Theory of Surface Forces in Multivalent ElectrolytesRahul Prasanna Misra,†,§ J. Pedro de Souza,†,§ Daniel Blankschtein,*,† and Martin Z. Bazant*,†,‡

†Department of Chemical Engineering, Massachusetts Institute of Technology, 25 Ames Street, Cambridge, Massachusetts 02142,United States‡Department of Mathematics, Massachusetts Institute of Technology, 182 Memorial Drive, Cambridge, Massachusetts 02142,United States

*S Supporting Information

ABSTRACT: Aqueous electrolyte solutions containing multivalent ions exhibit various intriguing properties, includingattraction between like-charged colloidal particles, which results from strong ion−ion correlations. In contrast, the classicalDerjaguin−Landau−Verwey−Overbeek theory of colloidal stability, based on the Poisson−Boltzmann mean-field theory,always predicts a repulsive electrostatic contribution to the disjoining pressure. Here, we formulate a general theory of surfaceforces, which predicts that the contribution to the disjoining pressure resulting from ion−ion correlations is always attractiveand can readily dominate over entropic-induced repulsions for solutions containing multivalent ions, leading to thephenomenon of like-charge attraction. Ion-specific short-range hydration interactions, as well as surface charge regulation, areshown to play an important role at smaller separation distances but do not fundamentally change these trends. The theory isable to predict the experimentally observed strong cohesive forces reported in cement pastes, which result from strong ion−ioncorrelations involving the divalent calcium ion.

■ INTRODUCTIONThe accurate prediction of forces between charged surfaces inaqueous electrolytes is of paramount importance in diversescientific disciplines, ranging from colloidal science tobiophysics and polymer chemistry.1 Surface forces ultimatelydetermine whether colloidal particles and macromolecules,such as DNA, proteins, and polymers, will aggregate or willremain stable in dispersions.2−6 Typically, two different typesof forces operate between charged surfaces in electrolytesolutions: short-range van der Waals forces, which arepredominantly attractive, and long-range electrostatic forces,which can be attractive or repulsive depending on the sign andmagnitude of the charge on the interacting surfaces.7−9 Themost widely used model of colloidal stability is the classicalDerjaguin−Landau−Verwey−Overbeek (DLVO) theory,10,11

in which the electrostatic contribution to the disjoiningpressure (force per unit area acting on the charged surfaces)is described using the Poisson−Boltzmann (PB) mean-fieldtheory, which assumes that the ions residing in the electricdouble layer (EDL) at the charged surface feel a meanelectrostatic potential from the smeared out (volume-averaged) charge density near the charged surface. The PB

model also assumes that the dielectric permittivity of thesolvent is constant and uniform, considers ions to be point-sized, and neglects any form of ion−ion correlations. Theprediction of the disjoining pressure by the DLVO theoryutilizing the PB model is in very good agreement1,12−25 withexperimental data in the so called weak-coupling limit, that is,the limit corresponding to a low surface charge density, highsolvent dielectric permittivity, low valency of counterions(those ions containing charge of sign opposite to that of thesurface), or high temperature. In the case of multivalentelectrolytes, ion−ion correlations become important due to thelarge valency of the counterions, and the PB model fails todescribe the disjoining pressure acting between chargedsurfaces even qualitatively.26−30 In particular, the mean-fieldPB theory always predicts a repulsive EDL pressurecontribution for like-charged surfaces, irrespective of themagnitude of the surface charge density, the ion concentration,or the valency of the counterions. This is in stark contrast to

Received: April 15, 2019Revised: July 10, 2019Published: July 16, 2019

Article

pubs.acs.org/LangmuirCite This: Langmuir 2019, 35, 11550−11565

© 2019 American Chemical Society 11550 DOI: 10.1021/acs.langmuir.9b01110Langmuir 2019, 35, 11550−11565

This is an open access article published under a Creative Commons Non-Commercial NoDerivative Works (CC-BY-NC-ND) Attribution License, which permits copying andredistribution of the article, and creation of adaptations, all for non-commercial purposes.

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the experimental results31−34 and Monte-Carlo simula-tions,29,35−37 which have provided ample evidence of thephenomenon of like-charge attraction mediated by multivalentions. This clear signature of electrostatic correlations isresponsible for many important interfacial phenomena,including cement cohesion,34,38 biopolymer aggregation,39

and colloidal coagulation,30,40 and nevertheless, it still lacks asimple mathematical description.Several theoretical approaches have been used in the past to

explain like-charge attraction, including the theory by Perel andShklovskii,41 the hypernetted-chain integral equationtheory,35,42 and the dressed-ion theory.43,44 Although thesetheoretical approaches have certainly improved our under-standing of ion−ion correlations, they lack the mathematicalsimplicity underlying the PB theory, which allows thedisjoining pressure to be directly related to various systemparameters, including the salt concentration, the ion valency,the solvent dielectric permittivity, the surface charge density,and the temperature. To this end, a simple and general theoryof electrostatic correlations based on a Landau−Ginzburg-typecontinuum framework was recently developed by Bazant,Storey, and Kornyshev (BSK),45 and initially used to explainscreening phenomena in solvent-free ionic liquids. Comparedto the PB mean-field theory, the BSK free-energy functionaladds an additional term containing the second derivative of theelectrostatic potential to account for ion−ion correlations

) { }∫∮

ρϕ φ ϕ

ϕ

= − ϵ [|∇ | + ∇ ] +

+

+ −l g c c

q

r

r

d2

( ) ( , )

d

V

S

2c

2 2 2

s (1)

where ρ = e(z+c+ − z−c−) is the mean charge density, z± and c±are the valencies and concentrations, respectively, of thecations and the anions, in the EDL of volume V, ϕ is theelectrostatic potential, qs is the surface charge density, ϵ is thedielectric permittivity of the solution, lc denotes the electro-static correlation length, and g(c+,c−) is the entropiccomponent of the total free energy and corresponds to theentropy density arising from the translational entropy of theions. Note that ϕ− |∇ |ϵ

22 and ρϕ represent the self-energy of

the electric field and the electrostatic potential energy of theions in this electric field, respectively. The potential gradientterm, ϕ− ∇ϵ l ( )

2 c2 2 2, is used to model an additional contribution

to the self-energy resulting from electrostatic correlationsbetween the ions. The terms excluding the entropiccontribution, g, arise via a charging process in which ions areadded to a system in which they interact with the electrostaticpotential, ϕ. Although the higher-order equation is still solvedin the mean-field of spatially averaged ϕ and ρ, the higher-order third term implies the nonlocal nature of the free energydue to electrostatic correlations. One can interpret this term asa phenomenological correction to the mean-field PB frame-work that is solved for in the mean field. This simple extensionof the PB theory is a useful first approximation of ion−ioncorrelation effects in ionic liquids46−50 and in so-called “water-in-salt” electrolytes.51

The BSK model has also been applied to electrolyticsolutions52 and shown to capture various ion correlationeffects, including electrophoretic mobility reversals of colloidalparticles in multivalent electrolytes,53 micelle formation,54 andion conduction through biological ion channels.55−58 The first

challenge in applying the BSK theory to different systems is tofind a suitable approximation of the electrostatic correlationlength, lc. In the original article, Bazant et al.45 suggested usingthe Bjerrum length, lb, for dilute electrolytes and the ionicdiameter for the opposite limit of room-temperature ionicliquids (where it is typically much larger than the Debyelength). In the general case of a z:1 primitive-modelelectrolyte, where z is the valency of the counterion, thecorrelation length parameter was recently found to scale as59

∼ | | − −l l q e c( / )c b1/4

s1/8

01/6

(2)

where = πϵl ek Tb 4

2

Bis the Bjerrum length,1 e is the unit of

elementary charge, and c0 is the bulk salt concentration. Notethat the Bjerrum length corresponds to the length scale atwhich the electrostatic interactions between two monovalentions in the solution is comparable to the thermal energy, kBT,where kB is the Boltzmann constant and T is the absolutetemperature. Physically, the correlation length, lc, scales withthe diameter of a correlation hole in the electrolyte, beyondwhich the mean-field equations are sufficient to describe theelectrostatic interactions between ions, as reflected in the PBmodel. In this article, we describe ion−ion correlations usingthe BSK model for electrolytes and study their influence on thedisjoining pressure acting between charged surfaces.The second challenge in applying the BSK model is the

choice of an additional boundary condition for the generalizedPoisson equation, which is a fourth-order ordinary differentialequation in our one-dimensional geometries. The second-orderPB equation requires only one condition on each boundingsurface, whereas the BSK equation requires two. Althoughmixed boundary conditions were first proposed,60 the standardchoice, advocated by BSK for ionic liquids, has been to set thethird derivative of the potential to zero, effectively cutting offion−ion correlations at the distance of closest approach. Here,we take advantage of the recent development of moreconsistent and general boundary conditions, which areobtained by equating the normal Maxwell stress at the chargedsurface to its form in the absence of ion−ion correlations givenby the contact theorem.59

We should stress here that the phenomenon of like-chargeattraction can be modeled using solely the higher order term inthe BSK functional eq 1 (see the Results section for details).However, for an accurate comparison of our theoreticallypredicted disjoining pressure with experimental data, we alsoincorporate some additional interactions, to arrive at a morecomplete generalization of the DLVO theory. In particular, thePB model assumes that ions are point-sized and are distributedthroughout the electrolyte according to the Boltzmanndistribution. This results in an exponential growth ofcounterions near the charged surface.60 In practice, however,the region closest to the charged surface is always hydrated bya condensed layer of water molecules. This layer is inaccessibleto the counterions and is referred to as the Stern layer.11,61

Recently, Bohinc et al.62 formulated the Poisson−Helmholtz−Boltzmann model to describe solvent-mediated,nonelectrostatic interactions between ions based on a pair-wiseYukawa potential. In this model, in addition to electrostaticinteractions, the ions experience a hydration (h) interactiongiven by, Uh/kBT = e−κh(r−lh)/r, where Uh is the hydrationenergy, r is the separation distance between the ions, κh is theinverse of the length scale corresponding to the decay ofordered water layers around an ion, and lh is the length scale at

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which Uh becomes comparable to the thermal energy, kBT. Inthis study, we follow the approach of Bohinc et al.62,63 toincorporate an additional solvent-mediated hydration inter-action into the model in eq 1. Our approach serves twoimportant purposes. First, we can self-consistently model boththe Stern and the diffuse layer regions of the EDL using asingle theory. Second, we can study the effect of the Stern layerregion on the disjoining pressure acting between chargedsurfaces. Recall that in the Stern layer region, counterions arerepelled from the charged surface due to the short-rangehydration interactions, whereas coions, which carry charge ofthe same sign as that of the charged surface, are repelled due tothe unfavorable repulsive electrostatic interactions with thecharged surface. We later show that the short-range repulsivehydration forces reported by Israelachvilli and Pashley,64 aswell as by LeNeveu and Rand,65 emerge naturally from theincorporation of hydration interactions into our model of theEDL.After incorporating ion−ion correlations and hydration

interactions into our EDL model, we implement a finalmodification by describing the phenomenon of surface chargeregulation (CR). This is the electrochemical mechanism bywhich surfaces in contact with aqueous electrolytes acquirecharge from the dissociation of surface species, such as thesilanol groups that typically terminate glass or silica surfaces.66

In this case, the degree of dissociation depends strongly on thepH of the solution. To validate our theoretical predictions onlike-charge attraction, we make use of the experimental data ofPlassard et al.34 who measured the disjoining pressure actingbetween calcium silicate hydrate (CSH) layers in a calciumhydroxide electrolyte solution. We incorporate the mechanismof surface CR into our EDL model by assuming that the chargeon the CSH layers results from the dissociation of silanolgroups (SiO−) at the CSH surface,34 which allows us to relatethe surface charge density on the CSH surface to the calciumhydroxide concentration in the solution.The end result after incorporation of ion−ion correlations,

hydration interactions, and surface CRphenomena which areall currently neglected in the DLVO theory, is a complete, self-consistent theory of the disjoining pressure for multivalentelectrolytes. Specifically, we demonstrate that our theory canbe used to describe the phenomenon of like-charge attraction,including predicting the disjoining pressure, in reasonablygood agreement with the experimental data of Plassard et al.34

for a broad range of salt concentrations.

■ MODEL AND METHODSWe begin our analysis by considering two similarly chargedsurfaces immersed in a z:1 electrolyte solution, where zdenotes the valency of the counterion. For reference, weassume that both surfaces are negatively charged, whichimplies that cations are the positively charged counterions inthis system. Because hydration interactions are short-ranged,we simplify our analysis by assuming that the hydrationinteractions are only relevant for the cations and negligible forthe anions, which are anyway repelled because of the repulsiveelectrostatic interactions with the negatively charged surface.Therefore, in our model, the cations interact with thenegatively charged surface through both electrostatic andhydration interactions, whereas the anions interact with thenegatively charged surface only through electrostatic inter-actions. Furthermore, as shown in Figure 1, the region closestto the negatively charged surface is accessible only to the water

molecules which serve as the source term for the boundarycondition for the hydration potential (see below). As discussedby Brown et al.,63 this is similar in spirit to the surface chargedensity at the charged surface serving as the boundarycondition for the electrostatic potential. The negativelycharged surfaces electrostatically attract cations from the bulkreservoir, resulting in the formation of an EDL of ions in theregion between the charged surfaces. As shown in Figure 1, theformation of the EDL results in the pressure felt by the chargedsurfaces, referred to as the disjoining pressure, beingsignificantly different than the osmotic pressure in the bulkreservoir.After incorporation of additional contributions from

hydration interactions into eq 1, we arrive at the followinggeneral functional for the total Helmholtz free energy of thesystem

) { }∫∮

∫

∮ ∫

ρϕ ϕ ϕ

ϕ

κ ψ ψπ

ψ

σ ψ μ μ

= − ϵ [|∇ | + ∇ ] +

+

+− − ∇

+ −

+ − +

κ

+ −

+

+ + − −

lmooonoooÄÇÅÅÅÅÅÅÅÅÅÅÅ

ÉÖÑÑÑÑÑÑÑÑÑÑÑ

|}ooo~ooo

l g c c

q

lk T c c k T

c c

r

r

r

r r

d2

( ) ( , )

d

d( )

8 e( )

d d ( )

V

S

V l

S V

2c

2 2 2

s

h2

h2

h2

hB 0 h B

h h

h h

(3)

where ψh is the dimensionless hydration potential, σh is thesurface density of water molecules at the charged surface whichmediate the interaction of cations with the charged surface, andμ+ and μ− are the chemical potentials of the cations andanions, respectively. The terms in the 1st and 2nd rows in eq 3are the same as those of eq 1 and correspond to the free energy

Figure 1. Schematic of our model problem: a multivalent z:1electrolyte is confined between two surfaces with surface chargedensity, qs. The disjoining pressure, Π, is defined as the difference inthe pressure (P) felt between the plates due to the presence of theEDL and the pressure in the bulk reservoir (P∞). As shown here, theStern layer which is the region closest to the charged surface isaccessible only to the water molecules (depicted as ellipsoids)modeled using the hydration potential, whereas the concentrationprofiles of the cations (depicted as red spheres) and the anions(depicted as blue spheres) in the EDL are influenced by ion−ioncorrelations, described in our theory using the BSK model.

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of the system after incorporating ion−ion correlations into thePB model. Furthermore, the terms in the 3rd row and the 1stterm in the 4th row in eq 3 result from the incorporation ofhydration effects into our model. Finally, the last term in eq 3couples the concentration of ions in the EDL confined by thecharged surfaces to that in the bulk reservoir. Although theterms in eq 3 resulting from ion−ion correlations andhydration interactions have been reported in refs 45 and 62,respectively, for completeness, a detailed derivation of theHelmholtz free energy expression in eq 3 is provided in theSupporting Information document.We note here that the incorporation of hydration

interactions automatically accounts for the finite size of theions. Therefore, there is no need to incorporate any additionalexcluded-volume interactions in our model, and we can retainthe original expression for the entropy density, g(c+,c−), fromthe PB model given by

∑= −=±

ÄÇÅÅÅÅÅÅÅÅÅÅÅ

ikjjjjj ikjjjjj y{zzzzz y

{zzzzzÉÖÑÑÑÑÑÑÑÑÑÑÑg k T c

cc

ln 1i

ii

iB

0 (4)

where c0i (i = ±) is the corresponding concentration ofcations/anions in the bulk reservoir.Next, we enforce the conditions of thermodynamic

equilibrium in the system by setting )δ δϕ =/ 0 for theelectrostatic potential and )δ δψ =/ 0h for the hydrationpotential in eq 3, respectively. For the electrostatic potential,we obtain a fourth-order Poisson equation and a boundarycondition, respectively, given by

ϕ ρϵ ∇ − ∇ = = −+ −l zec ec( 1) ( )c2 2 2

(5)

ϕ·ϵ ∇ − ∇ =n l q( 1)c2 2

s (6)

Similarly, the governing equations for the dimensionlesshydration potential, ψh, and the corresponding boundarycondition are given by

ψ κ ψ π∇ − = − [ − ]κ+l c c4 e l2

h h2

h h 0h h (7)

ψ π σ·∇ = − κn l4 e lh h h

h h (8)

We note here that eq 7 does not contain any term whichdepends on the concentration of anions, c−, which is consistentwith our assumption that the hydration potential acts onlybetween cations. As discussed by Brown et al.,63 eq 8 resultsfrom assuming that each ordered water molecule near thenegatively charged solid surface acts like a source term for thehydration interaction of the cations with the charged surface.At thermodynamic equilibrium, ) =δ

δ ±0

c, which results in the

following expressions for the chemical potential of thecounterions and the coions

μ ϕ ψ= + ++ +

+

ikjjjjj y{zzzzzk Tzek T

cc

lnB B

h0 (9)

μ ϕ= − +− −

−

ikjjjjj y{zzzzzk Te

k Tcc

lnB B 0 (10)

where c+0 and c−0 are the bulk concentration of the counterionsand the coions, respectively. To satisfy the condition ofelectroneutrality in the bulk, c+0 = c0 and c−0 = zc0 (c0 is theconcentration of the undissociated z:1 salt). Equating eq 10 at

any point in the EDL and in the bulk reservoir results in thefollowing expressions for the dimensionless counterion and thecoion densities in the EDL: c+ = e−zϕ+ψh, c− = eϕ, where,ϕ = ϕe

k TB, =+c

cc0and =−c c

zc0. For a numerical evaluation of

the electrostatic potential in the EDL, it is convenient toconvert eq 5 into a dimensionless form, given by

δ ϕ∇ − ∇ = − +

+ −c cz

( 1)1c

2 2 2

(11)

where x = x/λD, λD is the Debye−Huckel screening length,which for a z:1 electrolyte is given by the expression:λ = ϵ

+k T

z z c eD2

( 1)B

02 , ∇ = λD∇, and δc = lc/λD is the dimensionless

correlation length. The dimensionless correlation length for az:1 electrolyte is given by the following expression in ref 59

δ λ=−ikjjjjj y{zzzzz ikjjjjj y{zzzzzz l

lz l

0.35c

2b

GC

1/8 2b

D

2/3

(12)

In eq 12, the quantity, lGC, denotes the Gouy−Chapmanlength, which is the length scale at which the interaction of acounterion with a uniformly charged surface becomescomparable to the thermal energy, kBT, and is given by

π= | |l ezl q2GC

b s (13)

Similar to the electrostatic potential, the dimensionlessgoverning equation for the hydration potential is given by

ψ κ λ ψ π λ∇ − = [ − ]κ+l c c4 e 1l2

h h2

D2

h h 0 D2h h (14)

Note that the governing equation for the dimensionlesselectrostatic potential (eq 11) is fourth-order. Therefore, weneed four boundary conditions to solve for the dimensionlesselectrostatic potential. Assuming that the negatively chargedsurfaces are located at −d/2 − lh and d/2 + lh, where d is theseparation distance between the two negatively chargedsurfaces, corresponding to the region accessible to the ions,and lh is the hydration length parameter, eq 11 can be solvedusing the following four boundary conditions:

ϕ δ ϕλ

′ = −− − ′′′ = −

− = − ϵ

ikjjjj y{zzzz ikjjjj y{zzzzx d l x d lq e

k T(i)

2 2,h c

2h

s D

B

ϕ δ ϕλ

′ =+ − ′′′ =

+ = ϵ

ikjjjj y{zzzz ikjjjj y{zzzzx d l x d lq e

k T(ii)

2 2,h c

2h

s D

B

δ ϕ ϕ ′′′ = −− = ″ = −

− ikjjjj y{zzzz ikjjjj y{zzzzx d l x d l(iii)

2 2, andc h h

δ ϕ ϕ ′′′ =+ = − ″ =

+ ikjjjj y{zzzz ikjjjj y{zzzzx d l x d l(iv)

2 2.c h h

Note that boundary conditions (i) and (ii) result fromequating the electric displacement field at the charged surface,which is similar to the approach used in the previous studies toderive the boundary conditions for the PB model.Furthermore, the boundary conditions (iii) and (iv) resultfrom enforcing a force balance at contact, where ioniccorrelations must vanish.59 The boundary conditions to solvefor the dimensionless hydration potential using eq 14 are givenby:63

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ψ π λ σ′ = −− = − κikjjjj y{zzzzx d l l(i)

24 e andl

h h h D hh h

ψ π λ σ′ =+ = κikjjjj y{zzzzx d l l(ii)

24 e .l

h h h D hh h

Note that for a given d, the actual domain over which theequations are being solved changes depending on whether thehydration potential is included in the model. If the hydrationpotential is included (i.e., lh ≠ 0), then the electrolyte domainlies between x = −d/2 − lh and x = d/2 + lh. If not (lh = 0),then the distance of closest approach must be subtracted, suchthat the electrolyte domain consists of the region between x =−d/2 and x = d/2. Such a distinction is important when thepredicted disjoining pressure is compared using differentmodels, as shown in the Results section, where the separationdistance between the charged surfaces is denoted using d for allof the models considered.Next, we present our derivation of the disjoining pressure by

expressing the total free energy of the system as follows:) ∫ ∮ ∮ϕ ϕ ϕ ψ ψ ϕ σ ψ= ′ ″ ′ + +f qr r r( , , , , )d d d

V S Sh h s h h . At

thermodynamic equilibrium, ) =δδϕ 0 and ) =δ

δψ 0h

for the

bulk variation, which together with the mathematical identitiesof variational calculus can be used to obtain the followingrelation (see the detailed derivation in the SupportingInformation document)

ϕ ϕ ϕ ϕ ρϕ μ μ

ψκ ψ ψ

π

ϵ ′ + ϵ ″ − ϵ ′′′ ′ + + − −

+ − +− + ′

=κ

+ + − −

+

ÄÇÅÅÅÅÅÅÅÅÅÅÅ

ÉÖÑÑÑÑÑÑÑÑÑÑÑ

ll g c c

k T c c k Tl

2 2

( )8 e

constantl

2 c2

2c

2

h B 0 Bh

2h

2h

2

hh h

(15)

At thermodynamic equilibrium for a z:1 electrolyte, thefollowing additional relations apply

)δδ ϕ ψ

μ

= ∂∂ = ⇒ ∂

∂ = + + ∂∂

− =+ +

++

+ + ++

+ +

cfc

cfc

zec c k T cgc

c

0

0

h B

(16)

)δδ ϕ μ= ∂

∂ = ⇒ ∂∂ = − + ∂

∂ − =− −

−−

− −−

− −cf

cc

fc

ec cgc

c0 0

(17)

Adding eqs 16 and 17 and then subtracting the resultingexpression from eq 15 yields

∑

ϕ ϕ ϕ ϕ ψ

κ ψ ψπ

ϵ ′ + ϵ ″ − ϵ ′′′ ′ −

+− + ′

+ − ∂∂ =κ

=±

ÄÇÅÅÅÅÅÅÅÅÅÅÅ

ÉÖÑÑÑÑÑÑÑÑÑÑÑ

ll c k T

lk T g c

gc

2 2

8 econstantl

ii

i

2 c2

2c

20 h B

h2

h2

h2

hBh h

(18)

We note here that eq 18 is valid for any arbitrary separationdistance between the two charged surfaces. Furthermore, theleft hand side (LHS) of eq 18 is a constant, which can beevaluated at any point between two charged surfaces.Therefore, the constant in eq 18 should be in some wayrelated to the pressure acting between the charged surfaces.Indeed, in analogy to the constant in eq 18, the condition ofthermodynamic equilibrium ensures that the pressure is

uniform throughout the region confined between the twocharged surfaces. The relationship between the constant in eq18 and pressure becomes clearer when we consider the limitwhere the two charged surfaces are placed very far apart, suchthat the electrostatic potential, the hydration potential, and allhigher-order derivatives of the electrostatic and the hydrationpotentials reduce to zero at the mid plane between the twocharged surfaces. In this case, when evaluating the LHS of eq18 at the mid plane between the two charged surfaces, the firstfive terms involving the electrostatic and the hydrationpotentials, as well as their higher-order derivatives vanish andonly the last two terms remain finite. Further, the free-energyfunctional, f, becomes equal to the entropy density, g, andμ =±

∂∂ ±

gcfrom eqs 16 and 17, respectively. On the other hand,

the expression for the pressure in this case can be readilyobtained by using the thermodynamic relation: U/V = TS/V −P + ∑iμici, where U denotes the total internal energy of thesystem, S is the total entropy, and T, P, and V denote thetemperature, pressure, and total volume of the system,respectively. Because the Helmholtz free energy, F, can bewritten as, ) = −U TS, and the free-energy functional, f, isequal to the Helmholtz free energy per unit volume, that is f =F/V, we obtain, −P = ) V/ − μ∑ ci i i = f − μ∑ ci i i = g −∑ =±

∂∂ci i

gci. Therefore, the constant in eq 18 is equal to the

negative of the pressure, P, acting between the two chargedsurfaces, which according to eq 18 is given by

∑

ϕ ϕ ϕ ϕ

ψκ ψ ψ

π

= − ϵ ′ − ϵ ″ + ϵ ′′′ ′

+ +− ′

− − ∂∂

κ

=±

ÄÇÅÅÅÅÅÅÅÅÅÅÅ

ÉÖÑÑÑÑÑÑÑÑÑÑÑlmooonooo

|}ooo~ooo

P xl

l

c k Tl

k T

g cgc

( )2 2

8 e l

ii

i

2 c2

2c

2

0 h Bh

2h

2h

2

hB

h h

(19)

Using the expression for g from eq 4, and subtracting thepressure in the bulk reservoir obtained by setting theelectrostatic potential, the hydration potential, and all higher-order derivatives of the electrostatic and the hydrationpotentials to zero in eq 19, the disjoining pressure, Π = P −P∞, is given by

∑ϕ ϕ ϕ ϕ

ψκ ψ ψ

π

Π = − ϵ ′ − ϵ ″ + ϵ ′′′ ′ + [ − ]

+ +− ′

κ

=±ÄÇÅÅÅÅÅÅÅÅÅÅÅ

ÉÖÑÑÑÑÑÑÑÑÑÑÑ

ll k T c c

k Tc k Tl

2 2

8 e

ii i

l

2 c2

2c

2B 0

B 0 h Bh

2h

2h

2

hh h (20)

It is noteworthy that the mathematical framework presentedhere to derive the disjoining pressure, as expressed in eq 20, isvery general. In fact, in principle, this methodology can be usedto derive a closed-form analytical expression of the disjoiningpressure for any system given an expression for the free-energyfunctional, f, of the system. Indeed, a similar methodology wasused previously by Misra et al.9 to derive an expression of thedisjoining pressure for a different system where solventpolarization effects were included in the PB model.Next, we analyze in more detail each of the terms

contributing to the disjoining pressure predicted in eq 20.Specifically, the 1st term arises from the self-energy of the

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electrostatic field in the EDL, the 2nd and 3rd terms arise fromion−ion correlations, the 4th term is a contribution due to thetranslational entropy of the ions, and the 5th and 6th termsarise from the incorporation of hydration interactions into theEDL model. If hydration interactions are neglected, that is, ψh= 0 and ψh′ = 0, the 1st four terms in eq 20 yield the expressionof the disjoining pressure in the context of the BSK model,given by

∑ϕ ϕ ϕ ϕΠ = − ϵ ′ − ϵ ″ + ϵ ′′′ ′

+ [ − ]=±

ll

k T c c2 2

ii i

BSK2 c

22

c2

B 0(21)

Furthermore, if ion−ion correlations are also neglected, thatis if we set lc = 0 in eq 21, we recover the exact expression ofthe disjoining pressure for the PB model given by (see refs 11and 67)

∑ϕΠ = − ϵ ′ + [ − ]=±

k T c c2 i

i iPB2

B 0(22)

The disjoining pressure in eq 20 is uniform throughout theregion confined by the two charged surfaces. However, one canobtain additional insights about the nature of the termscontributing to the disjoining pressure by evaluating eq 20 atthe mid-plane between the two charged surfaces. For example,if the two surfaces carry charge of the same sign and magnitude(similarly charged), the 1st derivative of both the electrostaticand the hydration potentials are zero at the mid-plane betweenthe two charged surfaces due to symmetry conditions. In thiscase, the predicted disjoining pressure is given by

∑ϕ

ψκ ψ

π

Π| = − ϵ ″ | + [ | − ]

+ | +| κ

=±

−

lk T c c

k Tc k Tl

2

8e

ii i

l

centerc

22

center B center 0

B 0 h center Bh

2h

2center

h

h h

(23)

Equation 23 clearly shows that the contribution from thewater-mediated hydration interactions between the cations(last two terms in eq 23) to the disjoining pressure is alwaysrepulsive. The 2nd term in eq 23, which is the entropiccontribution to the disjoining pressure, is also always repulsive.Note that the 2nd term in eq 23 corresponds to the disjoiningpressure according to the PB model. This is the reason why thePB model always predicts a repulsive disjoining pressurebetween similarly charged surfaces, irrespective of the surfacecharge density, the salt concentration, and the valency of thecounterions. In contrast, the 1st term in eq 23, which arisesfrom the ion−ion correlations described by the BSK model, isalways negative. Therefore, eq 23 predicts that there can be anattractive EDL contribution to the disjoining pressure evenbetween similarly charged surfaces, as long as the attractivecontribution due to ion−ion correlations dominates over thecontributions due to the repulsive entropic and hydrationinteractions.In addition to electrostatic interactions, van der Waals

interactions between the two planar surfaces can contribute tothe disjoining pressure, although their range is significantlyshort-ranged. The van der Waals contribution to the disjoiningpressure can simply be added to eq 23, and is given by34

π= −

+P

Ad l6 ( 2 )vdW

h

h3

(24)

where Ah is the Hamaker constant and determines themagnitude of the van der Waals interaction between the twoplanar surfaces. It is noteworthy that the attractive disjoiningpressure associated with the van der Waals interactions canalso result in like-charge attraction, especially if the chargedsurface is neutralized by the multivalent counterions. However,the magnitude of the attraction resulting from the van derWaals interactions is highly system-specific. Indeed, theattractive contribution to the disjoining pressure resultingfrom the van der Waals interactions will depend on severalfactors: (i) the nature of the counterion neutralizing thecharged surface, (ii) the thickness of the EDL of counterionsneutralizing the charged surface, and (iii) the magnitude of theHamaker constant of the interacting charged surfaces in air. Allthese effects can be captured in our model by defining aneffective Hamaker constant to model the van der Waalsinteraction between charged surfaces in the presence of aconfined electrolyte. Note that unless specifically stated, mostof the results in this article will not include the contribution ineq 24 because our study is mainly focused on studying theEDL contribution to the disjoining pressure. However, we willinclude the van der Waals contribution to the disjoiningpressure when comparing the theoretically predicted disjoiningpressure to the experimentally measured one.Finally, we also include the contribution due to the

phenomenon of surface CR into our EDL model. Althoughthere are several theoretical approaches to model surfaceCR,68,69 we have adopted the formulation by Behrens andGrier,66 which is useful to model surface CR in cases where thesurface charge is related to the dissociation of silanol groupsfrom the surface. Note that in the case of CSH layers, thesurface dissociation of silanol groups is primarily responsiblefor the negative surface charge on the CSH layers, as reportedin the study by Plassard et al.70 To model the variation of thesurface charge density of CSH layers as a function of thesolution pH,70 we consider reactions of hydroxide ions withsilica sites, using the formulation of Behrens and Grier.66 Inbasic solutions involving hydroxide ions, the surface reactionproceeds as follows

F+ +− −SiOH OH SiO H O2 (25)

The surface is composed of negative (dissociated) andneutral (undissociated) silica sites determined by the surfacereaction equilibrium of the hydroxide ions interacting with theCSH layer. A total site balance on the Si atoms result in thefollowing equation

Γ = Γ + Γ−SiO SiOH (26)

where Γ is the total surface concentration of Si atoms, ΓSiO−

and ΓSiOH are the surface concentrations of the negative andthe neutral silica sites, respectively. Note that the surfacecharge density on the CSH layer can be expressed as theproduct of the number of negative silica sites times theircharge, that is, qs = −eΓSiO

−. Similar to the study by Behrensand Grier,66 we assume that the distribution of the hydroxideions near the CSH layer satisfies the Boltzmann distributiongiven by, cOH−

,s = cOH−,B exp(ϕs), where ϕs is the dimensionless

electrostatic potential at the CSH layer, and cOH−,B and cOH−

,sare the concentrations of hydroxide ions in the bulk and near

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the CSH layer, respectively. Furthermore, based on the surfacechemical reaction in eq 25, the following relation is obtained

= = ΓΓ

− −

−K

c10 K

bp SiO

OH ,s SiOH

b

(27)

where Kb is the base dissociation equilibrium constant, and pKbis the logarithmic dissociation constant characteristic of theCSH−water interface. Because the Stern layer is alreadymodeled self-consistently through the incorporation of water-mediated hydration interactions, the surface potential, ϕs canbe set equal to the potential at the surface located at (x = ±d/2± lh), without the need to account for the Stern capacitance, asdone in ref 66. Using the relation, qs = −eΓSiO

−, in conjunctionwith eqs 26 and 27, the expression for the surface chargedensity at the CSH layer is given by

ϕϕ

=− Γ

+ −

−−

−q

e cc

10 exp( )1 10 exp( )

K

Ks

pOH ,B s

pOH ,B s

b

b (28)

Note that Γ and pKb are parameters characteristic of theCSH−water interface and can be obtained through fitting toexperimental data. Because the correlation length is itself afunction of the surface charge density through its dependenceon lGC (see eqs 12 and 13), δc must be obtained in an iterativemanner. Specifically, the dimensionless electrostatic potential(see eq 11) is solved using an initial guess for qs, andsubsequently, the deduced value of ϕs is used to adjust qs forsubsequent iterations using eq 28. Finally, the converged valueof qs is used to obtain the dimensionless electrostatic potential(see eq 11), and the disjoining pressure is obtained using eq 23for various separation distances between the two similarlycharged surfaces.

■ RESULTSSeveral key predictions of our theory are presented below.First, we consider the role of ion−ion correlations on thedisjoining pressure acting between two similarly chargedsurfaces, including studying the dependence on the valencyof the counterions and coions, the bulk salt concentration, andthe surface charge density of the charged surfaces. Second, westudy the effect of incorporating hydration interactions andsurface CR on the disjoining pressure into our EDL model.Finally, after including the contribution of the van der Waalsinteractions to the disjoining pressure, we compare thepredictions of the disjoining pressure made using our completemodel with the experimental data of Plassard et al.34

Ion−Ion Correlations Cause Like-Charge Attraction.To isolate and properly quantify the effect of ion−ioncorrelations on the disjoining pressure, as described by theBSK theory, we initially neglect hydration interactions, surfaceCR, and van der Waals interactions in our EDL model. This isdone by setting ψh = 0 in eq 20 (thereby neglecting hydrationinteractions), assuming that qs is an independent parameterirrespective of the bulk salt concentration (thereby neglectingsurface CR), and setting Ah = 0 in eq 24 (thereby neglectingvan der Waals contributions to the disjoining pressure). InFigure 2a, we utilize eq 21 to plot the disjoining pressurepredicted using the BSK model as a function of the separationdistance between the two charged surfaces for the case of a 2:1electrolyte solution. Note that throughout the article, the waterdielectric permittivity, ϵ, is expressed as ϵ = ϵrϵ0, where ϵ0 isthe permittivity of vacuum (8.85 × 10−12 F m−1). In Figure 2a,

the disjoining pressures predicted using the BSK model (see eq21) and the PB model (see eq 22) are compared. Note that thegoverning equation for the dimensionless electrostaticpotential in the PB model can be obtained simply by settingδc = 0 in eq 11, including using boundary conditions (i) and(ii) in the set of four boundary conditions used to solve eq 11in the BSK model. As expected, the PB model always predicts arepulsive (positive) disjoining pressure irrespective of theseparation distance between the two charged surfaces.Furthermore, the repulsive disjoining pressure predicted bythe PB model decreases monotonically with an increase in theseparation distance between the two charged surfaces.However, in stark contrast to the result predicted by the PBmodel, it is possible to obtain an attractive (negative)disjoining pressure using the BSK model. For example, asshown in Figure 2a, the disjoining pressure curve predicted bythe BSK model displays an attractive well at a separationdistance of ∼0.8 nm between the two charged surfaces. Forseparation distances greater than 1 nm, where the disjoiningpressure predicted by the BSK model is negative andprogressively goes to zero at larger separation distances, theattractive pressure resulting from ion−ion correlationsdominates over the pressure arising from the entropicrepulsion.To better understand the above-mentioned competition, in

the inset of Figure 2a, we plot the contributions to the

Figure 2. Disjoining pressures operating between two similarlycharged surfaces separated by a 2:1 electrolyte solution predictedusing the BSK (eq 21) and the PB (eq 22) models for qs = −0.1 C/m2, c0 = 0.1 M, and ϵr = 80. (a) Variation of the disjoining pressure asa function of the separation distance between the two chargedsurfaces. The inset shows the contribution of the different terms in eq21 based on the BSK model. (b,c) Comparison of the charge densityand the dimensionless electrostatic potential profiles, respectively,predicted by the BSK model (eq 11) and the PB model (eq 11 with δcset to zero) when the separation distance between the two chargedsurfaces is 1 nm, and where x is the spatial coordinate perpendicularto the plane of the charged surface.

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disjoining pressure due to each of the four terms appearing ineq 21, all evaluated at the mid-plane between the two chargedsurfaces. Because we are considering similarly charged surfaces,the 1st derivative of the electrostatic potential is always zero atthe mid-plane between the two charged surfaces. Therefore,the 1st and the 3rd terms in eq 21 do not contribute to thepredicted disjoining pressure. Moreover, as shown in the insetof Figure 2a, the pressure resulting from ion−ion correlations(2nd term in eq 21) is always attractive (negative), whereas theentropic pressure (4th term in eq 21) is always repulsive(positive). For separation distances much smaller than 1 nm,the counterions get compressed between the two chargedsurfaces, thereby greatly enhancing the entropic contributionand making the overall disjoining pressure repulsive (positive).On the other hand, for separation distances larger than 1 nm,the disjoining pressure contribution from ion−ion correlationsalways dominates over the repulsive entropic contribution,which results in an overall attractive disjoining pressure asshown in Figure 2a. Interestingly, as Figure 2b shows, at aseparation distance of 1 nm between the two charged surfaces,the mean charge density (ρ = zec+ − ec−) and thedimensionless electrostatic potential (ϕ) profiles obtainedusing the PB and the BSK models are very similar. Thesimilarity in the two profiles suggests that the form of theHelmholtz free energy, F, and a self-consistent expression ofthe disjoining pressure, Π, derived from F, are essential inorder to obtain a negative disjoining pressure in the context ofthe BSK model. For example, even if one utilizes eq 11 to solvefor the electrostatic potential using the BSK model, use of eq22 which corresponds to the disjoining pressure from the PBmodel would still result in a repulsive disjoining pressure for allseparation distances between the similarly charged surfaces.Therefore, our study highlights the necessity to self-consistently incorporate terms in the predicted disjoiningpressure (see eq 21) arising from ion−ion correlations toexplain the phenomenon of like-charge attraction betweensimilarly charged surfaces.It is noteworthy that in the previous work, the BSK model

was shown to describe overscreening, or charge-inversion, inthe case of an EDL of solvent-free ionic liquids placed near acharged electrode.45 In this case, strong ion−ion correlationsgive rise to an oscillatory profile for the mean charge density, ρ,where the 1st layer of ions in the EDL overscreens the chargeon the electrode, and subsequently, the net charge of theelectrode together with the 1st layer of ions is progressivelyneutralized by additional layers of ions. In this study, we findthat although both the phenomena of charge-inversion andlike-charge attraction can result from strong ion−ioncorrelations, there is no direct correlation between them. Inother words, it is not necessary for the phenomenon of like-charge attraction to occur concurrently with the phenomenonof charge-inversion.Next, we study how the counterion valency affects the

disjoining pressure. As the counterion valency increases, theBSK theory predicts a stronger attractive contribution to thedisjoining pressure, as shown in Figure 3a. The qualitativeshape of the pressure profile also changes significantly as thecounterion valency increases from 1 to 3, with a deep attractivewell developing in the case of a 3:1 electrolyte. Moreover, theposition of the minimum in the disjoining pressure shiftstoward the left with an increase in the counterion valency,indicating an enhancement in the length scale over which anattractive disjoining pressure is observed due to ion−ion

correlations. The predicted attraction is consistent withexperimental observations of like-charge attraction andcolloidal coagulation in the presence of multivalent saltions.26−30 Physically, increasing the counterion valency, z,enhances ion−ion correlations, which is captured in our EDLmodel by a nonlinear dependence of the dimensionlesscorrelation length on the counterion valency (see eq 12).Increasing the value of z increases the dimensionlesscorrelation length parameter, which in turn enhances theattractive ion−ion correlation contribution to the disjoiningpressure (see 1st term in eq 23).In addition to studying the variation of the predicted

disjoining pressure with the counterion valency, it is alsointeresting to explore the dependence of the predicteddisjoining pressure on the coion valency, that is, for a 1:zelectrolyte solution. To this end, we utilize the same parametervalues used to generate Figure 3a, except that we change thesign of the surface charge density, qs, on the two surfaces. This,in turn, is equivalent to studying a 1:z electrolyte solutioninstead of the z:1 electrolyte solution considered earlier.Furthermore, note that we also set z = 1 in eqs 12 and 13. InFigure 3b, we show plots of the predicted disjoining pressure

Figure 3. Effect of the ion valency on the predicted disjoiningpressure acting between two similarly charged surfaces as a function ofthe separation distance between the two charged surfaces for z:1 and1:z electrolytes (z = 1, 2, and 3) solutions using the BSK model (eq21). To generate the various plots, we used the following parametervalues: |qs| = 0.1 C/m2, c0 = 0.1 M, and ϵr = 80. (a) Predicteddisjoining pressure of a z:1 electrolyte solution, corresponding to anegative charge on the two surfaces, qs = −0.1 C/m2. (b) Predicteddisjoining pressure of a 1:z electrolyte solution, corresponding to apositive surface charge on the two surfaces, qs = +0.1 C/m2.

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for a 1:z electrolyte, where the sign of qs is positive.Interestingly, the predicted disjoining pressure is not sensitiveto the coion valency, where the plots of the disjoining pressurepredicted using different coion valencies, z = 1, 2, and 3, are allvery similar to the predicted disjoining pressure plot for a 1:1electrolyte solution. This indicates that the phenomenon oflike-charge attraction is more strongly controlled by thevalency of the counterion than by that of the coion. Ourimportant finding that the coion valency plays an insignificantrole in controlling the surface forces between two chargedsurfaces is well supported by the recent experimental data ofUzelac et al.71 These authors measured surface forces betweencharged silica particles mediated by multivalent coions andreported that the valency of the counterions, and not thevalency of the coions, controls the surface forces operatingbetween the charged silica particles that they considered.Next, we explore the dependence of the predicted disjoining

pressure on the surface charge density and the saltconcentration of the bulk reservoir. As shown in Figure 4a,increasing the surface charge density of the two chargedsurfaces for a 2:1 electrolyte solution results in an enhance-ment of the attractive well for the disjoining pressure as the

magnitude of qs is increased from −0.01 to −0.2 C/m2. In thecase of an electrolyte confined between two charged surfaces,increasing the surface charge density of the two surfaces resultsin an enhancement of the overlap of the EDLs produced byeach surface because more counterions over an extendedregion are required to screen the increased charges on thesurfaces. This results in an increase in the curvature, or thesecond derivative, of the electrostatic potential. As a result, theattractive ion−ion correlation contribution to the predicteddisjoining pressure (2nd term in eq 21), which scales as thesquare of the 2nd derivative of the electrostatic potential,increases similarly. Consequently, increasing the surface chargedensity in the context of the BSK model results in apronounced enhancement of the attractive well for thepredicted disjoining pressure.Our findings clearly show that the disjoining pressure

predicted by the BSK model strongly disagrees with thatpredicted by the DLVO theory, which uses the PB model topredict the EDL contribution to the disjoining pressure actingbetween the two charged surfaces. Notably, increasing thesurface charge density in the PB model enhances the overlap ofthe EDLs originating from the two similarly charged surfaces,which in turn results in an increased entropic repulsion andconsequently, in an enhancement of the repulsive disjoiningpressure acting between the two similarly charged surfaces.Although it is certainly true that like in the PB model, themagnitude of the repulsive entropic term (4th term in eq 21)also increases with an increase in the surface charge density inthe BSK model, the attractive ion−ion correlation termcompletely dominates over the repulsive entropic term, leadingto a deepening of the attractive well for the predicteddisjoining pressure in the case of the BSK model. Our resultis in good agreement with that obtained by Plassard et al.34

who experimentally measured the disjoining pressure operatingbetween charged CSH layers in a calcium hydroxide saltsolution. Indeed, Plassard et al.34 reported an enhancement ofthe attractive well for the experimental disjoining pressureupon increasing the surface charge density on the CSH layers.The fact that the DLVO theory fails to predict theexperimentally observed trend, even at a qualitative level,demonstrates the importance and need for the availability of atheory such as the one presented here, which is capable ofaccurately modeling ion−ion correlations.As shown in Figure 4b, the predicted disjoining pressure also

depends strongly on the salt concentration of the bulkreservoir. Indeed, as the salt concentration increases, thedepth of the attractive well increases, reflecting stronger ion−ion correlations. In addition, the position of the disjoiningpressure minimum shifts toward the left, reflecting a decreasein the overlap of the EDLs originating from the two similarlycharged surfaces. We have explored a wide parameter space forthe dependence of the predicted disjoining pressure on saltconcentration and surface charge density, as shown in the twocontour plots in the Supporting Information document (seeFigure S1).We note that like-charge attraction can also be obtained in

the limit of a counterion-only system (also referred to as theone-component plasma limit),29 where the surface charge isscreened by a fixed number of counterions, with no coionspresent in the system. In this context, the BSK theory with theappropriate correlation length scale and boundary conditionshas been shown to exactly match the strong and weak couplinglimits for the counterion-only system.59 Furthermore, the BSK

Figure 4. Effect of the surface charge density and the saltconcentration for a 2:1 electrolyte solution on the predicted disjoiningpressure as a function of the separation distance between the twocharged surfaces. (a) Predicted disjoining pressures for three surfacecharge densities, where the following parameter values were used togenerate the three plots shown: c0 = 0.1 M and ϵr = 80. (b) Predicteddisjoining pressures for three salt concentrations in the bulk reservoir,where the following parameter values were used to generate the threeplots shown: qs = −0.1 C/m2 and ϵr = 80.

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theory can also accurately reproduce the intermediate couplinglimit, including describing correlation-induced like-chargeattraction in the case of counterion-only systems.59

Another interesting question is whether the BSK theory canpredict like-charge attraction in systems with monovalentcounterions, as was shown recently in simulations of chargednanoparticles.72 In Figure 5, we show the predicted disjoining

pressure curves for a 1:1 electrolyte solution with varyingsurface charge density and varying bulk salt concentration.Figure 5a clearly shows that, as the surface charge densityincreases from −0.1 to −0.4 C/m2, the disjoining pressurebetween two similarly charged surfaces predicted by the BSKtheory can become strongly attractive (negative), even in thecase of monovalent counterions.Furthermore, increasing the bulk salt concentration can

result in the appearance of an attractive well in the disjoiningpressure predicted by the BSK theory, similar to the onespredicted earlier in the case of a 2:1 electrolyte solution.However, very high bulk salt concentrations or moderate tohigh surface charge densities are needed to predict attractivedisjoining pressures in the case of a 1:1 electrolyte solution.Our BSK model predictions for the disjoining pressure suggestthat attractive interactions between two like-charged surfacesdo not result exclusively from the presence of multivalentcounterions. However, the presence of multivalent counterionscan significantly augment the attractive disjoining pressure

arising from ion−ion correlations, as shown earlier in Figure3a.

Contributions to the Disjoining Pressure fromHydration Interactions and Surface CR. Along with thecontributions from ion−ion correlations, it is also interestingto explore the role of short-range water-mediated hydrationinteractions between the cations on the predicted disjoiningpressure. As expected, water-mediated hydration interactionsprevent two cations from approaching each other too closely,where the distance of closest approach between any twointeracting cations is controlled by the parameter, lh, which isthe effective hydration size of the cation. In Figure 6, we plot

Figure 5. The disjoining pressure for a 1:1 electrolyte solutionpredicted by the BSK theory can be attractive depending on (a)surface charge density or (b) salt concentration. For (a), the bulk saltconcentration is fixed at c0 = 0.1 M. For (b), the surface chargedensity is fixed at qs = −0.3 C/m2.

Figure 6. (a) Variation of the disjoining pressure with the separationdistance between two similarly charged surfaces for a 2:1 electrolytesolution, predicted by the BSK and the PB models with inclusion/exclusion of the hydration potential. Note that the parameters used togenerate these results include: qs = −0.1 C/m2, c0 = 0.1 M, ϵr = 80, σh= 5/nm2 (from ref 63), κh

−1 = 0.3 nm (from ref 63), and lh = 0.2 nm.The parameters for the hydration potential (σh, κh

−1, and lh) can befurther fine-tuned to reproduce the experimental data. (b)Comparison of the counterion profiles predicted using the BSKmodel without hydration and the BSK model with hydration, for d = 1nm, as a function of x, where x is the spatial coordinate perpendicularto the plane of the charged surface. (c) Same as (b), but for aseparation distance of 2 nm between the two charged surfaces.

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the predicted disjoining pressure operating between twosimilarly charged surfaces for a 2:1 electrolyte solution,where we incorporated hydration interactions into both thePB model and the BSK model. As discussed earlier,incorporation of hydration interactions allows us to modelthe Stern layer self-consistently in both the BSK and PBmodels. However, for an appropriate comparison between theBSK model without hydration and the BSK model withhydration, we need to shift the plot obtained in the case of theBSK model with hydration toward the left by 2lh to beconsistent with the distance of closest approach in the twomodels. Similarly, the plot for the PB model with hydration isalso shifted leftward by 2lh.Figure 6a shows that incorporation of the hydration

potential into both the PB and the BSK models results in astrong repulsive (positive) contribution to the predicteddisjoining pressure. In the case of the PB model, addition ofthe hydration potential makes the predicted repulsivedisjoining pressure even more repulsive. In the case of theBSK model, addition of the hydration potential results in adisappearance of the attractive (negative) well, with a veryweak attractive (negative) disjoining pressure visible at ∼3 nmseparation distance between the two charged surfaces. Notethat the plots shown in Figure 6a were obtained using qs =−0.1 C/m2, corresponding to low to moderate surface chargedensities, typically encountered on the two charged surfaces.However, as shown later in this section, surface chargedensities reported in some experimental studies can exceed thisvalue. In such cases, we will be able to recover the attractivewell in the disjoining pressure predicted using the BSK model,where hydration interactions are included. Note that weselected qs = −0.1 C/m2 to carry out these calculations todemonstrate the key role of the hydration interactions inmodulating the predicted disjoining pressure, especially at lowto moderate surface charge densities on the two chargedsurfaces. Moreover, if the EDL is decomposed into Stern anddiffuse layer regions, with the diffuse layer region modeledusing an EDL theory such as the PB model, it follows that thecontribution of the Stern layer to the predicted disjoiningpressure cannot be modeled self-consistently. In this study, weuse the hydration potential to model the Stern layer and self-consistently incorporate the repulsive contribution from thehydration potential to the predicted disjoining pressure (see3rd and 4th terms in eq 23), which are currently neglected inthe PB model. Furthermore, the predicted repulsive disjoiningpressures resulting from the hydration potential can alsoqualitatively explain the repulsive pressures reported in theexperimental data of Israelachvilli and Pashley64 and ofLeNeveu and Rand.65

To further elucidate the role of hydration interactions inmodulating the predicted disjoining pressure, in Figure 6b,c,we compare the counterion profiles predicted using the BSKmodel without hydration and the BSK model with hydration,when the two charged surfaces are separated by a distance of 1and 2 nm, respectively. Note that the distances of minimumapproach for the BSK model and BSK model with hydrationare d and d + 2lh, respectively. This is the reason why thecounterion profiles in the context of the BSK model withhydration extend beyond d in Figure 6b,c. An analysis of thecounterion profiles in Figure 6b,c shows that in the case of theBSK model without hydration, the predicted counterionconcentrations near the two charged surfaces (at x = ±0.5nm and x = ±1 nm, where x is the spatial coordinate

perpendicular to the charged surfaces) are unbounded andseveral folds higher than those predicted in the case of the BSKmodel with hydration. As discussed earlier, hydrationinteractions limit the counterion concentrations near the twocharged surfaces to replicate a Stern layer region which isdevoid of counterions. Consequently, to neutralize the chargeon the two surfaces, the concentration of counterions at themid-plane between the two charged surfaces (located at x = 0in Figure 6b,c) is higher in the BSK model with hydration thanin the BSK model without hydration. This results in anenhancement of the repulsive entropic contribution to thepredicted disjoining pressure (2nd term in eq 23) in the BSKmodel with hydration, which together with the new repulsiveterms arising from the incorporation of the hydrationinteractions (3rd and 4th terms in eq 23) contributesignificantly toward making the overall predicted disjoiningpressure repulsive, especially at smaller separation distances. AsFigure 6b,c shows, increasing the separation distance betweenthe two charged surfaces from 1 to 2 nm weakens thecontribution from the hydration potential, such that thecounterion concentration near the midplane (at x = 0)between the two charged surfaces, predicted using the BSKmodel with hydration (dotted blue curve), approaches thatpredicted using the BSK model without hydration (solid bluecurve).Finally, to complete our analysis, we consider the effect of

surface CR on the predicted disjoining pressure and thepredicted surface charge density, where qs is no longer anindependent parameter but instead, depends on the bulk saltconcentration. As a representative example, we consider twosimilarly charged CSH layers immersed in a calcium hydroxidesalt solution. We consider this system because, in the nextsection, we will compare the disjoining pressure predicted byour theory for this system with the experimental data ofPlassard et al.34 As discussed in the Model and Methodssection, the value of qs in eq 28 depends on the bulkconcentration of OH− ions as well as on the dimensionlesssurface potential. As a result, qs needs to be obtained in aniterative manner, such that eqs 11 and 7, together with theboundary conditions, and eq 28 are simultaneously satisfied.In Figure 7a, we plot the disjoining pressure as a function of

the surface separation distance, d, predicted by first includinghydration interactions into the PB and BSK models and,subsequently, by including surface CR in the BSK and PBmodels. The disjoining pressure versus the separation distanceprofiles were predicted for a bulk salt concentration of 19.1mM, a typical concentration for which the experimental data ofPlassard et al.34 is available. To predict the disjoining pressurewhen CR is neglected (e.g., BSK model and BSK model withhydration), we need to assign a value to the surface chargedensity, qs, which serves as an independent parameter. To thisend, we choose qs = −0.63 C/m2, obtained by solving eq 28using c0 = cOH−

,B = 19.1 mM. Figure 7a shows that choosing avalue of qs which is much larger than the qs value used topredict the results in Figure 6a, we predict large attractive(negative) disjoining pressures using the BSK model. Includinghydration interactions in the BSK model results in a largerepulsive (positive) pressure contribution to the disjoiningpressure. However, for qs = −0.63 C/m2, the predictedattractive (negative) disjoining pressure contribution resultingfrom ion−ion correlations dominates over the repulsive(positive) contributions resulting from the entropy of theions as well as from the hydration interactions (2nd, 3rd, and

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4th terms in eq 23). This results in an attractive (negative) wellat ∼1 nm separation distance between the two chargedsurfaces. For comparison, we also plotted the disjoiningpressure profiles predicted using the PB model. It is clear that ifion−ion correlations are neglected, it will be impossible topredict an attractive (negative) well for the disjoining pressureusing the PB model.

Interestingly, after inclusion of the hydration interactions inthe PB and BSK models, further inclusion of CR in the twomodels does not affect the predicted disjoining pressureprofiles at all. This indicates that for both the BSK and PBmodels, the surface charge density is almost insensitive to theseparation distance between the two charged surfaces. Indeed,as shown in Figure 7b, the surface charge density varies withthe separation distance between the two charged surfaces onlyat very small separation distances and remains constant beyonda separation distance of 1 nm. In addition, there is not muchdifference between the predictions made using the PB and theBSK models, indicating that the dependence of the surfacecharge density on the separation distance between the twocharged surfaces is insensitive to the modeling of ion−ioncorrelations. In fact, the surface charge density shows a verystrong dependence on the pH of the solution. Recall that theconcentration of OH− ions present in the calcium hydroxidesalt solution in the bulk reservoir can be related to the pH ofthe solution using the relation, cOH−

,B = 10pH‑14. Therefore,increasing the pH of the solution results in a larger dissociationof silanol groups at the CSH layer (see eq 25), which in turnresults in an increase in the surface charge density at the CSHlayer. Once again, the striking similarity of the variation of qswith the pH of the solution, obtained using the PB and BSKmodels, indicates that the variation of the surface chargedensity with the pH of the solution is insensitive to themodeling of ion−ion correlations. Finally, we also note that thevariation of qs with the solution pH, as predicted in Figure 7c,is in good qualitative agreement with that reportedexperimentally by Plassard et al.34 This finding is reassuringin terms of the values of the surface CR parameters, Γ and pKb,used to model the variation of qs with the solution pH inFigure 7c. For example, Plassard et al. have reported thatincreasing the solution pH from 10 to 12.5 results in amonotonic enhancement in the surface charge density, qs, atthe CSH surface from −0.08 to −0.69 C/m2. Similarly, inFigure 7c, for a similar variation of the solution pH, we predicta monotonic enhancement of qs from −0.02 to −0.58 C/m2,for both the BSK and the PB models.

Comparison of the Disjoining Pressures Predictedfrom Our Complete Theory with the ExperimentalData. Plassard et al.34 used atomic force microscopy (AFM) tostudy the surface forces responsible for the strong cohesivestrength of cement pastes. These authors observed strongattractive (negative) disjoining pressures operating betweenthe CSH layers in the presence of a Ca(OH)2 electrolytesolution. They attributed the observed behavior to the strongion−ion correlations between the divalent calcium ions.Plassard et al. noted that because the DLVO theory neglectsion−ion correlations, the disjoining pressure predicted by theDLVO theory is not consistent with their experimental findingseven qualitatively. In Figure 8, we compare the disjoiningpressure versus the surface separation distance profilespredicted by our complete theory (eqs 23 plus 24) to thosemeasured by Plassard et al. (see Figure 5 in ref 34) for fiveCa(OH)2 salt concentrations. Plassard et al. measured theforces operating between the two CSH layers, and weconverted their experimental surface force data into disjoiningpressure data by dividing the reported surface forces by thesurface area of the CSH layers, estimated to be 64 nm2 in theirstudy.34

Note that in the study by Plassard et al., one of the CSHlayers was coated over an AFM tip, whereas the other CSH

Figure 7. (a) Disjoining pressure vs the surface separation distanceprofiles predicted by the BSK model without hydration, the BSKmodel with hydration, and the BSK model with hydration and withCR are compared with those predicted by the PB model withouthydration, the PB model with hydration, and the PB model withhydration and with CR. The parameters used to generate the resultsshown here are: c0 = 19.1 mM, ϵr = 80, κh

−1 = 0.3 nm, σh = 5/nm2, lh =0.2 nm, Γ = 1 × 1018 m−2, and pKb = 4. Note that when CR isneglected, we used qs = −0.63 C/m2. (b) Surface charge density vs thesurface separation distance profiles predicted using the BSK modelwith hydration and CR (blue dotted line) and the PB model withhydration and CR (black dotted line). Note that the parameters: c0, ϵr,κh

−1, σh, lh, Γ, and pKb are the same as those used in (a), and that dueto the incorporation of CR, qs varies with the separation distancebetween the two charged surfaces. (c) Surface charge density vs thesolution pH profiles predicted using the BSK model with hydrationand CR (blue dotted line) and the PB model with hydration and CR(black dotted line) for a separation distance of 4 nm between the twocharged surfaces. Note that both surfaces are assumed to be negativelycharged and to possess identical surface charge densities.Furthermore, the parameters used to generate the surface chargedensity profiles are the same as those used in (b).

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layer was coated over a calcite crystal. A key assumption madeby Plassard et al. (also implemented here) is that theinteraction of the two CSH layers can be approximated asthe interaction between two flat surfaces. Note that to solve forthe disjoining pressure, we used the following parameters: lh =0.1 nm, κh−1 = 0.3 nm, σh = 5/nm2, Γ = 1 × 1018 m−2, pKb = 4,and Ah = 14 × 10−20 J. The following parameters were used inour predictions: (i) the Hamaker constant, Ah, used in thisstudy is the same as that determined by Plassard et al. bymeasuring the surface forces operating between two CSHlayers in air, (ii) the values of the hydration parameters, σh andκh

−1, used here are identical to those reported by Brown etal.,63 (iii) the value of the surface site density, Γ, is identical tothat reported by Behrens and Grier,66 (iv) the pKb is the sameas that used in Figure 7c, where we showed that thecombination of the surface CR parameters, Γ and pKb, isable to predict the variation of qs with the solution pH inreasonably good agreement with the experimental data ofPlassard et al., and (v) the lh parameter is obtained by fitting tothe experimental disjoining pressure data. We note here thatthe choice of the parameters for the hydration potential, σh,κh

−1, and lh, used to predict the disjoining pressure curvesreported in Figure 8, is not unique. Although the values of σhand κh−1 are quite reasonable as noted in the study by Brown etal.,63 the value of lh (0.1 nm) used here is smaller than thehydration shell diameter of 0.48 nm reported for the Ca2+ ion(see the tabulated hydration shell radius in Table V of ref 73).We note here that, in principle, one can choose a larger valueof lh to be closer to the corresponding experimental value ofthe hydration shell diameter and still obtain similar disjoiningpressure curves reported here by also concurrently changingthe values of σh and κh

−1. However, we found that followingsuch an approach does not change any of the overall trends ofthe disjoining pressure curves reported here.As shown in Figure 8, the disjoining pressures predicted by

our complete theory are in good qualitative agreement with theexperimental data of Plassard et al. for the five Ca(OH)2concentrations considered. For the 0.2 mM Ca(OH)2 saltconcentration, the predicted disjoining pressure is purelyrepulsive (positive) for all values of the surface separationdistance (see the solid pink line) which agrees very well with

the corresponding experimental data (see pink circles).However, as the Ca(OH)2 salt concentration increases, theexperimental disjoining pressures become increasingly attrac-tive (negative; see the red, green, black, and blue circles inFigure 8), a trend which is captured reasonably well by ourtheory (see the red, green, black, and blue solid lines in Figure8). Note that the Hamaker constant for CSH reported byPlassard et al. is large when compared to the value for silica (Ah= 2 × 10−21 J) reported in the literature.74,75 However, in theSupporting Information document we show that, even if theHamaker constant is reduced to Ah = 2 × 10−21 J, our theorycan still capture the experimental trends with minormodification in the value of σh, as shown in Figure S2.Finally, it is noteworthy that although our complete theory

for the disjoining pressure (eqs 23 and 24) contains severalparameters: σh, κh−1, lh, Γ, and pKb, these parameters areintroduced in order to incorporate new physics into the EDLmodel, including hydration interactions and surface CR.However, most importantly, no fitting parameter is introducedin our theory to model ion−ion correlations because thedimensionless correlation length is determined directly basedon eq 12. As Figure 8 clearly shows, accurate modeling of ion−ion correlations is essential to predict the attractive electro-static contribution to disjoining pressures between similarlycharged surfaces. This is, in fact, the main reason why theDLVO theory, which neglects ion−ion correlations, fails tomodel surface forces/disjoining pressures even qualitatively,especially in solutions containing multivalent salt ions.

■ CONCLUSIONSIn this article, we formulated a general theory of the disjoiningpressure operating between two charged surfaces in amultivalent electrolyte solution. We used the BSK frameworkto model ion−ion correlations and derived a closed-formexpression for the contribution resulting from ion−ioncorrelations to the disjoining pressure. In stark contrast tothe predictions of the PB mean-field theory, which alwayspredicts a repulsive (positive) disjoining pressure operatingbetween two similarly charged surfaces, we showed here thatthe attractive (negative) disjoining pressure resulting fromion−ion correlations can dominate over an entropic repulsion,and thereby cause the overall disjoining pressure to beattractive (negative). Our theory predicts that both themagnitude as well as the sign of the disjoining pressure arestrong functions of the counterion valency. Indeed, increasingthe valency of the counterion significantly enhances theattractive pressure resulting from ion−ion correlations, therebypromoting attractions between similarly charged surfaces inaqueous solutions containing divalent and trivalent counter-ions. On the other hand, we found that the dependence of thedisjoining pressure on the valency of the coion is insignificant.We also showed that the disjoining pressure in the case ofmonovalent ions can still be attractive, provided that thecharge densities on the two charged surfaces are sufficientlyhigh to induce significant ion−ion correlations.Another important limitation of the DLVO theory of

colloidal stability, and the PB model on which it is based, isthat it predicts an enhancement of the repulsive disjoiningpressure operating between two similarly charged surfaceswhen the surface charge densities of the two surfaces increase.This is in stark contrast with the available experimental data,34

which conclusively show that the disjoining pressure operatingbetween two similarly charged surfaces becomes attractive

Figure 8. Comparison of the disjoining pressure vs the surfaceseparation distance predicted by our complete theory (eqs 23 and 24,solid lines) with the experimental data of Plassard et al.34 (circles) forthe five Ca(OH)2 salt concentrations shown.

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upon increasing the surface charge densities of the twosurfaces. Our new theory of the disjoining pressure can explainthis experimental finding based on the existence of ion−ioncorrelations, which are neglected in the DLVO theory.In addition to modeling ion−ion correlations, we also

included water-mediated hydration interactions and surfaceCR in our theory to allow a direct comparison of the disjoiningpressure predicted by our theory with the experimentallyavailable disjoining pressure data. To this end, we showed thatmodeling the Stern layer self-consistently through water-mediated hydration interactions results in a significantrepulsive contribution to the disjoining pressure, althoughthe attractive pressure resulting from ion−ion correlations canstill dominate over the repulsive pressures resulting from theentropy of the ions as well as from the hydration interactions.Furthermore, incorporation of surface CR into our modelenabled us to relate the surface charge density at the chargedsurface directly to the salt concentration in the electrolytesolution. Finally, we demonstrated that the disjoining pressureoperating between the CSH layers in a calcium hydroxide saltsolution predicted by our theory is in reasonable qualitativeagreement with the experimental data of Plassard et al.34

Therefore, we believe that the complete theory of surfaceforces presented here shows promise in overcoming the knownlimitations of the DLVO theory, especially for multivalentcounterions.In terms of future work, although the BSK model can

describe electrostatic correlation effects at an interface,currently, it does not capture the Bjerrum pair formation,which can decrease the effective ion concentration in the bulkreservoir.76 Furthermore, the BSK model does not capturelong-range oscillations in the charge density at high electrolyteconcentrations.77 This can be corrected by consideringweighted concentrations in the expression for the entropy ofions (i.e., the g(c+,c−) term in eq 3), as implemented in varioustheoretical approaches, including the classical density func-tional theory.77,78

Finally, at separation distances of ∼1 nm between the twocharged surfaces, in addition to an attractive disjoiningpressure resulting from ion−ion correlations, short-rangedattractive van der Waals interactions as well as repulsivehydration interactions can play important roles. A betterestimation of the parameters used to model the van der Waalsinteractions, the hydration interactions, as well as of the scalingproposed for the correlation length may be carried out througha systematic comparison of the disjoining pressure predicted byour theory with that available experimentally, as well as thatobtained using the Monte Carlo and molecular dynamicssimulations.

■ ASSOCIATED CONTENT

*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.lang-muir.9b01110.

Derivation of the expression for the Helmholtz freeenergy, derivation of eq 15 in the article describing therelation between the electrostatic potential, hydrationpotential, and their higher-order derivatives, contourplots showing the dependence of the disjoining pressureon the salt concentration and surface charge density, and

comparison to experimental data for reduced Hamakerconstant (PDF)

■ AUTHOR INFORMATIONCorresponding Authors*E-mail: dblank@mit.edu (D.B.).*E-mail: bazant@mit.edu (M.Z.B.).ORCIDRahul Prasanna Misra: 0000-0001-5574-2384Daniel Blankschtein: 0000-0002-7836-415XMartin Z. Bazant: 0000-0002-8200-4501Author Contributions§ R.P.M. and J.P.d.S. contributed equally to this work.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSR.P.M., J.P.d.S., D.B., and M.Z.B. acknowledge the financialsupport received as part of the Center for EnhancedNanofluidic Transport (CENT), an Energy Frontier ResearchCenter funded by the U.S. Department of Energy, Office ofScience, Basic Energy Sciences under award # DE-SC0019112,for formulating the theory of surface forces and theaccompanying analysis presented in this paper. J.P.d.S. andM.Z.B. also acknowledge the Amar G. Bose Research Grant forpartial funding on the modeling of ion−ion correlations usingthe BSK model. J.P.d.S. also acknowledges support in the formof a National Science Foundation Graduate Research Fellow-ship under grant no. 1122374.

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