in Tables I and II. Other parameters are T = 300 K,"w = 80 (water), and "a = 1 (air).

tension to leading order in d is

8⇡kBT��1'�

"w�"a"w+"a

!2D

2

266664ln

12DlB

!� ln

12

lB⇤

!

�2⇥"wd

�⇠2� 2

D

�⇤22D("2

w�"2a)

ln�D⇤

�1�377775 , (53)

with the Bjerrum length defined as lB= �e2/"w and ⇠2 definedin Eq. (18). The first term in ��1 is the well-known OSresult5,23,31 and it varies as ⇠2

D ln(DlB), the second term is acorrection due to the ion minimal distance of approach with⇤= 2

p⇡/a, while the third term is a correction related to the

adhesivity parameters, ↵±, through ⇠2⌘ 2D

�e��↵�+e��↵+

�/2.

For �↵±⌧ 1, the latter term is negligible and the derivedsurface tension agrees well with the OS result, as expected.

For salt concentration larger than ⇠0.3kBT , the analyticalapproximation shown in Eq. (53) deviates from the fullnumerical solution of Eq. (51). It is nevertheless possibleto solve Eq. (51) analytically to order O(d2). Along these lineswe present in Appendix B, an analytical solution to orderO(1/⇤), Eq. (B1), where we do not neglect constants (withrespect to ⇤). This solution is almost equivalent to the fullnumerical solution of Eq. (51).

V. COMPARISON TO EXPERIMENTS

We now compare our results for the surface tension,�� =��0+��1, with experimental data. The surface tensionprediction is obtained from ��0 in Eq. (49) and the numericalsolution, ��1 of Eq. (51).

For simplicity we take d, the thickness of the proximallayer, to be equal to a, the average minimal distance betweencations and anions, a = rhyd

+ + rhyd� . Furthermore, we treat

↵± as fit parameters. We note that the obtained results for thesurface tension, �� =��0+��1, are symmetric with respectto anion ! cation exchange. This is important because itmeans that two parameter fit with ↵± will always give twoequivalent results, ↵+$ ↵�. Furthermore, for �↵±⌧ 1, onecan define ↵⇤= ↵�+↵+ as a single fit parameter, and the fit

with ↵⇤ produces almost equivalent results to the fit with thetwo independent parameters.

We start by presenting the fits of the experimental datawith ↵⇤. In Fig. 2, we compare our theory at the air/waterinterface for (a) three di↵erent ionic solutions with K+ astheir cation and (b) four di↵erent ionic solutions with Na+

as their cation. The fits in (a) are in very good agreementwith experiments. In (b), for the larger Br� and I� anions(with respect to their crystallographic size), the fit agrees wellfor the entire concentration range up to ⇠1M, while for thesmaller F� and Cl� anions, some deviations at concentrationslarger than ⇠0.8M can be seen.

Our model can also be applied successfully to other typesof liquid interfaces, such as oil/water, and to more complexanions such as oxy anions, which are defined by the genericformula, AxO�y , or acids such as HCl and HClO4. As hydrogencan form complexes with water molecules, the H+ representsall of these complexes. We do not need to know the specificcomplexation of the hydrogen in water but only its e↵ectiveradius.48

In Fig. 3, we compare the fits for oil/water interface,where in the experiments dodecane is used as the oil. The fitsare done for three di↵erent salts having in common the K+cation. The fits for KCl and KBr are in very good agreementwith experiments, while the fit for KI is not as good. Thesurface tension of KI shows a very small ��, which is almostindependent of the salt concentration and, hence, harder to fit.In Fig. 4, we fitted yet another series of four di↵erent oxyanions (with Na+ as their cation), and the fits in the figure arein excellent agreement with experiments.

It is reasonable to assume that for the same interface, theadhesivities of the anions/cations will not di↵er significantlyfor di↵erent combinations of anion-cation pairs. For example,we can calculate ↵⇤KI even when we use ↵⇤ as the onlyfit parameter. This can be done with a simple substitution,↵⇤KI' ↵⇤NaI�↵⇤NaBr+↵

⇤KBr' 0.1071. Using this procedure, it is

possible to obtain reasonable predictions for ↵⇤ for additionalsalts.

We also tried to fit the experimental data with twoindependent parameters for anions (↵�) and cations (↵+).



FIG. 3. Comparison of the fitted excess surface tension for KX electrolytes,where X stands for one of the Halogen anions Cl�, Br�, and I�, with ex-perimental data from Ref. 45, as function of ionic concentration, nb, fordodecane/water. The fitted adhesivity values, ↵⇤, are shown in Table I. Allother parameters are as in Fig. 2, beside the dielectric constant of dodecane,"d = 2.

FIG. 4. Comparison of the fitted excess surface tension at the air/waterinterface for NaX electrolytes, where X = IO�3, BrO�3, ClO�3, and NO�3 standsfor one of the oxy anions, with experimental data from Refs. 49 and 50,as function of ionic concentration, nb. The fitted adhesivity values, ↵⇤, areshown in Table II. All other parameters are as in Fig. 2.

Just taking the “best fit” is not su�cient for these fits. Thefirst problem is that our results do not distinguish betweenanions and cations, as explained earlier. This implies that thecation and anion adhesivities have to be attributed from otherexternal considerations.

TABLE II. Fitted values for ↵⇤ and ↵± at the air/water interface. The ↵± areobtained by the procedure elaborated in the text and include a prediction forHClO4 at the air/water interface. The radii for all ions except H+ are obtainedfrom Ref. 47. The H+ e↵ective radius is obtained from Ref. 48.

d (Å) ↵⇤ (kBT ) ↵� (kBT ) ↵+ (kBT )

NaF 7.1 0.2813 0.1515 0.1111NaCl 6.9 0.2052 0.0851 0.1111NaBr 6.88 0.0991 �0.0101 0.1111NaI 6.89 0.0330 �0.0707 0.1111

NaIO3 7.32 0.2673 0.1391 0.1111NaBrO3 7.09 0.1852 0.0671 0.1111NaClO3 6.99 0.0651 �0.0410 0.1111NaNO3 6.93 0.0591 �0.0470 0.1111

HCl 4.32 �0.6116 0.0851 �0.6537HClO4 4.38 �0.9898 �0.5696 �0.6537

The other, and more significant issue, is the fact thatmany pairs of ↵± give similar excellent fits. We first fit all theelectrolytes at the air/water interface in the following way. (i)We fit NaF and choose from the “best fit” the larger adhesivityto be ↵+ as it is known9,21 that F� is more hydrated thanNa+. (ii) We then use the Na+ adhesivity and fit all other ionsin the NaX series. (iii) We continue by using the fitted ↵�for I� adhesivity and fit the KI electrolyte. (iv) With the K+adhesivity and the adhesivities obtained from the NaX fits, wecan make predictions for the air/water surface tension of KBrand KCl.

The same procedure was also applied to two acids, HCland HClO�4 (see Table II). First, we used the Cl� adhesivityto fit HCl and then the H+ adhesivity to fit HClO�4 . Finally,for the K+ fits at the oil/water interface, we first fitted KI andthen used the adhesivity of K+ to fit all other KX salts at theoil/water interface. The results of these fits are all presentedin Tables I and II. We do not show these fits in the figures asthey are almost identical to the fits in Figs. 2-5, where a single↵⇤ parameter is utilized.

In Fig. 5, we show the di↵erence of our prediction to KBrfrom the “best fit” of the data. The predicted surface tensionis excellent. The di↵erence from the “best fit” is very small,and up to 0.8M, it is almost unnoticeable. We do not showthe prediction of KCl, but it is just as good. It is importantto note that other fitting strategies would have given di↵erentresults for ↵± and probably would result in good fits. Ourmodel gives good estimates of the adhesivity parameters forphysical systems, and we can consistently fit a large numberof experiments and even make reasonable predictions.

TABLE I. Fitted values for ↵⇤ and ↵± for the air/water and oil/water interfaces. The ↵± are obtained by theprocedure elaborated in the text and include also predictions for KCl and KBr at the air/water interface. The ionradii are obtained from Ref. 47.

Air Oil

d (Å) ↵⇤ (kBT ) ↵� (kBT ) ↵+ (kBT ) d (Å) ↵⇤ (kBT ) ↵� (kBT ) ↵+ (kBT )

KCl 6.63 0.2112 0.0851 0.1652 6.63 0.1171 �0.0050 0.1313KBr 6.61 0.1732 �0.0101 0.1652 6.61 �0.0390 �0.1451 0.1313KI 6.62 0.0811 �0.0707 0.1652 6.62 �0.3333 �0.4141 0.1313



FIG. 5. Comparison of the “best fit” to the data (dashed line) and the pre-dicted (red solid line) excess surface tension for KBr at the air/water interfaceas function of ionic concentration, nb. The fit parameter values, ↵±, areshown in Table I.

In Fig. 6, we plot the excess surface tension at the air/waterinterface, ��, the sum of Eqs. (49) and (51), as function ofionic concentration, nb, for di↵erent values of ↵. The OSresult is recovered almost exactly for ↵ ' �0.2kBT . Noticethat the surface tension has an upper bound for ↵!1, butthis bound is practically reached for ↵ & 5kBT . This happensfor infinite repulsion when all the ions are expelled from theproximal layer. In addition, in order to obtain this bound, werequire that the ionic concentration at the surface cannot benegative. This is not the case for ↵!�1, which leads toan infinite amount of anions to be adsorbed on the surface.This is not physical because the actual upper bound is theclose packing concentration (not accounted by us), even foran infinite attraction. In this paper, we expect, and obtain,only small values of adhesivities. Hence, this non-physicalsituation is not relevant.

We can now arrange the various ions in an extendedreverse Hofmeister series with decreasing adhesivitystrength at the air/water interface. The anions series is

FIG. 6. Excess surface tension at the air/water interface as function of ionicconcentration, nb, for di↵erent values of ↵ (in units of kBT ). The OS resultis obtained for ↵ ' �0.2kBT . All other parameters are as in Fig. 2.

F� > IO3� > Cl� > BrO3

� > Br� > ClO3� > NO3

� > I�, whilefor cations it is K+> Na+.

At the oil/water interface, Fig. 3, the same reversedHofmeister series emerges and the interaction becomesmore attractive. This e↵ect is substantially stronger for theanions and might be connected with the stronger dispersionforces at the oil/water interface, related directly to thelarge anion polarizabilities.30 We denote the di↵erence inadhesivity between the air/water and oil/water interfaces by�↵ = ↵(a/w)�↵(o/w). From the fitted values of ↵⇤, �↵ isdi↵erent for each anion and �↵I > �↵Br > �↵Cl. This can beexplained by a change in the water-surface interaction. If thewater-water interaction (hydrogen bonds) becomes weaker inthe vicinity of the surface, the larger ions (with respect to theircrystallographic size) will be more attracted to the surface.

VI. CONCLUDING REMARKS

Our work presents a general self-consistent theory forcalculating free energies up to one-loop order for ionicsolutions with a surface proximal layer of finite width. Inthis layer, we consider a slowly varying ion-specific surfacepotential. The loop expansion we use can be computedsystematically to higher orders, and re-summing the loopterms is actually equivalent to the cumulant expansion donein Refs. 22 and 23.

The calculation of the excess surface tension, ��, isbased on the free-energy di↵erence between a system with anair/water interface on one hand and two semi-infinite systems(electrolyte and air) with no interface on the other hand.The same calculation can be applied to other liquid/liquidinterfaces such as oil/water, simply by using "a as the dielectricconstant of the oil instead of air. This calculation method isequivalent to the Gibbs adsorption isotherm method,51 but itis mathematically more accessible as it avoids the explicitcalculation of the ionic densities. It is also possible to use thegrand potential instead of the free-energy, but the latter doesnot simplify the calculation, because one has to consider theone-loop correction to the fugacities (Appendix C).

We have computed the linearized MFT electrostaticpotential as well as the MFT and one-loop free energies,utilizing the secular determinant method, where we haveextended the secular determinant method of Ref. 39 for thethree boundaries as in our system.

The one-loop excess surface tension, ��1, is calculatedand the OS surface tension result was naturally recoveredand extended. In fact, we showed that the OS result isobtained by considering the thermodynamic fluctuations ofthe electrostatic potential around its MFT solution, while thevolume fluctuations recovered the known DH correction tothe MFT free energy. Our surface-tension result is analyticaland interpolates between several known limits: the result ofRef. 38 for d! 0, the OS one for ↵±! 0 or d = 0, and a Sternlayer for ↵±!1. A wide variety of monovalent ions at theair/water and oil/water interfaces were fitted by our model, alltaken on a common and unified ground.

Within some approximations, we obtain an analyticaldependence of the excess surface tension on the salt



concentration. The fits for �� agree well with experimentsand show clearly the reversed Hofmeister series (F� > Cl� >Br� > I�) both at the air/water and oil/water interfaces. Thevarious fits reveal an even more extended (Hofmeister)series: F� > IO3

� > Cl� > BrO3� > Br� > ClO3

� > NO3� > I�.

In Ref. 50, a di↵erent series was obtained for anions:IO3

� > F� > BrO3� > Cl� > NO3

� > Br� > ClO3� > I�. At

present, it is not clear which of these two predictions is moreaccurate. Other experimental measurements, such as surfacepotential, might shed light on this discrepancy. We intend tofurther investigate this issue in the future by calculating thesurface potential to one-loop order.

In the weak-coupling linear regime, where our theoryis valid, fluctuations dominate over the MFT contribution,and the cation and anion adhesivities are roughly equal,↵+' ↵�. This then leads to a small MFT contribution, whilethe one-loop contribution has terms proportional to ln(⇤) andindependent on ↵±. These terms arise from the image chargeinteraction and, indeed, give the OS result with an ionic-sizecorrection (minimal distance of approach between ions).

Two important limitations of our theory are related to theion finite size and to the linearization of the MFT electrostaticequations. For large adhesivities, where the ion density on thesurface is high and does not correspond to the dilute solutionlimit, our theory is expected to fail. Instead, other theories,such as the modified PB theory,52 can be utilized as theytake into account explicitly the ion finite size. Considering,for example, the surface tension of acids, such as HCl andHClO4, the fitted ↵’s from our theory are found to be ratherlarge (see Table II) and are not expected to be as reliable, for thereasons mentioned above. Furthermore, when |↵+�↵�| ' kBT ,the linear approximation fails and one should solve the fullMFT electrostatic equation.

Note that the image charge term (or other ion-interfaceinteractions) cannot be simply added in the Boltzmann weightfactor (potential of mean force). Since the PB equation isa MFT equation that follows from a certain free-energyminimization, a consistent way to generalize it should be basedon an augmented free-energy functional, which then gives ageneralized PB equation. In this way, double-counting ofdi↵erent electrostatic contributions is prevented, and remedya common ambiguity where the image charge or other ion-interface interactions are added to the PB equation in an adhoc fashion.

Our model provides a self-consistent way to calculate thefluctuations around the MFT free energy. The image chargecontribution naturally arises from the fluctuations hence ↵±originate only from solvent (short-range) interactions and theproblem of double counting is avoided. An augmented freeenergy (“action”), Eq. (10), is obtained and its minimizationindeed leads to a modified PB equation.

The microscopic origin of the adhesivity is still notvery well understood. Nevertheless, several possibilities havebeen suggested recently. For the special case of silica/waterinterface,12,13 the orientation of water molecules in the vicinityof the interface was proposed to lead to changes in thehydrogen bond strength at the interface. This surface e↵ectcan be identified as a possible microscopic source of ↵±,whose value is proportional to the di↵erence in free energy

between a single ion solvated in the bulk as compared to itspartially solvated state at the surface. Another possible originfor the adhesivity is the cavitation energy.21 The ion interfereswith the water structure by breaking hydrogen bonds. Thiscosts energy, leading to an e↵ective attraction of the ion tothe interface. Other e↵ects such as the ion polarizability21 ordispersion forces30 might also play a role in the ion-surfaceinteraction. Since all proposed origins of the adhesivity areshort range (their range is comparable to the ionic radius) andour theory is a coarse-grained one, it is appropriate to averageover their degrees of freedom and obtain ↵±, whose magnitudeis reasonably obtained from the fits to the experimental data.

Our model is applicable to many systems that exhibita spatial region characterized by ionic-specific and slowlyvarying interactions. Two examples of such systems are adense layer of polymer brushes that are attached to a solidsurface or a polyelectrolyte gel occupying a finite volumeof solution. Our model is ionic-specific and can be used toobtain simple and analytical predictions also for such complexinhomogeneous systems.

ACKNOWLEDGMENTS

This work was supported in part by the Israel ScienceFoundation (ISF) under Grant No. 438/12 and the U.S.-Israel Binational Science Foundation (BSF) under GrantNo. 2012/060. We thank M. Biesheuvel, Y. Ben-Yakar, A.Cohen, H. Diamant, R. Netz, H. Orland, and Y. Tsori fornumerous suggestions, and D. Frydel, Y. Levin, and W. Kunzfor useful discussions. One of us (R.P.) would like to thankTel Aviv University for its hospitality during his stay thereand acknowledges the support of the ARRS through GrantNo. P1-0055.

APPENDIX A: SECULAR DETERMINANT

The 4 ⇥ 4 coe�cient matrices A, B, and C for theboundary condition equation, as discussed in Eq. (28), are

A=*.....,

�"ak "w 0 00 0 0 00 0 0 00 0 0 0

+/////-,

B=*.....,

0 0 0 01 0 �1 00 1 0 �10 0 0 0

+/////-, (A1)

C=*.....,

0 0 0 00 0 0 00 0 0 00 0 0 1

+/////-.

In general, for a system with two boundaries, the seculardeterminant can be written as39

D⌫(k)= det⇥A+B �⌫(a)��1

⌫ (0)+C �⌫(L)��1⌫ (0)

⇤, (A2)



with �⌫(z) computed at the spatial point z and is defined asanother 4 ⇥ 4 matrix

�⌫(z)=*......,

h(2)⌫ g(2)⌫ 0 0

@zh(2)⌫ @zg

(2)⌫ 0 0

0 0 h(3)⌫ g(3)⌫

0 0 @zh(3)⌫ @zg

(3)⌫

+//////-.

In the equation above, we have used the notations h(i)⌫ (z) and

g(i)⌫ (z) with i = 2, 3 for the two independent solutions of theeigenvalue equation, Eq. (26), in the proximal (0 z d)and distal (z > d) regions, respectively. It is then convenientto choose these independent solutions such that �(0) = 1.The secular determinant expression, Eq. (A2), can be thensimplified as is used in Eq. (29).

APPENDIX B: ONE-LOOP SURFACE TENSION TOORDER O(1/⇤)

The analytical solution to the integral in Eq. (51) in thelimit ⇤!1 can be computed to order O(d2) by neglectingterms of order O(1/⇤), while retaining constant terms (withrespect to ⇤)

��1'�kBT8⇡

"w�"a"w+"a

!2

D

4

2666642 ln(D⇤�1)�1

+4"w

("w�"a)2*,"a ln 2�"w ln

"w+"a"w+!�1

D

+-377775

+kBT8⇡

!D

"w+"a

2666664� "w"w�"a

� !

2D("w+"a)

+!

D("w�"a)*,

2"a"w�"a

ln 2� ln"w+"a"w+!�1

D

+-

+!

D("w+"a)ln(D⇤

�1)

+4"w"a

⇥2

D

�"2w�"2

a

��!2⇤ 1

2

D("w�"a)2("w+"a)

⇥ arctan*..,

q2

D("2w�"2

a)�!2

D("w+"a)+!+//-

37777775, (B1)

with ! ⌘ "wd�⇠2� 2

D

�. The first term above (proportional to

2D) represents a correction to the OS result. In the case where↵±= 0, we recover exactly the result obtained in Ref. 31 (apartfrom an inconsequential typo). The second term (proportionalto !D) is the ion-specific term, with leading behavior as inEq. (53).

APPENDIX C: FUGACITIES ONE-LOOP CORRECTION

The grand-potential is written in terms of the fugacities,�i, with i = ±, for the anions and cations, respectively. Thefugacities are related to the bulk densities by

n(i)b =��i

�

V@⌦

@�i. (C1)

It is clear from the above equation that if one modifies thegrand potential with quadratic fluctuations, the fugacities willbe modified as well. Writing the grand potential to one-looporder yields

⌦'⌦0(�i)+C⌦1(�i), (C2)

where we introduce the parameter C to keep track of theexpansion terms. After finishing the calculation, this parameterwill be set to unity. Substituting Eq. (C2) into Eq. (C1) gives,�i,1, the one-loop correction to the fugacities

�i'�i,0+C�i,1

=�266664

V n(i)b

�@⌦0/@�i+C�i,0

@⌦1/@�i@⌦0/@�i

377775�i=�i,0, (C3)

where �i,0 is the zeroth-order fugacity. We now expand thegrand-potential correction,⌦1(�i), around �i,0 and use Eq. (C3)for �i,1 to obtain

⌦'⌦0(�i,0)

+C266664⌦1(�i,0)�

X

i=±�i,0

@⌦1(�i)@�i

377775�i=�i,0. (C4)

The electro-neutrality conditionP

i�iqi = 0, for symmetricelectrolytes, imposes �+ = �� ⌘ �. Using Eq. (C1) and thedefinition of the fugacities, Eq. (5), we obtain the intrinsicchemical potential

µi(r)'�12

e2u(r, r)+ kBT"ln

��i,0a3�+C �i,1

�i,0

#. (C5)

In the above equation, we have also expanded the fugacitiesaround �i,0. We now calculate the free energy from Eqs. (38)and (C4), with the chemical potential obtained in Eq. (C5),

F 'F0+CF1=⌦0(�i,0)�e2

2n(i)b Vub(r, r)

+V kBTX

i=±n(i)b ln

��i,0a3� + C

"⌦1(�i,0)

+X

i=±�i,1

@⌦0(�i)@�i

!

�i=�i,0

+ V kBTX

i=±n(i)b

�i,1�i,0

#. (C6)

The second and third terms in F1 (the square brackets) canceleach other exactly (from the definition of �0, Eq. (C3)). Thus,the free-energy to one-loop order yields

F '⌦0(�i,0)�12

e2n(i)b Vub(r, r)

+V kBTX

i=±n(i)b ln

��i,0a3�+⌦1(�i,0). (C7)

This result gives rise to a useful simplification, where to firstorder in a loop expansion of the free-energy, the fugacities canbe taken as �i,0 and are equal to the bulk densities. Note thatthis is not the case for the grand potential,⌦. For the latter, thefugacities must be calculated consistently to one-loop orderas shown in Eq. (C3).

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⌘+ p

⇣1+�e�2kd

⌘#,

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Pi µ

toti Ni is the Helmholtz

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our Eq. (36) by identifying ! = "wd�⇠2�2

D�.

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Oxford, New York, 1980), Vol. 4, Chap. 15, p. 1.52I. Borukhov, D. Andelman, and H. Orland, Electrochim. Acta 46, 221 (2000).


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