Electrical Double Layers: Effects of Asymmetry in Electrolyte Valenceon Steric Effects, Dielectric Decrement, and Ion−Ion CorrelationsAnkur Gupta and Howard A. Stone*

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, United States

ABSTRACT: We study the effects of asymmetry in electrolytevalence (i.e., non z:z electrolytes) on mean field theory of theelectrical double layer. Specifically, we study the effect of valenceasymmetry on finite ion-size effects, the dielectric decrement, andion−ion correlations. For a model configuration of an electrolytenear a charged surface in equilibrium, we present comprehensiveanalytical and numerical results for the potential distribution,electrode charge density, capacitance, and dimensionless salt uptake.We emphasize that the asymmetry in electrolyte valence significantlyinfluences the diffuse-charge relations, and prior results reported inthe literature are readily extended to non z:z electrolytes. We developscaling relations and invoke physical arguments to examine theimportance of asymmetry in electrolyte valence on the aforementioned effects. We conclude by providing implications of ourfindings on diffuse-charge dynamics and other electrokinetic phenomena.

■ INTRODUCTION

Diffuse charge refers to the distribution of ions in an electrolytesolution adjacent to a charged solid surface. The charge profile iscritical to a variety of applications; in colloid science andmicrofluidic applications, diffuse charge is important in electro-phoresis,1−3 electroosmosis,1−5 and diffusiophoresis,6,7 while inenergy storage devices, the charged layers form the basis ofelectrochemical capacitors8,9 and more recently semisolid flowcapacitors.10 The properties of diffuse charge are typicallydetermined by the widely used Gouy−Chapman (GC) theory,which describes the dependence of surface charge density q andcapacitance on the potential drop ψD across the diffuse-chargedregion.11−16

The classical GC results are based on a mean-fieldapproximation and provide analytical expressions for q(ψD)and ψ( )D . Due to their relatively simple nature, GC resultscontinue to be widely used, even though it is well-recognized thatthey suffer from several limitations.17−23 To improve thepredictions of the GC results, several modifications have beensuggested in the literature while retaining the mean-fieldframework of the classical GC results. In this article, we focuson three such modifications for valence asymmetric (or non z:z)electrolytes: finite ion-size effects, dielectric decrement, and ion−ion correlations, and we indicate below earlier work in each area.We first discuss finite ion-size effects, also commonly known as

steric effects. The classical GC results are obtained by solving thePoisson−Boltzmann equations where ions are treated as pointcharges. Therefore, the classical GC results predict an unphysicaloutcome that ion concentration can increase indefinitely withincrease in |ψD|. This limitation was recognized already 90 yearsago by Stern,17 Bikerman18 and Freise,19 and these authorsaccounted for the finite ion-size effects by assuming a simple

hard-sphere model with a finite diameter of ions. A vast literatureis available in this area, and we only discuss a few reports in detail.For a more in-depth review of the literature on finite ion-sizeeffects, including more sophisticated models for spherical ions,we refer the readers to refs 20, 21, and 24−26. For example,Borukhov et al.27 derived an expression for the concentration ofions with a finite ion size for a valence symmetric (or z:z), as wellas valence asymmetric (or non z:z) electrolyte near a chargedsubstrate. Kilic et al.21,28 and Kornyshev20 derived a modifiedPoisson−Boltzmann equation, and provided explicit expressionsfor q(ψD) and ψ( )D for z:z electrolytes while assuming that thecation and anion diameters are equal. In these articles, theauthors also discussed the implications of including steric effectson the dynamics of diffuse charge. More recently, Han et al.29

derived the results for q(ψD) and ψ( )D for z:z electrolytes butallowed the cation and anion diameters to be unequal, thusextending the results of Kilic et al.21,28 and Kornyshev.20 Weemphasize that equality of cation and anion valence is a commonassumption. In this article, we relax this assumption and wefurther extend the above-mentioned results to non z:zelectrolytes. We show that asymmetry in cation and anionvalence significantly influences the behavior of q(ψD) and ψ( )D .Second, we focus on the dielectric decrement effect, which

refers to the decrease in the dielectric constant due to a reductionin the orientational polarizability of the hydrated ions withincrease in electrolyte concentration. A decrease in the dielectricconstant lowers the ability to store charge in the double layer.The dielectric decrement has been recognized in several

Received: June 18, 2018Revised: August 26, 2018Published: August 28, 2018

Article

pubs.acs.org/LangmuirCite This: Langmuir 2018, 34, 11971−11985

© 2018 American Chemical Society 11971 DOI: 10.1021/acs.langmuir.8b02064Langmuir 2018, 34, 11971−11985

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reports.18,22,26,30−34 Here, we focus on the recent results ofNakayama and Andelman,32 which describe the interplaybetween finite ion-size effects (with equal cation and aniondiameters) and the dielectric decrement for z:z electrolytes. Inparticular, we derive general results for the dielectric decrementfor non z:z electrolytes while also allowing for unequal cation andanion diameters. We then focus on the effect of valenceasymmetry and show that it strongly impacts the double layerproperties.The ion−ion correlation effect, also known as the over-

screening effect, relates to the interaction between nearby ions.This effect can be accounted for in the mean-field framework bydefining a screening length and including an additional fourth-order term in the modified Poisson−Boltzmann equation, asrecently derived by Bazant, Storey, and Kornyshev,23,35 whodiscussed the competition between steric effects and ion−ioncorrelations. The authors showed that ion−ion correlations giverise to oscillations in charge density profiles, especially for largescreening lengths.23,35 We note that the article by Bazant andStorey35 considers 1:1 and 1:2 electrolytes with equal cation andanion diameters. In this article, we extend these results for nonz:z electrolytes with different cation and anion diameters andinvestigate the effect of electrolyte valence on ion−ioncorrelations.Before proceeeding further, we acknowledge that we are

certainly not the first to investigate non z:z electrolytes. Theeffect of electrolyte valence has been investigated for the classicalPoisson−Boltzmann equation11,36−41 (e.g., Gouy,11 Levine andJones,37 and Grahame36 analyzed the scenarios of z−/z+ = 2 orz−/z+ = 1/2). On the other hand, Lyklema38 and Levie40

described ψ( )D for a general combination of z+ and z−, thoughthe results are presented in an awkward dimensional form.However, the equilibrium relationships for non z:z electrolytes inthe modified Poisson−Boltzmann description are not readilyavailable. Therefore, the aim of this article is to investigate theimpacts of asymmetry in electrolyte valence on the modifiedPoisson−Boltzmann equation. We find that inclusion ofasymmetry in electrolyte valence is critical as it affects all of theeffects mentioned above.We present physical arguments to broadly highlight the

importance of asymmetry in electrolyte valence. Since themagnitude of valence dictates the force experienced by the ions,when cations and anions are of different valence, the magnitudeof forces experienced by the cations and anions are unequal,which creates an asymmetry in double layer properties, or q(ψD)≠ q(−ψD) and ψ ψ≠ − ( ) ( )D D . This apparent breaking ofsymmetry can have significant implications. For instance, severalexperimental data sets published in the supercapacitor literatureutilize valence asymmetric electrolytes such as Na2SO4 andCaCl2.

42,43 However, the modeling approaches in this area arestill typically restricted to z:z electrolytes,44 and thus studies havenot exploited valence asymmetry as a way to tune the energy andpower density of supercapacitors. Similarly, valence asymmetrycan be important for capacitive deionization,3,45 a process wheretoxic ions migrate move from the bulk to the double layer. Here,capacitance would influence the quantity of toxic ions depleted aswell as the time required for the depletion.3 Since

ψ ψ≠ − ( ) ( )D D for valence asymmetric ions, the direction ofthe potential drop will impact the efficacy of the process.We also emphasize that our analysis is general since we

consider a combination of valence asymmetry with other effects(e.g., finite ion sizes, dielectric decrement, and ion−ion

correlations). To highlight the relative importance of simulta-neous effects, we now present a physical argument for thescenario where the combined effects of valence asymmetry andfinite ion sizes are relevant. Let us assume that in an electrolyte,anions have a higher valence than the cations. At the same time,the anions have a significantly smaller ion size than the cations.When such an electrolyte comes in contact with a positivelycharged surface, the anions migrate toward the charged surfaceand cations move away from the surface. The higher valence ofthe anions leads to a rapid increase in the anion concentrationwith increase inψD until there is no longer space to accommodatemore anions. Therefore, a higher valence implies that the doublelayer saturates with anions at a smaller value of ψD. In contrast,the smaller anion size implies that each ion occupies a smallervolume, and thus saturation of the double layer with anionsoccurs at larger values of ψD. Therefore, valence asymmetry cancompete or cooperate with other additional effects and canprovide flexibility in design of processes when the additionaleffects are significant.In this article, we study the influence of asymmetry in

electrolyte valence on finite ion-size effects (with unequal cationand anion diameters), dielectric decrement, and ion−ioncorrelations. For each of these effects, we first derive thediffuse-charge relations for a general valence electrolyte andprovide analytical and numerical results for the potentialdistribution, q(ψD), ψ( )D and dimensionless salt uptakeα(ψD). Under appropriate assumptions, we recover previouslyreported results for q(ψD) and ψ( )D , thus highlighting the

generality of the proposed relations. Since ψ( )D has multiplelocal extrema, we provide scaling relations to better explain thedependence of the extrema on different parameters. Lastly, wediscuss the implications of our results on diffuse-charge dynamicsand dimensionless parameters that govern the electrokineticphenomena. We conclude by providing limitations of the modelsand directions for future research.

■ PROBLEM SETUP

We consider an electrolyte in equilibrium with a charged surface(Figure 1). Due to electrostatic attraction, oppositely charged

Figure 1. An electrolyte with cation valence z+ and anion valence z− isnear a charged surface. The Stern layer thickness is denoted λS, and theDebye length is denoted λD. The potential at the electrode is taken asψ =ψ0, the potential at the boundary between the Stern layer and diffuselayer is ψ = ψD, and the potential in bulk is zero.

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ions (also referred as counterions) migrate toward the chargedsurface, and compete with thermal or entropic effects to create aregion of diffuse charge. The typical thickness of the region ofexcess charge, or double layer, is given by the Debye lengthλD

15,16,38,46

λε ε

=∑

k Te z c

s

i i iD

0 B2 2

0, (1)

where ε0 is the electrical permittivity of vacuum, εs is thedielectric constant of the solution without the electrolyte, kB isthe Boltzmann constant, T is temperature, zi is the valence of theith ion type, c0,i is the bulk concentration of the i

th ion type, and e isthe charge on an electron. The sum is over all ionic speciespresent in the solution. The region close to the electrode whereions are adsorbed at the surface is known as the Stern layer(Figure 1). The (molecular) thickness of the Stern layer isdenoted as λS.To be specific, we consider an electrolyte with one cation type

and one anion type. The cation and anion valences are denotedz+ and z−, respectively. Since the electrolyte in bulk is neutral, theion concentrations in bulk are c+ = z−c0 and c− = z+c0. Forinstance, K2SO4 is denoted by z+ = 1, z− = 2, c+ = 2c0, and c− = c0.Therefore, for a z+:z− electrolyte, according to equation 1, λD isgiven as

λε ε

=++ − − +

k Tz z z z e c( )

sD

0 B2

0 (2)

■ FINITE ION-SIZE EFFECTSIn this section, we consider finite ion-size effects and specificallyfocus on the effect of valence asymmetry on double layerproperties. For simplicity, we assume that the relativepermittivity ε is independent of c±, or ε(c±) = εs. However, wediscuss the effect of a change in dielectric constant ε(c±)separately in the next section. Similarly, we also exclude the ion−ion correlations in this section but discuss their effect in thesubsequent section.Derivation. Potential Distribution and Charge Accumu-

lated. To include the finite ion-size effects, we assume a hard-sphere model where ions are spherical and are described with aneffective diameter. Depending on the interaction betweendifferent ion types, the effective diameter may or may not bethe same as the ion diameter. We refer the readers to refs 20, 21,and 28 for a more detailed discussion on effective diameters andthe length scale of interactions. Here, we denote the effectivediameter of a cation as a+. Typically, = −+

−a (10 10)1 nm,and thus the concentration of cations cannot exceed

= −+

(10 10 )a1 24 30

3 m−3.21 We also allow for asymmetry in

the effective diameter of anions and cations and define theeffective diameter of an anion as a−. Therefore, the concentrationof anions cannot exceed

−a1

3 .

The free energy of the system per unit volume F is defined as

= −F U TS (3)

where U is the internal energy per unit volume, S is the entropyper unit volume, and F(ψ,c±). U is defined as20,21,32

ε ε ψ ψ ψ= − + −+ + − −Ux

z ec z ec2

dd

s02

(4)

where ψ(x) is the potential at a location x relative to a referencepotential at x → ∞, or ψ(∞) = 0. The first term in equation 4represents the energy stored in the electric field, and theremaining two terms account for the potential energy of the ions.To evaluate S, we use Boltzmann’s formula of mixing, S = kB lnω,where ω is the number of microstates. To estimate ω, we firstestimate the number of ways to arrange the larger ions and thenmultiply with the number of ways to arrange the smaller ions.Here, for convenience, we assume that a+ ≥ a−. We estimateentropy as

− = +−

− +

−+

− −

− −−

+ + ++ +

++ +

−− −

+ +

+ + − −

−

+ + − −

+ +

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

Sk

c a ca c

aa c

ca c

a ca c a c

aa c a c

a c

ln( )1

ln(1 )

ln1

(1 )

ln1

1

B

33

33

3

3

3 3

3

3 3

3(5)

An equivalent expression of entropy for a− ≥ a+ can beestimated by switching the positive and negative subscripts. WithF(c±) given by equations 3−5, we evaluate the chemicalpotentials μ± as

μ

ψ

= ∂∂

= +−

−− −

−

++

++ +

+ +

+

−

+ + − −

+ +

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟⎤⎦⎥⎥

Fc

z e k Ta c

a caa

a c a ca c

ln1

ln1

1B

3

3

3

3

3 3

3

(6a)

μ ψ= ∂∂

= − +− −−

−−

− −

+ + − −

⎛⎝⎜

⎞⎠⎟

Fc

z e k Ta c

a c a cln

1B

3

3 3(6b)

We define dimensionless concentrations as =++

−n c

z c0,

=−−

+n c

z c0, and the dimensionless electric potential as Ψ = ψe

k TB.

At equilibrium, the chemical potential is constant for all x, μ±(x)= μ±(∞), and we obtain

=− ΨΨ+

+nz

gexp( )

( ) (7a)

=Ψ ΨΨ−

−nz fg

exp( ) ( )( ) (7b)

Ψ = Ψ + − Ψ − Ψ

+ Ψ Ψ −− + +

+ − −

g f z a c z f

z a c f z

( ) ( ) [exp( ) ( )]

( )[exp( ) 1]

30

30 (7c)

Ψ = +Ψ −

−+ − −

− +

−+ −⎛⎝⎜

⎞⎠⎟f

z a c zz a c

( ) 1(exp( ) 1)

1

a a30

30

/ 13 3

(7d)

Physically, g(Ψ) accounts for the reduction in concentrationdue to finite ion sizes, and f(Ψ) accounts for the change inconcentration due to the contrast in ion sizes [note f(Ψ)=1 for a−= a+]. For a± → 0 in equation 7, we recover the Boltzmanndistribution. For a+ = a− and z+ = z−, we recover the standardresult in refs 21, 27, and 28. Furthermore, for a+≠ a− and z+ = z−,we recover the known result in ref 29. To solve for c± and ψ, wecouple equation 7 with Gauss’s law,

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ε ε ψ = −− − + +xe z c z c

dd

( )s0

2

2 (8)

which is to be solved with boundary conditions

ψ ψ λ= − ψ

=(0)

x x0 Sdd 0

and ψ(∞) = 0, where ψ(0) = ψD is the

potential drop across the diffuse layer (Figure 1). We note that

the boundary condition at the electrode assumes a thin Stern

layer.3 We nondimensionalize with Ψ = ψek TB

, =λ

X x

D, and

Λ = λλS

S

Dto obtain

Ψ =−+

− +

+ −Xn nz z

dd

2

2(9)

with two boundary conditions

Table 1. Summary of ΨQ ( )D and Ψ( )D Relations That Account for Ion Valence and Finite Ion Sizea

conditions diffuse charge relations refs

= =+ −z z z→±a 0 =

ΨQ

zz2

sinh2

D 11 and 12

=Ψ

z

cosh2

D

≠+ −z z→±a 0 = Ψ

Ψ + − Ψ+

−+ −

+ − − +

+ −

⎛⎝⎜

⎞⎠⎟Q

z zz z z z

z zsgn( )

2 exp( ) exp( )1D

D D 38 and 40

=Ψ − − Ψ

+ | |− +

+ −

z zz z Q

exp( ) exp( )( )

D D

= =+ −z z z= =+ −a a a = Ψ +

Ψ⎛⎝⎜

⎞⎠⎟Q

z a cza c

zsgn( )

1ln 1 4 sinh

2D 3 30

30

2 D 20, 21, and 28

=| Ψ |

+ | |Ψ

( )z

z a Q

sinh

1 4z sinh zD

30

22

D

= =+ −z z z≠+ −a a =

− ΨΨ

=Ψ ΨΨ+ −n

zg

nz fg

exp( )( )

,exp( ) ( )

( )29

Ψ = +Ψ −

−−

+

−+ −⎛⎝⎜

⎞⎠⎟f

za c zza c

( ) 1(exp( ) 1)1

a a30

30

/ 13 3

Ψ = Ψ + − Ψ − Ψ + Ψ Ψ −+ −g f za c z f za c f z( ) ( ) (exp( ) ( )) ( )(exp( ) 1)30

30

= Ψ Ψ+

Qz a c

gsgn( )1

ln ( )D 3 30

D

=| Ψ − Ψ |

| |− +

n nz Q

( ) ( )2

D D

≠+ −z z≠+ −a a =

− ΨΨ

=Ψ ΨΨ+

+−

−nz

gn

z fg

exp( )( )

,exp( ) ( )

( )this work

Ψ = +Ψ −

−+ − −

− +

−+ −⎛⎝⎜

⎞⎠⎟f

z a c zz a c

( ) 1(exp( ) 1)

1

a a30

30

/ 13 3

Ψ = Ψ + − Ψ − Ψ + Ψ Ψ −− + + + − −g f z a c z f z a c f z( ) ( ) (exp( ) ( )) ( )(exp( ) 1)30

30

= Ψ+

Ψ+ − + − +

Qz z z z a c

gsgn( )2

( )ln ( )DD 3

0

=| Ψ − Ψ |

+ | |− +

+ −

n nz z Q

( ) ( )( )

D D

aFor simplicity, we assume ΛS = 0.

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Ψ = Ψ − Λ Ψ

=XddD

X0 S

0 (10a)

Ψ ∞ =( ) 0 (10b)

Equations 7 and 9 are governed by six dimensionlessparameters: z+, z−, a+

3c0, a+3/a−

3 , Λs and Ψ0 (the potentialmeasured on the solid boundary). To solve for Ψ and n±, weassume z+, z−, a+

3c0, a+3/a−

3 , ΛS, and Ψ0 are specified. Typically,= −±

− −a c (10 10 )30

10 1 . For a more detailed discussion on thephysical interpretation of a±

3 c0 and the range of possible values ofa±3 c0, we refer the reader to ref 21. We multiply both sides ofequation 9 by Ψ

Xdd

and integrate once (using equation 10b) to find

Ψ = − Ψ+

Ψ+ − + − +X z z z z a c

gdd

sgn( )2

( )ln ( )3

0 (11)

where sgn(Ψ) is the sign function. For a+3c0→ 0, a+/a− = 1, and z+= z− = z, equation 11 becomes = −Ψ Ψ2 sinh

Xzd

d 2, and another

i n t e g r a t i o n y i e l d s t h e w e l l - k n o w n r e l a t i o n

= −Ψ Ψ Xtanh tanh exp( )z z4 4

D ,11 where ΨD is related to Ψ0

through the boundary condition in equation 10a. For a+ ≠ 0,a+/a− ≠ 1, and z+ ≠ z−, we numerically integrate equation 11with the Stern layer boundary condition (equation 10a) to obtainΨ(X), and the results are discussed later.Next, we evaluate the surface charge density on the electrode

as ε ε= − ψ

=q s x x

0dd 0

. Similarly, we can also calculate the

capacitance (i.e., the charged stored in the electrode per unit

total potential drop, or =ψ

qd

d 0). It is convenient to

nondimensionalize =λ++ − + −

Q qz z z z e c( ) D 0

and = λε ε

D

s0, such that

= − Ψ

=Q

X X

dd 0

and = ΨQd

d 0. Thus, from equation 11, we find

the dimensionless surface charge density

= Ψ+

Ψ+ − + − +

Qz z z z a c

gsgn( )2

( )ln ( )D D3

0 (12)

Capacitance.To calculate = ΨQd

d 0, we write the Stern layer

boundary condition in equation 10a as Ψ0 = ΨD + ΛSQ.Differentiating this relation with respect to ΨD, we get

= + ΛΨΨ Ψ1 S

Qdd

ddD D

0 . Thus, = ΨΨΨ

Qdd

ddD

D

0, or

=Ψ

+ Λ−−⎛

⎝⎜⎞⎠⎟

Qdd S

1

D

1

(13)

so that

=+ Ψ

| Ψ − Ψ |+ Λ− + −

+ − + − +

z z gz z a c n n

2( )ln ( ) 1( ) ( ) S

1 D3

0 D D

(14)

Equations 13 and 14 demonstrate the well-known result thatwe can characterize the system as an electrical circuit with acapacitor representing the Stern layer and a capacitorrepresenting the diffuse layer in series. For a+ → 0, a+/a− = 1,and z+ = z− = z, equations 12 and 14 take the form

=Ψ

Qz

z2sinh

2D

(15a)

=Ψ

+ Λ−z

sech2

1 DS (15b)

which are the classical GC relations.11,12 For a+ → 0 and a+/a− =1 but different ion valences, equations 12 and 14 become

= ΨΨ + − Ψ

+−

+ −

+ − − +

+ −

⎛⎝⎜

⎞⎠⎟Q

z zz z z z

z zsgn( )

2 exp( ) exp( )1D

D D

(16a)

=+

| Ψ − − Ψ |

×Ψ + − Ψ

+− + Λ

− + −

− +

+ −

+ − − +

+ −

⎛⎝⎜

⎞⎠⎟

z z

z z

z zz z z z

z z

exp( ) exp( )

2 exp( ) exp( )1

1

D D

D DS

(16b)

which are consistent with the results reported in ref 38. However,the results analogous to equation 16 reported in ref 38 arepresented in an awkward dimensional form. Taking the limit of z+= z− = z in equation 16, it is easy to recover the GC relations inequation 15. Lastly, for z+ = z− = z and a+ = a− = a, equations 12and 14 are evaluated as

= Ψ +Ψ⎛

⎝⎜⎞⎠⎟Q

z a cza c

zsgn( )

1ln 1 4 sinh

2D 3 30

30

2 D

(17a)

=+

| Ψ |

× +Ψ

+ Λ

−Ψ

⎛⎝⎜

⎞⎠⎟

za c

z

za cza c

z

1 4 sinh

sinh

1ln 1 4 sinh

2

z1

30

22

D

30

30

2 DS

D

(17b)

which agree with the relations presented in refs 21 and 28. Wesummarize the validity of the aforementioned diffuse chargerelations for Q(ΨD) and C(ΨD) in Table 1. To the best of ourknowledge, equations 12 and 14 are the most general charge andcapacitance relations reported in the literature accounting for ionvalence and finite ion size.

Salt Uptake. Both Q and are measures of the net chargeinside the double layer. However, the formation of a double layeralso depletes salt from the bulk. As noted in refs 3 and 45, theamount of salt uptake directly dictates the dynamics of thedouble layer formation process, as explained later. We define thelength scale of the bulk as L and estimate a dimensionlessmeasure of the excess salt uptake as

∫α =

+ − +

++ − + −

+ −

c c z z c x

z z c L

( ( ) )d

( )

L

0 0

0 (18)

When α ≪ 1, the solution to double layer charging for anelectrolyte between two parallel plates can be reliablyapproximated as an electrical circuit.3 However, if α = (1),this approximation is no longer reliable. Therefore, estimation ofα is important for time-dependent problems. To estimate α, weutilize equations 7 and 11 to obtain

∫αλ

=+

− + −Ψ

Ψ+ +

+ −

Ψ− + + −

La c zz z

z n z n

g2( )( 1) ( 1)

ln ( )dD

30

0

D

(19)

where n+, n−, and g(Ψ) are evaluated from equation 7. In the limita+ → 0, a+/a− = 1, and z+ = z− = z, we recover the well-

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documented result3 of α = λ ΨsinhL

z4 24

D D . We numerically

integrate equation 19 to evaluate the dependence of α ondifferent parameters, and the results are discussed later.Potential Distribution. We now discuss the numerical

solution to equation 11 with the Stern layer boundary condition(equation 10a). We first report the effect of changes in a+

3c0 onΨ(X) withΨ0 = 5, z+ = 1, z− = 3, a+/a− = 1, andΛS = 0; see Figure2a. Physically, for a larger value of a+

3c0 (i.e., larger steric effects)

|Q| is smaller. Since = − Ψ

=Q

X X

dd 0

, a larger a+3c0 implies a more

gradual decrease in Ψ with X.

The effect of change in z+ and z− on Ψ(X) with Ψ0 = 5, a+3c0 =

0.2, a+/a− = 1, and ΛS = 0 is provided in Figure 2b. The trendshows that the fastest decay in Ψ occurs for z+ = z− = 1, whereasthe decay is slowest for z+ = 2, z− = 1. Physically, for a+

3c0 = 0.2, theion concentration is high even in the bulk, and thus, finite ion-sizeeffects are important. We assume that for Ψ ≈ 5, c− ≈ 1/a−

3 ,z+c+(0) ≪ z−c−(0), and it can be estimated that

≈ −Ψ Ψ++ + − −X z z z a c

dd

2( ) 3

0; see equation 9. This approximation

explains the trend we observe in Figure 2b. Similarly, for z+ = 1,z− = 3, a+

3c0 = 0.2, and ΛS = 0, we find that a smaller a− increases

the magnitude of ΨX

dd

, and thus the change inΨ is more rapid for

a smaller a−, as observed in Figure 2c.The effect of ΛS on Ψ(X) enters through the boundary

condition (equation 10a). We present the results of changes inΛS onΨ(X) withΨ0 = 5, a+

3c0 = 0.2, z+ = 1, z− = 3, and a+/a− = 1in Figure 2d. A thicker Stern layer, or a larger ΛS, implies a largerpotential drop across the Stern layer. Therefore, we see thatΨD =Ψ(0) decreases for an increase in ΛS. Further, a smaller ΨDindicates a lower |Q| (see below), and thus for a largerΛS, the rateof decay of Ψ is smaller.Charge Accumulated. We present the dependence of

accumulated charge Q on different parameters according toequation 12. Figure 3a shows the dependence of Q with ΨD fordifferent values of a+

3c0 with z+ = 1, z− = 3, and a+/a− = 1. A larger

a+3c0 implies that steric effects are stronger, and thus Q is smaller.Increasing ΨD increases the concentration of ions, and thus Qincreases. For a large ΨD, steric effects start to become moreimportant, and the increase in Q is smaller since the rate ofchange in ion concentration is lower.Next, we consider valence asymmetry for a+

3c0 = 0.2 and a+/a−= 1. Since finite ion-size effects are significant here, forΨD > 0, weassume c−(0) ≈ 1/a−

3 and z+c+(0) ≪ z−c−(0) to obtain

= − Ψ ≈Ψ

+= + + − −Q

X z z z a cdd

2( )x 0

D3

0 (20)

We find good agreement between computed values fromequation 12 and approximate values from equation 20, especiallyfor large |ΨD|, since concentration approximations are moreaccurate for large |ΨD|.Q is highest for z+ = z− = 1 followed by z+= 1, z− = 3, and z+ = 2, z− = 1 (see Figure 3b). Similarly, for a+

3c0 =0.2, z+ = 1, and z− = 3, a smaller a− leads to a largerQ, as predictedby equation 20. This observation is corroborated in Figure 3c.Lastly, ΛS does not influence the variation of Q versus ΨD.However, for the same value of Ψ0, values of ΨD will be smallerfor a larger ΛS (equation 10a, see Figure 2d).

Capacitance. Capacitance is a measure of the amount of

charge stored per unit total potential drop (i.e., ΨdQd 0

).We discuss

the dependence of on different parameters based on equation14. Figure 4a plots the variation of withΨD for different valuesof a+

3c0 with z+ = 1, z− = 3, a+/a− = 1, and ΛS = 0 as constants.Depending on the value of a+

3c0, capacitance exhibits differentbehavior.For dilute ion concentrations in the bulk (i.e., ≲+

−a c (10 )30

2 ), (ΨD) displays a camel shape with one local minimum and twolocal maxima. Physically, this occurs because for small values of|ΨD|, counterion concentration increases with increase in |ΨD|.For large values of |ΨD|, the counterion concentration saturatesaround ΨD = ΨD,max, beyond which the capacitance decreases.Figure 4a shows that the curves are asymmetric when the cationand anion valences are not equal, or −Ψ ≠ Ψ ( ) ( )D D for z+ ≠z−. We note that the location of the minimum ΨD = ΨD,min ≠ 0for z+ = 1 and z− = 3, unlike valence symmetric electrolytes.Similarly, the location of two maxima are not equal and oppositefor unequal cation and anion valences.For large bulk ion concentrations (i.e., ≳+

−a c (10 )30

1 ), Ψ( )Dcurves show a bell shape with no local minimum and one localmaximum. Since the ion concentration is high even in bulk, thelocal minimum disappears and only one local maximum remains.We find that since cation and anion valences are unequal, ΨD,max≠ 0. Though the camel shape and bell shape curves have been

Figure 2. Effect of asymmetry in electrolyte valence on finite ion-sizeeffects for Ψ(X), as given by the numerical solution of equation 11 withboundary condition (equation 10a) forΨ0 = 5. (a) Different a+

3c0 with z+= 1, z− = 3, a+/a− = 1, andΛS = 0. (b) Different z+ and z−with a+

3c0 = 0.2,a+/a− = 1, and ΛS = 0. (c) Different a+

3/a−3 with z+ = 1, z− = 3, a+

3c0 = 0.2,andΛS = 0. (d) DifferentΛS with z+ = 1, z− = 3, a+

3c0 = 0.2, and a+3/a−

3 = 1.

Figure 3. Effect of asymmetry in electrolyte valence on finite ion-sizeeffects for Q(ΨD), as given by equations 12 and 20. The solid linesrepresent results from equation 12, and the dotted lines denote resultsfrom equation 20. (a) Different a+

3c0 with z+ = 1, z− = 3, and a+/a− = 1.(b) Different z+ and z−with a+

3c0 = 0.2 and a+/a− = 1. (c) Different a+3/a−

3

with z+ = 1, z− = 3, and a+3c0 = 0.2.

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reported previously,20 here we emphasize that the shapes and theΨD,max andΨD,min are significantly influenced by z+ and z−. In thesubsequent subsection, we use a scaling analysis to detail a morequantitative estimate of extrema and their dependence on z+ andz−.Next, we present the results for a+

3c0 = 0.2, a+/a− = 1, andΛS =0 but with different cation and anion valences in Figure 4b. Wefind that the position of a local maximum in the bell shapecapacitance is also dictated by the valence andΨD,min > 0 when z−> z+ andΨD,min < 0 when z+ > z−. We note that an approximationfor is possible by assuming c+(0)≈ 1/a+

3, z−c−(0)≪ z+c+(0) forΨD < 0 and c−(0) ≈ 1/a−

3 , z+c+(0)≪ z−c−(0) forΨD > 0. We canestimate by differentiating equation 20 to obtain

≈ |Ψ | + Ψ <− − + +− z z z a c(2 ( ) ) for 0D

30

1/2D (21a)

≈ |Ψ | + Ψ >+ − + −− z z z a c(2 ( ) ) for 0D

30

1/2D (21b)

Equation 21 is more accurate for large a ±3 c0 and |ΨD| since the

assumptions for ion concentrations are more readily satisfied.Therefore, equation 21 predicts a decrease in with an increasein |ΨD| and does not predict the extrema nearΨD = 0. However, itcorrectly captures the trends and relative position of reportedin Figure 4b for |Ψ | ≳ (1)D for different combinations of z+ andz−.We note that equation 21 suggests that also depends on a+

3/a−3 . Typical results are presented in Figure 4c with a+

3c0 = 0.2, z+ =1, z− = 3, andΛS = 0 for different a+

3/a−3 . Equation 21 explains the

collapse of curves for ΨD < 0 and the increase in for highera+3/a−

3 for ΨD > 0.Lastly, we discuss the effect of ΛS. For different values of ΛS,

Figure 4d presents the variation of with ΨD for different witha+3c0 = 0.2, z+ = 1, z− = 3, and a+/a− = 1. We find that decreaseswith an increase in ΛS and becomes almost independent of ΨDfor larger values of ΛS. This change in behavior occurs since theStern layer capacitor and the diffuse layer capacitor are in series;

an increase inΛS results in the effective capacitance to be dictatedby the Stern layer capacitance.

Salt Uptake. In this section, we describe the variation of thedimensionless salt uptake α(ΨD) as given by equation 19. Asnoted previously, α dictates the dynamics of double layercharging. Figure 5a shows the dependence of α for different a+

3c0

with z+ = 1, z− = 3, and a+3/a3− = 1. As expected, an increase in

a+3c0 decreases salt uptake since the ion concentration saturatesdue to finite ion-size effects.In Figure 5b, we present the dependence of valence on salt

uptake for a+3c0 = 3 and a+/a− = 1. We find that αλD/L is largest

for z+ = 1, z− = 1 followed by z+ = 2, z− = 1, and z+ = 1, z− = 3. Thistrend occurs since for ΨD > 0, anion concentration saturatesinside the double layer. This saturation occurs for smaller valuesofΨD for z+ = 1, z− = 3, and thus, lower salt is depleted from thebulk when compared to z+ = 2, z− = 1, and z+ = z− = 1. Further,though these two cases might achieve anion concentrationsaturation at similar values ofΨD, a larger number of anions in thebulk for z+ = 2, z− = 1 leads to a lower level of salt depletion.Figure 5c summarizes the effect of a+

3/a−3 for a+

3c0 = 0.2, z+ = 1, andz− = 3. We find that salt depletion is maximum for smaller valuesof a− since the saturation concentration of anions is larger, andthus more salt can be depleted. A quantitative prediction ofα(ΨD), similar to equations 20 and 21 forQ and, is challengingsince we need to find an approximate description for dψ/dx forall x (and not just at x = 0 as in equations 20 and 21); see equation18.

Scaling Anlaysis. A unique feature of the derived diffuse-charge relations is the presence of extrema in the dependence of with ΨD and their dependence on the values of z+ and z−. Wenow present physical arguments to predict the location of localextrema. Local maxima occur when the ion concentration insidethe diffuse layer is on the order of 1/a±

3 . ForΨD > 0, negative ionswill be attracted and the condition for a local maximum implies

=− −c a(1/ )3 . On the other hand, forΨD < 0, the condition for alocal maximum becomes =+ +c a(1/ )3 . Thus, assuming the c±follow the Boltzmann distribution, we estimate

− Ψ = Ψ ≤− + +z c z aexp( ) (1/ ) for 00 D,max3

D (22a)

Ψ = Ψ ≥+ − −z c z aexp( ) (1/ ) for 00 D,max3

D (22b)

or

Ψ = Ψ ≤+−

− +z z a c( ln( )) for 0D,max1 3

0 D (23a)

Ψ = − Ψ ≥−−

+ −z z a c( ln( )) for 0D,max1 3

0 D (23b)

Equation 23 demonstrates thatΨD,max is strongly influenced byz+ and z−. We observe a good quantitative agreement between

Figure 4. Effect of asymmetry in electrolyte valence on finite ion-sizeeffects for Ψ( )D , as given by equations 14 and 21. The solid linesrepresent results from equation 14, and the dotted lines denote resultsfrom equation 21. (a) Different a+

3c0 with z+ = 1, z− = 3, a+/a− = 1, andΛS= 0. (b) Different z+ and z− with a+

3c0 = 0.2, a+/a− = 1, and ΛS = 0. (c)Different a+

3/a−3 with z+ = 1, z− = 3, a+

3c0 = 0.2, and ΛS = 0. (d) DifferentΛS with z+ = 1, z− = 3, a+

3c0 = 0.2, and a+3/a−

3 = 1.

Figure 5. Effect of asymmetry in electrolyte valence on finite ion-sizeeffects for α(ΨD), as given by equation 19. (a) Different a+

3c0 with z+ = 1,z− = 3, and a+/a− = 1. (b) Different z+ and z−with a+

3c0 = 0.2 and a+/a− =1. (c) Different a+

3/a−3 with z+ = 1, z− = 3, and a+

3c0 = 0.2.

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the predictions of equation 23 and the computed values(obtained from equation 14), as illustrated in Figure 6 (panels

a and b). The scaling relations accurately capture the behavior,especially for the camel-shape capacitance curves (i.e., with twolocal maxima and one local minima). However, the scalingrelation is not as accurate for the bell shape capacitance curves(i.e., only one local maximum and no local minimum), since theassumption that concentration follows a Boltzmann distributionis less accurate. Nevertheless, equation 23 correctly captures thedependence of ΨD,max on z+, z−, a+

3c0, and a+/a−.We now present a physical argument to predict ΨD,min, which

is only observed in the camel shape capacitance curves. We knowthat the magnitude of charge per unit volume carried by thecations and anions is proportional to z+c+ and z−c−, respectively.For instance, in bulk, by definition, z+c+ = z−c−, and the chargesbalance. However, the magnitude of the rate of change in chargewith ΨD of cations and anions is proportionate to z+

2c+ and z−2 c−

(assuming the Boltzmann distribution, see equation 7).We argue

that ΨD,min can be estimated when =+ +

− −(1)z c

z c

2

2 . Assuming the

Boltzmann distribution, this condition yields

Ψ =++ −

+ −

⎛⎝⎜

⎞⎠⎟

z zz z

ln( / )D,min

(24)

Therefore, physically, ΨD,min is the potential at which the rateof change of both positive and negative charges with ΨD areequal. Due to symmetry, this occurs atΨD = 0 for z+ = z− = 1. Weshow that predictions of equation 24 are in good agreement withcomputed results (obtained from equation 14) in Figure 6c.Physical Significance. We now discuss the physical

significance and implications of valence asymmetry of finiteion-size effects. For this discussion, we briefly restore dimensionsfor charge q(ψD) and capacitance ψ( )D . By converting equations20 and 21 to dimensional form, we obtain

ε ε ψ ψ≈ − | | <+ +−q z a(2 ) for 0s0

3D

1/2D (25a)

ε ε ψ ψ≈ | | >− −−q z a(2 ) for 0s0

3D

1/2D (25b)

ε ε ψ ψ≈ | | <+ +− − z a(0.5 ) for 0s0

3D

1 1/2D (25c)

ε ε ψ ψ≈ | | >− −− − z a(0.5 ) for 0s0

3D

1 1/2D (25d)

Equation 25 clearly shows that q(ψD) = q(−ψD) andψ ψ= − ( ) ( )D D only when z+a+

−3 = z−a−−3. Physically speaking,

z+/a+3 and z−/a−

3 are, respectively, a measure of the maximum

positive charge density and negative charge density that can bestored inside the double layers. Therefore, the condition z+a+

−3 =z−a−

−3 implies that the double layer formation is symmetric onlywhen the magnitude of maximum charge densities accumulatedinside the double layer are the same irrespective of the sign of thepotential drop. Moreover, the individual factors z+a+

−3 and z−a−−3

combine the relative importance of ion valence and finite ion sizeand suggest that a higher valence and a lower ionic diameterincreases the amount of charge stored and the capacitance. Thisresult is consistent with physical intuition since increasing thevalence increases the charge stored per ion, and a smaller ion sizeallows for a larger number of ions per unit volume to accumulatein the double layer. Moreover, the square root dependencehighlights a quantitative feature that is important for practicalapplications. As an example, we consider the case of CaCl2, whichis a possible electrolyte candidate for supercapacitors,43 amongmany applications. For this salt, z+ = 2, z− = 1, a+ = 0.11 nm, and

a− = 0.17 nm. Here, the factor =+ +−

− −−( ) 2.71z a

z a

1/23

3 , which

indicates that for the same magnitude potential drop across thecharged surface, when the surface is negatively charged, thedouble layer will accumulate almost thrice as much net charge.This break of symmetry is significant for supercapacitorapplications where the amount of charge stored dictates theenergy density. We note that the asymmetry in the double layerproperties that arise when z+≠ z− is the novel aspect of this workand creates opportunities for future research.

■ DIELECTRIC DECREMENT EFFECTIn this section, we relax the assumption of a constant dielectricconstant (i.e., ε(c±) = εs). As previously discussed, changes in thedielectric constant with ion concentration can reduce the abilityto store charge inside a double layer, and thus can have a majorimpact on diffuse layer properties. Here, we assume that thedecrease in the dielectric constant is linear with ionconcentrations, or

ε ε γ γ= − −± + + − −c c c( ) s (26)

where γ± are constants. Though equation 26 is not obtained froma theoretical derivation, experiments have shown that thisdependence works reasonably well for ion concentration up to afew molars.32,34,47 Typical values of γ± range fromγ = −±

− −(10 10 )27 26 m−3 (see reference 32).Derivation. To derive charge and capacitance relationships

for a variable dielectric constant, equations 3−5 remain identicalexcept εs is replaced by ε(c±), as given by equation 26. Due to thedependence of the dielectric constant on c±, the chemicalpotentials μ± also have a dependence on ε, or

με ε ψ ψ= ∂

∂= − ∂

∂+ +

−−

− −−

++ +

++ +

+ +

+

−

+ + − −

+ +

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟⎤⎦⎥⎥

Fc c x

z e k Ta c

a caa

a c a ca c

2dd

ln1

ln1

1

02

B

3

3

3

3

3 3

3(27a)

με ε ψ ψ= ∂

∂= − ∂

∂− +

− −−− −

−− −

+ + − −

⎛⎝⎜

⎞⎠⎟

Fc c x

z e k T a ca c a c2

dd

ln1

02

B

3

3 3

(27b)

Though it is possible to find an explicit relationship for

concentration by equating μ±(x) = μ±(∞), the presence of ψx

dd

2

in equation 27makes the expression inconvenient. Therefore, we

Figure 6. Scaling analysis of finite ion-size effects. Comparison ofcomputed ΨD,max with eq 23 for (a) ΨD < 0 and (b) ΨD > 0. (c)Comparison of computed ΨD,min with equation 24. The red points arecases with the camel shape (i.e., two local maxima and one localminima), and the blue points are cases with the bell shape (i.e., one localmaximum); see Figure 4. For computations, 10−6 ≤ a+

3c0 ≤ 0.2, 1 ≤ a+3/

a−3 ≤ 8, 1 ≤ z+ ≤ 3, and 1 ≤ z− ≤ 3.

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exploit the relationships developed in refs 32−34 for osmoticpressure π(x), i.e.

πε ε ψ ψ ψ μ μ= − − + + + ++ + − − + + − −x

xz ec z ec TS c c( )

2dd

02

(28)

Using equations 5, 26, and 27 in equation 28 yields

πε

ε γ γ ψ= − − −

− − − + − −

+ + − −

−+ + − −

+ −+ +

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟

x c cx

k Ta

a c a ca a

a c

( )2

( 2 2 )dd

1ln(1 )

1 1ln(1 )

s0

2

B 33 3

3 33

(29)

Furthermore, utilizing the equilibrium requirements π(x) =π(∞) and μ±(x) = μ±(∞), we obtain three equations to relate

c±(x), ψ(x), andψx

dd

2. Thus, using these three equations at x = 0,

we can evaluate c±(ψD) andψx

dd

(ψD), and by extension evaluate

q(ψD) and ψ( )D . To simplify our calculations, we assume thatsteric effects and Stern layer effects are negligible. However, asclear from the above derivation, no such restriction is necessaryand the results can also be evaluated for the general scenario. Wenote that since we have used osmotic pressure to generate anadditional relationship, we have not utilized Gauss’s law withvariable dielectric constant.We first evaluate π(x) = π(∞) and μ±(x) = μ±(∞) in the

absence of steric effects (i.e., in the limit a+3c0→ 0 and a+/a− = 1).

Consistent with the earlier discussion, we define dimensionlessvariables as =+

+

−n c

z c0, =−

−

+n c

z c0, =

λX x

D, and Ψ = ψe

k TB, and

obtain

εε γ γ

Ψ =− + −

+ − −− + + −

+ − + − + − + − + −Xz n z n

z z z z c z n c z ndd

2 ( ( 1) ( 1))( ) ( 2 2 )

s

s

2

0 0

(30a)

γε γ γ

− + −− −

+ Ψ + =+− + + −

+ − + − + −+ +

⎛⎝⎜⎜

⎞⎠⎟⎟c

z n z nc z n c z n

z n( 1) ( 1)

2 2ln( ) 0

s0

0 0

(30b)

γε γ γ

− + −− −

− Ψ + =−− + + −

+ − + − + −− −

⎛⎝⎜⎜

⎞⎠⎟⎟c

z n z nc z n c z n

z n( 1) ( 1)

2 2ln( ) 0

s0

0 0

(30c)

Equation 30 is governed by the dimensionless parameters z±,γ+c0, γ+/γ−, and εs. We solve equation 30 numerically for specifiedvalues ofΨ(0) =ΨD, z±, γ+c0, γ+/γ−, and εs = 80 (typical of water)

and obtain the functions n±(ΨD) and ΨΨ ( )X

dd D .

Once we have obtained ΨΨ ( )X

dd D , it is straightforward to

obtain Ψ(X) through numerical integration. Next, we evaluate

= − ε γ γε

− − Ψ

=+ − + − + −Q

c z n c z n

X X

( ) dd 0

s

s

0 0 . Moreover, since we assume

ΛS = 0, then Ψ0 =ΨD (see equation 10a), and = ΨQd

d D, which

is evaluated through numerical differentiation. We also numeri-ca l ly ca lculate the dimens ionless sa l t uptake as

∫α = ΨλΨ

− + −

+− + + −

+ −Ψ d

Lz n z n

z z0

( 1) ( 1)

( )X

DD

dd

. We note that though

equation 30 has been derived in the limit a± → 0 and a+=a−,the results are readily extended to the general case. Also, we findthat in the limit γ+c0 → 0 and γ+/γ− = 1 (the absence of thedielectric decrement), equation 30 gives results consistent withequation 16.

Potential Distribution. We present the variation of Ψ(X)forΨD = 5 in Figure 7. First, we discuss the effect of the change in

γ+c0 with z+ = 1, z− = 3, and γ+/γ− = 1. The change inΨ(X) is lessrapid with an increase in γ+c0 as shown in Figure 7a. In thedielectric decrement effect, much like the finite ion-size effect, theconcentration of the counterion saturates beyond some |ΨD|.Here, the saturation occurs because a lower dielectric constantimplies that the charge storage capacity of the solution is reduced,and thus the concentration of the counterion saturates. Since it isdifficult to parse the dependence of different parameters fromequation 30, we present a simplified model to understand theeffect of the dielectric decrement.Since ΨD > 0, we assume that z+c+(0) ≪ z−c−(0) and

==

− 0cx x

dd 0

. These assumptions physically imply that the

majority of the repelled ions have been depleted and that thedielectric decrement leads to a saturation of the counterions atthe surface, and thus the gradient of the counterion vanishes.These assumptions allow us to simplify Gauss’s law at x = 0 as

ε ε ψ

ε ε ψ

= − |

⇒ ≈ |

±=

− − + + =

=− − =

⎜ ⎟⎛⎝

⎞⎠x

cx

e z c z c

xz ec

dd

( )dd

( )

dd

xx

xx

00

0

0 s

2

20

0(31)

Next, we invoke the chemical potential equality μ−(x) =μ−(∞) with a± → 0 in equation 26 and 27 to get

ε γ ψ ψ− + ≈−−

−

+

⎛⎝⎜

⎞⎠⎟x

z e k Tc xz c2

dd

ln( )

002

D B0 (32)

We differentiate equation 32 with respect to x and utilize

equation 31 to evaluate c−(0) andψ

=x x

dd 0

asεγ

≈−−

c (0)2

s

(33a)

ε ε≈−c( (0)) /2s (33b)

ψ ψψ

ε γ≈ −

− εγ

=

−

−

− +⎜ ⎟⎛⎝

⎞⎠( )

x

z e k Tdd

sgn( )2 ln

x

z c

0D

D B 2

0

s

0

(33c)

Figure 7. Effect of asymmetry in electrolyte valence on dielectricdecrement forΨ(X), as given by the solution of equation 30 forΨD = 5.(a) Effect of γ+c0 with z+ = 1, z− = 3, and γ+/γ− = 1. (b) Effect of z+ and z−with γ+c0 = 3 and γ+/γ− = 1. (c) Effect of γ+/γ− with γ+c0 = 3 with z+ = 1and z− = 3. εs = 80 is assumed for all calculations.

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Nondimensionalizing equation 33c, we arrive at

ε

γΨ ≈ − Ψ

Ψ −

+

εγ

= + + − −

− − +⎜ ⎟⎛⎝

⎞⎠( )

X z z z cdd

sgn( )2 ln

( )X

z z c

0D

s D1

2

0

s

0

(34)

We emphasize that equation 34 is an approximation and

assumes that z+c+(0)≪ z−c−(0) and ==

− 0cx x

dd 0

. Moreover, the

argument inside the square root needs to be positive, or

Ψ > εγ− − +( )ln

z z cD1

2s

0, and thus the relation is only applicable for

large ΨD. However, there are useful insights to be gained fromequation 34. The equation suggests that increasing γ−c0 leads to amore gradual decay in Ψ, consistent with the numericalobservations in Figure 7a.The variation ofΨ with z+ and z− is presented in Figure 7b for

γ+c0 = 3 and γ+/γ− = 1. From equation 34, we learn that Ψ

=X X

dd 0

is

largest for z+ = 1, z− = 1, followed by z+ = 1, z− = 3, and z+ = 2, z−= 1. This trend is consistent with the results shown in Figure 7b.However, equation 34 is only valid at X = 0 and the variation inΨfor other values of X are not captured in equation 34.Nonetheless, the results in Figure 7b and equation 34 clearlyshow that Ψ(X) depends on the cation and anion valence. Next,we discuss the effect of γ+/γ− with γ+c0 = 3, z+ = 1, and z− = 3. Weobserve in Figure 7c that an increase in γ+/γ− leads to a morerapid decay in Ψ with X, although the difference is relativelyminor. This observation is consistent with the prediction ofequation 34.Charge Accumulated. We now discuss the dependence of

the charge accumulated Q onΨD. Since the dielectric decrementsaturates the counterion concentration, an increase in γ+c0reduces Q; see Figure 8a where the trends are presented for z+= 1, z− = 3, and γ+/γ− = 1. By utilizing equation 33, we predict

ε

γε

γ≈ Ψ

Ψ −

+Ψ >

εγ

+ + − − − − +

− − +⎜ ⎟⎛⎝

⎞⎠ ⎛

⎝⎜⎞⎠⎟

( )Q

z z z c z z csgn( )

ln

2 ( )for

1ln

2

z z cD

s D1

2

0D

s

0

s

0

(35)

Equation 35 shows that increasing γ− reduces Q, consistentwith the trends observed in Figure 8a. The dependence of Q onz+ and z− is presented in Figure 8b.We find qualitative agreementbetween the computed values from equation 30 with thepredictions of equation 35. Figure 8c shows the variation ofQ onγ+/γ− for γ+c0 = 3, z+ = 1, and z− = 3. The results suggest a largerQfor a smaller γ−, qualitatively consistent with the prediction of

equation 35. The disagreement between the solutions fromequations 30 and 35 occur since the approximation of z+c+(0)≪z−c−(0) and ≈ ε

γ−−

c (0)2

s is more accurate for Ψ = (10)D .

Nonetheless, equation 35 provides a convenient analyticalexpression to infer the dependence of different parameters on Q.

Capacitance. We now focus on the dependence ofcapacitance on ΨD. Figure 9a shows the dependence of

on γ+c0 for z+ = 1, z− = 3. and γ+/γ = −1. First, we note that forγ+c0 = 0 (i.e., no dielectric decrement), has only one localminimum at ΨD = ΨD,min < 0. This response has been describedin detail in the previous section; see equation 24.We observe thatincrease in γ+c0 leads to a decrease in . In addition, with finitedielectric decrement, we start observing a maximum in for ΨD= ΨD,max, leading to the camel shape curves. For very large γ+c0,we find thatΨD,min disappears and only one of the maximaΨD,maxremains, leading to a bell shape curve, similar to the finite ion-sizeeffects (see Figure 4a). However, increase in γ+c0 also influencesthe at ΨD = 0 unlike the increase in a+

3c0 for finite ion-sizeeffects; see Figure 4a.To understand the capacitance response more quantitatively,

we build on our simplified model for dielectric decrement. Here,we assume that for ΨD > 0, z+c+(0) ≪ z−c−(0) and ≈ ε

γ−−

c (0)2

s

(see equation 33), and thus by extension for ΨD < 0, z−c−(0)≪z+c+(0) and ≈ ε

γ++

c (0)2

s . On the basis of these assumptions, we

previously derived Q(ΨD) (see equation 35) and = ΨQd

d Dis

thus calculated as

ε γε

γ

εγ

ε γε

γε

γ

≈ + |Ψ | −

− Ψ >

≈ + |Ψ | −

Ψ >

−− + − +

+ + −

+ + −

−+ + − −

− − +

− − +

⎛⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

z z z cz z c

z z c

z z z cz z c

z z c

8 ( )1

ln2

for

1ln

2,

8 ( )1

ln2

for

1ln

2

s2 1

0 Ds

0

Ds

0

s2 1

0 Ds

0

Ds

0 (36)

We investigate the effect of z+ and z− on for γ+c0 = 3 andγ+/γ− = 1 in Figure 9b. We find that changes in z+ and z− createsasymmetry in the capacitance curves. We are able to qualitativelycapture the asymmetry and relative positions for differentcombinations of z+ and z− in equation 36. However, equation 36does not predict a maximum and suggests that is a strictly

Figure 8. Effect of asymmetry in electrolyte valence on dielectricdecrement for Q(ΨD), as given by the solution of equations 30 and 35.The solid lines represent results from equation 30, and the dotted linesdenote results from equation 35. (a) Effect of γ+c0 with z+ = 1, z− = 3, andγ+/γ− = 1. (b) Effect of z+ and z−with γ+c0 = 3 and γ+/γ− = 1. (c) Effect ofγ+/γ− with γ+c0 = 3 with z+ = 1 and z− = 1. εs = 80 is assumed for allcalculations.

Figure 9. Effect of asymmetry in electrolyte valence on dielectricdecrement for Ψ( )D , as given by the solution of equations 30 and 36.The solid lines represent results from equation 30, and the dotted linesdenote results from equation 36. (a) Effect of γ+c0 with z+ = 1, z− = 3, andγ+/γ− = 1. (b) Effect of z+ and z−with γ+c0 = 3 and γ+/γ− = 1. (c) Effect ofγ+/γ− with γ+c0 = 3 with z+ = 1 and z− = 1. εs = 80 is assumed for allcalculations.

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decreasing function with |ΨD|. This discrepancy between thecomputed results from equation 30 and approximate results fromequation 36 arise due since the assumptions of cation and anionconcentration are less accurate for |Ψ | ≲ (1)D . In Figure 9c, wedescribe the dependence of withΨD for different γ+/γ−, γ+c0 =3, z+ = 1, and z− = 3. We find that decreasing γ− increases forΨD > 0, whereas keeping γ+ constant collapses curves forΨD < 0,consistent with the prediction of equation 36.Salt Uptake. An overview of the effect of dielectric

decrement on dimensionless salt uptake is provided in Figure10. First, we focus on the effect of γ+c0 for z+ = 1, z− = 3, and γ+/γ−

= 1. We observe in Figure 10a that an increase in γ+c0 decreasesαL/λD. This response is expected since an increase in dielectricdecrement leads to a larger saturation of ion concentration, andthus less salt is absorbed in the diffuse layer. Figure 10b presentsthe variation of αL/λD on z+ and z−. On the basis of our analysisfor finite ion-size effects, here also, we expect αL/λD to be lowestfor z+ = 1, z− = 3 since saturation would occur at the smallestvalue of ΨD. However, we find a different trend in Figure 10b.Though αL/λD for z+ = 1, z− = 3 does become lowest for largeΨD, it is maximum for small ΨD. Furthermore, αL/λD for z+ = 2,z− = 1 is higher than z+ = 1, z− = 1, in contrast to finite ion-sizeeffects. These differences arise due to trends in Ψ

Xdd

; see equation

30a and Figure 7b.The effect of γ+/γ− on αL/λD is summarized in Figure 7c for

γ+c0 = 3, z+ = 1, and z− = 1. The dimensionless salt uptake αL/λDincreases for decrease in γ− since for ΨD > 0, the saturationconcentration of anions is larger; see equation 33. Therefore, alarger amount of salt can be taken up by the double layer.Scaling Analysis.We now develop scaling relations to better

analyze the effect of the dielectric decrement. To estimate thelocation of extrema in versus ΨD and their dependence on thevalues of z+ and z−, we invoke physical arguments. For ΨD > 0,negative ions will be attracted to the surface and the condition for

a local maximum implies = εγ−−

( )c2

s . In contrast, for ΨD < 0,

the condition for a local maximum becomes = εγ++

( )c2

s . Thus,

assuming the c± follows the Boltzmann distribution, we get

εγ

εγ

− Ψ = Ψ ≤

Ψ = Ψ ≥

− ++

+ −−

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

z c z

z c z

exp( )2

for 0,

exp( )2

for 0,

0 D,maxs

D

0 D,maxs

D(37)

or

εγ

εγ

− Ψ = Ψ ≤

Ψ = Ψ ≥

+−

− +

−−

+ −

⎛⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟⎞⎠⎟⎟

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟⎞⎠⎟⎟

zz c

zz c

ln2

for 0,

ln2

for 0

D,max1 s

0D

D,max1 s

0D

(38)

We hypothesize that to predict a local minimum follows thesame argument as before; see equation 24. We summarize ourresults from the scaling analysis in Figure 11. We find that bothequations 38 and 24 are in good agreement with the computedvalues.

In this section, we evaluated the influence of electrolytevalence on the dielectric decrement effect. Though we analyzedthe case of a linear dielectric decrement, it is straightforward toextend this analysis to the case of nonlinear dielectric decrement.We refer the readers to ref 32 for more details.

Physical Significance. We now present physical argumentsto explain the significance of electrolyte valence on the dielectricdecrement, which relates to the decrease in dielectric constantdue to reduction in orientational polarizability. Simply put, adecrease in dielectric constant relates to reduction in the ability ofthe electrolyte to accumulate charge. The dielectric constantdecreases with increase in ion concentration, and in this article,we assume a linear decrement; see equation 26. When theelectrolyte comes in contact with a charged surface, due toelectrostatic attraction, the concentration of counterionsincreases closer to the surface. Consequently, the dielectricconstant, and the ability to store charge, decrease closer to thesurface. For a large potential drop across the double layer, thesetwo effects are comparable and result in saturation of thecounterions near the surface; see equation 33. It might appearthat this effect is very similar to those from finite ion size wherethe concentration of counterions also saturates, and thusequivalent expressions can be derived by replacing the maximumion concentration

−a1

3 with εγ−2s and

+a1

3 with εγ+2s . However, upon

comparison of equations 20 and 35, we note two differences, (i) areduction by a factor of 2 and (ii) an apparent decrease in ΨD.The reduction by a factor of 2 occurs because the dielectricconstant is reduced by a factor of 2 at the surface; see equation 33.On the other hand, the apparent decrease in ΨD occurs becausewhen the dielectric decrement is included, the energy stored inelectric field also varies with ion concentration. Therefore, tokeep the double layer in equilibrium, the electric field energynegates the potential energy, leading to a smaller effectivepotential drop across the double layer.

Figure 10. Effect of asymmetry in electrolyte valence on dielectricdecrement for α(ΨD), as given by the solution of equation 30. (a) Effectof γ+c0 with z+ = 1, z− = 3, and γ+/γ− = 1. (b) Effect of z+ and z−with γ+c0= 3 and γ+/γ− = 1. (c) Effect of γ+/γ−with γ+c0 = 3, z+ = 1, and z− = 1. εs =80 is assumed for all calculations.

Figure 11. Scaling analysis of dielectric decrement effects. Comparisonof computedΨD,max with equation 38 for (a)ΨD < 0 and (b)ΨD > 0. (c)Comparison of computed ΨD,min with equation 24. The red points arecases with the camel shape (i.e., two local maxima and one localminima), and the blue points are cases with the bell shape i.e., one localminimum); see Figure 9. For computations, 0.25≤ γ+c0≤ 3, 1≤ γ+/γ− ≤4, 1 ≤ z+ ≤ 3, and 1 ≤ z− ≤ 3. εs = 80 is assumed for all calculations.

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The effect of asymmetry in electrolyte valence is nontrivial, asshown in Figure 8b and Figure 9b. We observe that the effect ofelectric field energy dominates for small potential drops such thatQ for z+ = 1, z− = 1 is lower thanQ for z+ = 1, z− = 3, unlike Figure3b. Furthermore, inclusion of valence asymmetry leads toasymmetry in capacitance values, see Figure 9b, similar to thecase with finite ion size. Therefore, regardless of which effect (thefinite ion size or the dielectric decrement) dominates, theinclusion of valence asymmetry leads to asymmetry incapacitance values. As noted before, the asymmetry incapacitance is valuable since capacitance influences the chargestorage capacity and time scale of double layer formation.Therefore, for applications such as capacitive deionization,3,45 itwill be crucial to account for valence asymmetry since itsignificantly impacts the process variables.In summary, dielectric decrement leads to the counterion

saturation, the decrease in dielectric constant at the surface, andthe decrease in the effective potential drop across the doublelayer due to variation in electric field energy with ionconcentration. The effect of valence is nontrivial, especially forsmall ΨD, when the effect of variation in electric field energy isdominant. Furthermore, an asymmetry in electrolyte valenceresults in asymmetric double layer properties. Next, we analyzethe effect on electrolyte valence of ion-ion correlations on doublelayer properties.

■ ION−ION CORRELATIONSIn this section, we consider the combined effect of ion−ioncorrelations and the steric effect. For simplicity, we do notconsider the effect of the dielectric decrement and Stern layer inthis section. For ion−ion correlations, we build on the work ofBazant, Storey, and Kornyshev,23,35 and we refer the readers tothese references for the derivation of the modified Gauss law,which is given as

ε ε ψ ψ− = −+ + − −

⎛⎝⎜

⎞⎠⎟l

x xz ec z ec

dd

dd0 s c

24

4

2

2(39)

where lc is the correlation length that quantifies the effect of ion−ion correlations. Equation 39 is to be solved with boundary

conditions ψ(0) = ψD, =ψ

=0

x x

dd 0

3

3 , and ψ(∞) = 0. We

nondimensionalize equation 39 with =λ

X x

D, =+

+

−n c

z c0,

=−−

+n c

z c0, =

λL l

cc

Dto arrive at

Ψ − Ψ =−+

+ −

+ −L

X Xn nz z

dd

ddc

24

4

2

2(40)

where n± are given by equation 7. We numerically integrateequation 40 to find Ψ(X) and n±(X). From the numericalintegration, we evaluate the dimensionless charge and salt uptakeas

∫=−+

∞ − +

+ −

⎛⎝⎜

⎞⎠⎟Q

n nz z

Xd0 (41a)

∫αλ

=++

−∞ − + + −

+ −

⎛⎝⎜

⎞⎠⎟

L z n z nz z

X1 dD 0 (41b)

Lastly, we can also numerically evaluate the dimensionless

capacitance as = ΨQd

d D. We also note that analytical solutions

of equation 40 are only possible for |ΨD| ≪ 1, and we refer thereaders to ref 35 for more details.

Potential Distribution.We first the discuss the effect of Lc =lcλD

−1 on the potentialΨ(X). For a+3c0 = 0.2, a+/a− = 1, z+ = 1, andz− = 3, we find that for small values of Lc (i.e., Lc ≤ 0.5), thevariations in Ψ(X) are not significant. However, for larger valuesof Lc, we start to see oscillations inΨ(X), as previously describedby Bazant and co-workers.23,35

Similar to finite ion-size and dielectric decrement effects, theeffect of asymmetry in electrolyte valence is significant for ion−ion correlation effects. Figure 12b shows the effect of change in z+

and z− for Lc = 1, a+3c0 = 0.2, and a+/a− = 1. We find that

combination of cation and anion valences also influence theΨ(X) and the degree of oscillations. The parameters a+

3c0 anda+/a− can also be varied. However, these effects have alreadybeen discussed in detail in the previous sections, and we expectthe qualitative trend to remain the same even with the inclusionof ion−ion correlations.

Charge Accumulated.The oscillations inΨ(X) due to ion−ion correlations impact the charge accumulated inside the doublelayer. The effect of Lc onQ for a+

3c0 = 0.2, a+/a− = 1, z+ = 1, and z−= 3 is provided in Figure 13a. We expect oscillations in Ψ(X) to

be larger for larger value of Lc. Therefore, Q decreases withincrease in the value of Lc. However, we emphasize that the shapeof the curves are similar to the scenario without ion−ioncorrelations and for Lc ≤ 0.5. Equation 20 can be used as a first-order approximation of Q, as shown in Figure 13a.

Figure 12. Effect of asymmetry in electrolyte valence on ion−ioncorrelations. Variation of Ψ with X as given by the solution of equation40 forΨD = 5. (a) Effect of Lc with a+

3c0 = 0.2, a+/a− = 1, z+ = 1, and z− =3. (b) Effect of z+ and z− with Lc = 1, a+

3c0 = 0.2, and a+/a− = 1.

Figure 13. Effect of asymmetry in electrolyte valence on ion−ioncorrelations for Q(ΨD), as given by the solution of equations 40 and 41,plotted here in solid lines. We also represent the approximate solution asgiven by equation 20 for Lc = 0 with dotted lines. (a) Effect of Lc witha+3c0 = 0.2, a+/a− = 1, z+ = 1, and z− = 3. (b) Effect of z+ and z− with Lc =1, a+

3c0 = 0.2, and a+/a− = 1.

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The effect of z+ and z− onΨ(X) for Lc = 1, a+3c0 = 0.2, a+/a− = 1is provided in Figure 13b. The trends inQ are qualitatively similarto the results discussed in Figure 3b and equation 20. However,since Lc = 1, the magnitude of Q is slightly lower; see Figure 13a.Capacitance. We now discuss the effect of ion−ion

correlations on the capacitance . Since measures the amountof charge per unit potential drop, we expect the ion−ioncorrelations to have a similar impact on as it has on Q. Figure14a describes the variation of for different Lc and a+

3c0 = 0.2,a+/a− = 1, z+ = 1, and z− = 3. We note that since a+

3c0 = 0.2,

versus ΨD is a bell shaped curve with the maximum at ΨD > 0,similar to Figure 4a. Upon increase in the value of Lc, the shape ofthe curve does not change. However, the values of decreasewith increase in ΨD, because an increase in Lc leads tofluctuations in charge density profiles, which in turn leads to asmaller amount of charge stored. We note that for Lc ≤ 0.5, thechange in Ψ( )D is not significant and we can use equation 21 toapproximate Ψ( )D , especially for large |ΨD|, as shown.As mentioned previously, Lc does not significantly influence

the shape of versus ΨD. Therefore, the effect of z+ and z−should be similar to the results reported in Figure 4b, which isindeed the case as shown in Figure 14b where values arecalculated for different z+ and z−with Lc = 1, a+

3c0 = 0.2, and a+/a−= 1. As expected, we find that the shape of versusΨD remains abell shape. However, the position of a local maximum shifts dueto the change in z+ and z−. For z+ = 1 and z− = 1, versus ΨD issymmetric and has a maximum at ΨD = 0. However, for z+ = 2and z− = 1, the maximum shifts to ΨD < 0, similar to the resultsshown in Figure 4b. We also show that we can approximate thecapacitance for large ΨD using equation 21.Salt Uptake. Due to the oscillations in potential arising from

ion−ion correlations, both charge Q and capacitance decreasewith increase in the correlation length Lc since a larger amount ofrepelled ions migrate inside the double layer. Therefore, theseoscillations could result in an increase in the total amount ofdimensionless salt uptake, which is in fact what we observe inFigure 15a where αL/λD is measured for different Lc with a+

3c0 =0.2, a+/a− = 1, z+ = 1, and z− = 3. Here, we see the increase inαL/λD is relatively insignificant for Lc ≤ 0.5. However, for largervalues of Lc, we find that the increase in αL/λD is quite significant.We now discuss the effect of z+ and z− on αL/λD with Lc = 1,

a+3c0 = 0.2, and a+ = 1/a− = 1, see Figure 15b. The trends of z+ andz−with the ion−ion correlations is the same as without finite ion-

size effects; see Figure 5b. For instance, when we compare z+ = 1,z− = 3 with z+ = 1, z− = 1. SinceΨD > 0,ΨD at which ions saturateis lower for z+ = 1 and z− = 3 than z+ = 1 and z− = 1 and thus,αL/λD is lower for z+ = 1 and z− = 3. In other words, the trends ofα(ΨD) remain similar even though αL/λD is higher due to ion−ion correlations.

Scaling Analysis. As mentioned above, the effect of ion−ioncorrelations typically do not significantly influence the shape ofQ, and αL/λD but does affect the magnitude of theseproperties. Therefore, we predict that ΨD,max and ΨD,min followequations 23 and 24. We summarize a comparison of predictedvalues using equations 23 and 24, and computed values in Figure

16, where our predictions are in good agreement with thecomputations. We note that though the location of ΨD,max andΨD,min is similar to the scenarios without ion−ion correlations,the magnitude of Q and are different upon inclusion of ion−ion correlations.

Physical Significance. As noted before, the inclusion ofion−ion correlations leads to oscillations in charge densityprofiles. Therefore, the values of Q and decrease with increasein correlation length since the repelled ions are present in thedouble layer at higher concentration. The effect of z+ and z− issimilar to finite ion size and dielectric decrement (i.e., theasymmetry in valence leads to asymmetry in double layerproperties). As shown in Figure 14b, even when ion−ioncorrelations are significant, the asymmetry in electrolyte valencebreaks the symmetry of the capacitance, which underscores theneed to incorporate the effect for more accurate predictions.

Figure 14. Effect of asymmetry in electrolyte valence on ion−ioncorrelations for Ψ( )D , as given by the solution of equations 40 and 41,plotted here in solid lines. We also represent the approximate solution asgiven by equation 21 for Lc = 0 with dotted lines. (a) Effect of Lc witha+3c0 = 0.2, a+/a− = 1, z+ = 1, and z− = 3. (b) Effect of z+ and z− with Lc =1, a+

3c0 = 0.2, and a+/a− = 1.

Figure 15. Effect of asymmetry in electrolyte valence on ion−ioncorrelations. Variation of αL/λD with ΨD as given by numericalintegration of solution of equation 40 and 41. (a) Effect of Lc with a+

3c0 =0.2, a+/a− = 1, z+ = 1, and z− = 3. (b) Effect of z+ and z− with Lc = 1, a+

3c0= 0.2, and a+/a− = 1.

Figure 16. Scaling analysis of ion−ion correlation effects. Comparisonof computedΨD,max with equation 23 for (a)ΨD < 0 and (b)ΨD > 0. (c)Comparison of computed ΨD,min with equation 24. The red points arecases with the came shape (i.e., two local maxima and one local minima),and the blue points are cases with the bell shape (i.e., one localmaximum). For computations, 0.02 ≤ a+

3c0 ≤ 0.2, 1 ≤ a+3/a−

3 ≤ 4, 1 ≤ z+≤ 3, 1 ≤ z− ≤ 3, and 0.1 ≤ Lc ≤ 5.

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■ SUMMARY AND OUTLOOK

In this article, we studied the effect of asymmetry in electrolytevalence on finite ion size, dielectric decrement, and ion−ioncorrelation effects. For finite ion size, we considered the scenariowith different cation and anion diameters. We analyticallyderived expressions for surface charge density Q(ΨD),capacitance Ψ( )D and dimensionless salt α(ΨD), and numeri-cally solved for the variation of the potential distribution Ψ(X).These results have been discussed in equations 12, 13, and 19. Tothe best of our knowledge, these are the most general expressionsfor diffuse-layer properties with steric effects. Wherever possible,we derived simplified expression for Q(ΨD) and Ψ( )D , andthese results are provided in equations 20 and 21. Furthermore,we developed scaling relations for local extrema such as ΨD,maxand ΨD,min observed in the behavior of versus ΨD, and theseresults are presented in equations 23 and 24.For the dielectric decrement, we considered the scenario of a

linear dielectric decrement. We derived the fundamentalequations that enable us to numerically solve for Ψ(X),Q(ΨD), Ψ( )D , and α(ΨD) for non z:z electrolytes. Theseresults are presented in equation 30. We also present simplifiedexpressions for Q(ΨD) and Ψ( )D in equations 35 and 36. Thescaling relations for dielectric decrement results are presented inequations 38 and 24.Lastly, for ion−ion correlations, we numerically solve equation

40 to obtain Ψ(X), Q(ΨD), Ψ( )D , and α(ΨD). Here, weemphasize that equations 20 and 21 provide a good startingapproximation for small values of the correlation length Lc. Wealso show that scaling relations for finite ion-size effects (i.e.,equations 23 and 24), are applicable even after the inclusion ofion−ion correlations.Our findings are important for mean field double layer theory,

diffuse-charge dynamics, and electrokinetics. We show thatasymmetry in cation and anion valences play an important role indiffuse-layer properties and should be included in calculations ofQ(ΨD), Ψ( )D , and α(ΨD). Going forward, we expect our resultsto be also useful in time-dependent problems. For instance, sincez+ and z− influence α(ΨD), asymmetry in electrolyte valence willalso influence the charging−discharging dynamics of theelectrolyte between electrodes. Charging−discharging dynamics,which is relevant for the design of energy storage devices,8,9 isespecially important at large |ΨD| where finite ion-size anddielectric decrement effects are pronounced, and valenceasymmetry effects can be potentially exploited to control theseeffects. We also expect that our description of chemical potentialscan be directly used to derive modified Nernst-Planck equationsin future studies. Moreover, the dimensionless salt uptake α(ΨD)could influence the adsorption quality and time scale for toxicheavy metal ion separations in capacitive deionization.3,45

We note that though we incorporated several modifications tothe classical Poisson−Boltzmann model for valence asymmetricelectrolytes, there are some limitations in our analysis, and futurework should focus on overcoming these limitations. For instance,we considered a simple hard-sphere model to include the stericeffects, and future work can focus on extending the analysis ofasymmetry in electrolyte valence onmore accurate approaches toinclude steric effects.25,48,49 Further, we assume a linear dielectricdecrement model. Though the linear model works reasonably forelectrolyte concentrations up to a few molars,47 it is possible toextend our results to the scenario of nonlinear decrement, andthe future efforts should be directed in this direction.32 Our

treatment of many body interactions in ion−ion correlations islimited to the derivative of the potential, and more sophisticatedmodels could be considered in subsequent research.50

Furthermore, a direct comparison between mean-field theoryand sophisticated models such as density functional theory,Monte Carlo simulations, and molecular dynamics simula-tions,48,51,52 will help test the validity of the mean-field resultsand should be considered by researchers in the field. Lastly, wedo not consider important effects such as specific forces53 andhydration forces,54,55 which have been a topic of recent interestand should also be included in future studies.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: hastone@princeton.edu.ORCIDAnkur Gupta: 0000-0003-3474-9522Howard A. Stone: 0000-0002-9670-0639NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe are thankful for support from the Andlinger Center forEnergy and the Environment at Princeton University and theNational Science Foundation (CBET - 1702693). The authorsthank Pawel J. Zuk and Guang Chen for useful discussions. Wealso thank the anonymous referees for very helpful feedback thatimproved the paper.

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