Chemistry Publications Chemistry

2011

A Molecular Debye-Hückel Theory and ItsApplications to Electrolyte SolutionsTiejun XiaoIowa State University

Xueyu SongIowa State University, xsong@iastate.edu

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A Molecular Debye-Hückel Theory and Its Applications to ElectrolyteSolutions

AbstractIn this report, a molecular Debye-Hückel theory for ionic fluids is developed. Starting from the macroscopicMaxwell equations for bulk systems, the dispersion relation leads to a generalized Debye-Hückel theory whichis related to the dressed ion theory in the static case. Due to the multi-pole structure of dielectric function ofionic fluids, the electric potential around a single ion has a multi-Yukawa form. Given the dielectric function,the multi-Yukawa potential can be determined from our molecular Debye-Hückel theory, hence, theelectrostatic contributions to thermodynamic properties of ionic fluids can be obtained. Applications tobinary as well as multi-component primitive models of electrolyte solutions demonstrated the accuracy of ourapproach. More importantly, for electrolyte solution models with soft short-ranged interactions, it is shownthat the traditional perturbation theory can be extended to ionic fluids successfully just as the perturbationtheory has been successfully used for short-ranged systems.

KeywordsElectrolytes, Electrostatics, Thermodynamic properties, Dielectric function, Perturbation theory

DisciplinesChemistry

CommentsThe following article appeared in Journal of Chemical Physics 135 (2011): 104104, and may be found atdoi:10.1063/1.3632052.

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A molecular Debye-Hückel theory and its applications to electrolyte solutionsTiejun Xiao and Xueyu Song Citation: The Journal of Chemical Physics 135, 104104 (2011); doi: 10.1063/1.3632052 View online: http://dx.doi.org/10.1063/1.3632052 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/135/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quadrupole terms in the Maxwell equations: Debye-Hückel theory in quadrupolarizable solvent and self-salting-out of electrolytes J. Chem. Phys. 140, 164510 (2014); 10.1063/1.4871661 Symmetry effects in electrostatic interactions between two arbitrarily charged spherical shells in the Debye-Hückel approximation J. Chem. Phys. 138, 074902 (2013); 10.1063/1.4790576 Debye–Hückel theory for mixtures of rigid rodlike ions and salt J. Chem. Phys. 134, 074111 (2011); 10.1063/1.3552226 Dressed ion theory of size-asymmetric electrolytes: Effective ionic charges and the decay length of screenedCoulomb potential and pair correlations J. Chem. Phys. 122, 064502 (2005); 10.1063/1.1843811 Integral equation theory of flexible polyelectrolytes. I. Debye–Hückel approach J. Chem. Phys. 109, 4659 (1998); 10.1063/1.477071

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THE JOURNAL OF CHEMICAL PHYSICS 135, 104104 (2011)

A molecular Debye-Hückel theory and its applicationsto electrolyte solutions

Tiejun Xiao and Xueyu Songa)

Department of Chemistry, Iowa State University, Ames, Iowa 50011, USA

(Received 8 June 2011; accepted 12 August 2011; published online 9 September 2011)

In this report, a molecular Debye-Hückel theory for ionic fluids is developed. Starting from themacroscopic Maxwell equations for bulk systems, the dispersion relation leads to a generalizedDebye-Hückel theory which is related to the dressed ion theory in the static case. Due to themulti-pole structure of dielectric function of ionic fluids, the electric potential around a single ionhas a multi-Yukawa form. Given the dielectric function, the multi-Yukawa potential can be deter-mined from our molecular Debye-Hückel theory, hence, the electrostatic contributions to thermody-namic properties of ionic fluids can be obtained. Applications to binary as well as multi-componentprimitive models of electrolyte solutions demonstrated the accuracy of our approach. More impor-tantly, for electrolyte solution models with soft short-ranged interactions, it is shown that the tradi-tional perturbation theory can be extended to ionic fluids successfully just as the perturbation the-ory has been successfully used for short-ranged systems. © 2011 American Institute of Physics.[doi:10.1063/1.3632052]

I. INTRODUCTION

Coulomb interaction is ubiquitous in nature, rangingfrom physical, chemical to biological systems, such asplasma, electrolyte solutions, and protein-DNA complexes.The long-ranged nature of Coulomb interaction makes itdifficult to tackle compared with short-ranged interactions.On the other hand, the microscopic neutrality conditionnaturally leads to the Debye screening, which dramaticallysimplified the description of systems with Coulomb inter-actions as demonstrated in the Debye-Hückel (DH) theory.1

This traditional treatment is based on the Poisson equationin combination with a mean field approximation for theradial distribution function of ionic species, which lead tothe familiar Poisson-Boltzmann (PB) equation. Since PBequation is highly nonlinear, analytical solutions are hard toattain except some simple geometries, and even numericalsolutions of PB are highly nontrivial. In the low couplinglimit, linearization of PB equation leads to the Debye-Hückeltheory1 of electrolyte solutions. The DH theory is elegant inthe sense that it reveals the universality of Debye screeningin systems with Coulomb interactions. The electric potentialin the traditional DH theory has one Yukawa term, and thescreening parameter is simply determined by the Debyescreening length, which leads to straightforward calculationsof thermodynamic properties contributed from the Coulombinteractions.

Due to the simplicity of the DH theory, it became the hall-mark of theoretical treatments for Coulomb systems.2 Sincethe traditional DH is valid only for low coupling limit, nat-urally there are numerous works to improve this approach.For example, Kirkwood and Poirier3 utilized the conceptof potential of mean force, which could be expanded in a

a)Author to whom correspondence should be addressed. Electronic mail:xsong@iastate.edu.

power series in terms of a charging parameter, and the co-efficient of the expansion of potential of mean force couldbe derived in a systematic way using the method of semi-invariants. They demonstrated that the DH theory is a rig-orous statistical mechanics theory in the low coupling limit.Outhwaite and co-workers4–7 had provided a modified PBapproach by considering the so-called fluctuation potential,which could be solved using superposition principle. Attardand co-workers8–10 had improved the traditional DH in a self-consistent way, with the Debye screening length replacedby an effective screening length which could be determinedby the Stillinger-Lovett relation.11 A particular relevant ap-proach to the current work is the dressed ion theory (DIT) byKjellander and Mitchell,12, 13 which keeps the Yukawa formof electrical potential as in the traditional DH theory by intro-ducing renormalized charges and effective screening lengths,which are determined from the dielectric function of the sys-tem. Thus, a formally exact theory of Coulomb systems canbe formulated using a combination of linearized PB solutions.A comprehensive review of the DH theory and its extensionscan be found in Ref. 14.

Essentially, all of these works are for the static caseand there is no clear route to extend to dynamical cases,where the frequency-dependent screening may play impor-tant roles such as protein-protein interaction models in elec-trolyte solutions15 and solvation dynamics in ionic fluids.16

With these motivations in mind, we began to develop not onlystatic extension of the traditional DH theory, but also system-atic extension to the dynamical case.17 In our previous work,it was demonstrated that the dispersion relation of the dielec-tric function naturally leads to the effective screening lengths,which reduced to the screening lengths in the dressed ion the-ory for the static case. Furthermore, such dynamical screen-ing lengths provide a simple theory for solvation dynamics inionic fluids.17

0021-9606/2011/135(10)/104104/14/$30.00 © 2011 American Institute of Physics135, 104104-1

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104104-2 T. Xiao and X. Song J. Chem. Phys. 135, 104104 (2011)

There are two important concepts in the dressed iontheory,12, 13 namely, roots of the dielectric function lead tothe effective screening lengths (see also Ref. 32), and theeffective charge of the dressed ion reflects the nonlinearresponse of the environment to the tagged ion. The dielectricfunction is a macroscopic property of the bulk system andcould be obtained from molecular simulations or experimen-tal measurements, while the effective charge is a microscopicproperty which is related to the local molecular correlations.For applications of the dressed ion theory, the radial distribu-tion function of the system is necessary to find the effectivecharges and the effective screening lengths, which contain theinformation that is missing in the traditional DH theory. Forcomplex systems, such as proteins in an electrolyte solution,the radial distribution functions of the whole system eitherfrom molecular simulations or from the integral equationsare difficult to obtain. Furthermore, as the electric potentialis expressed in terms of an asymptotic series of Yukawaterms, accurate evaluation, which is crucial for accuratethermodynamic properties, requires summation of many suchterms.

Our strategy in this report is to develop a molecularDH theory, which keeps the simplicity of the traditional DHtheory and at the same time thermodynamic properties ofproteins can be evaluated accurately given the macroscopicdielectric function of the electrolyte solution and a molecularmodel of the protein. To this end, we have found that for aprimitive model of electrolyte solutions, general relationsbetween the parameters of multi-Yukawa potential formand the dielectric function can be established, where thefirst two relations are just the charge neutrality conditionand the Stillinger-Lovett relation. From these relations, wecan determine the parameters of Yukawa potential forms,and then accurate excess thermodynamic properties couldbe determined. Our theory is also applicable to models ofelectrolyte solutions with soft short-ranged interactions byintroducing effective hard sphere sizes using perturbationtheory. It is shown that the widely successful perturbationtheory used for short-ranged interaction systems can beextended to Coulomb systems as long as our molecularDH theory is used to calculate the Coulomb contribution tothermodynamic properties. Applications to 1:1, 2:2, and 2:1primitive models as well as models of electrolyte solutionswith inverse power potential or Lennerd-Jones potential areused to demonstrate the validity of our approach.

The paper is organized as follows. In Sec. II, the con-nection between the dispersion relations from the Maxwellequations and an extended DH theory is established. InSec. II, a molecular DH theory is introduced to calculate themean electric potential of a tagged ion in terms of multi-Yukawa forms. Calculations of the electrostatic contributionto thermodynamic properties from the mean electric poten-tial are discussed in Sec. III. Applications to primitive modelsand models with soft short-ranged interactions are presentedin Sec. IV. Some concluding remarks are given in Sec. V. Forthe static case, a rigorous statistical mechanics derivation andconnection with the extended DH theory is presented in theAppendix to clarify the connection between our dispersionrelation formulation and the dressed ion theory.

II. DISPERSION RELATIONS AND AN EXTENDEDDEBYE-HüCKEL THEORY OF IONIC FLUIDS

Recently, we have developed a theoretical framework toconnect the Debye-Hückel theory to the dispersion relationsin electrodynamics.17 This connection between the dispersionrelation (the functional relationship between wave-vector andfrequency) and an extended Debye-Hückel theory will be pre-sented here. The purpose of this section is to rigorously derivean extended dynamical Poisson-Boltzmann, Eq. (14), whichis the starting point of any extended Debye-Hückel theory.

Let’s start with the Maxwell equations of a non-magneticmacroscopic system in Systeme International (SI) unit,18

∇ · B = 0,

∇ × E + ∂B∂t

= 0,

∇ · D = ρ,

∇ × B − ∂D∂t

= J, (1)

where ρ and J are the macroscopic (free) charge density andcurrent density in the system. After a Fourier transformationwith eiωt−ik·r, all quantities are transformed from X(r, t) toX(k, ω), where

X(k, ω) =∫ ∞

−∞dt

∫∫∫ ∞

−∞dr exp(iωt − ik · r)X(r, t),

and the inverse transform is defined as

X(r, t) = 1

(2π )

∫ ∞

−∞dω

1

(2π )3

×∫∫∫ ∞

−∞dk exp(−iωt + ik · r)X(k, ω).

For brevity, we omit the arguments unless it is not clear fromthe context. Then the Maxwell equations become

k · B = 0,

k × E − ωB = 0,

−ik · D = ρ,

−ik × B − iωD = J. (2)

For a conducting medium such as an ionic fluid, the constitu-tive relations according to the linear response are

D = �(k, ω) · E, (3)

where �(k, ω) = ε(k, ω) + iωσ (k, ω) is the generalized di-

electric tensor, ε(k, ω) is the dielectric tensor, and σ (k, ω) isthe conductivity tensor.19 If there is no external charge andcurrent as the induced charge and current have been includedin the generalized dielectric tensor �(k, ω), a combination ofthe equations in Eq. (2) leads to

ω2D = ω2� · E = Ek2 − k(k · E). (4)

The condition for this set of linear homogeneous equations ofE to have nontrivial solutions is the vanishing of the followingdeterminant: ∣∣∣∣k2

(I − kk

k2

)− ω2�(k, ω)

∣∣∣∣ = 0. (5)

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104104-3 Molecular Debye-Hückel theory J. Chem. Phys. 135, 104104 (2011)

This is the dispersion relation in electrodynamics,20 and I isthe rank 2 identity tensor. For an isotropic system, the longitu-dinal and transverse components of the generalized dielectrictensor are scalars,19

�(k, ω) = �t (k, ω)

(I − kk

k2

)+ �l(k, ω)

kkk2

. (6)

The dispersion relations are reduced to

k2

ω2− �t (k, ω) = 0, (7)

�l(k, ω) = 0. (8)

Now let us find the solution of Eq. (1) in terms of a scalarpotential �(r, t) and a vector potential A(r, t) , i.e.,

B(r, t) = ∇ × A(r, t), (9)

E(r, t) = −∇�(r, t) − ∂A(r, t)∂t

. (10)

Note that from the charge density equation we have

ρ = −ik · D = �l(k, ω)(k2� − ωk · A), (11)

and for the current we have

J = −ik × B − iωD = [k2I − ω2�(k, ω)] · A

+ω�l(k, ω)k� − kk · A. (12)

It is noted that the solution will be unchanged under gaugetransformations.18 For the Coulomb gauge, we have

∇ · A = 0, (13)

and then the scalar potential satisfies the Poisson equation,

k2�l(k, ω)�(k, ω) = ρ(k, ω). (14)

If the current is split into a longitudinal part Jl and a transversepart Jt , i.e., ∇ × Jl = 0 and ∇ · Jt = 0, then the vector poten-tial satisfies the following equation if the continuity equationfor the charge density is used

[k2 − ω2�t (k, ω)]A(k, ω) = Jt (k, ω). (15)

Therefore, these two potentials are decoupled, and thescalarand vector potentials are related to the longitudinal andtransverse property, respectively.

For the static case ω = 0, application of the disper-sion relation �l(k) = 0 to ionic fluids leads to extendedDebye-Hückel theories as in the DIT.12–14 In general, theroots of �l(k) = 0 are complex and they appear in pairs as�l(ω, k) = �∗

l (−ω, k).20 Define an electric potential �(r)via E(r) = −∇�(r), then the electric potential satisfies thefollowing equation:

k2�l(k)�(k) = ρ(k), (16)

which is the starting point of any extended Debye-Hückeltheory as exploited in the dressed ion theory12–14 as long asa microscopic expression of ρ(k) is given. Therefore, aboveresults Eqs. (14) and (15) provide a natural extension of the

Debye-Hückel theory to the dynamical domain. In order tofurther clarify the connection between the above formulationand the traditional Debye-Hückel theory, we will present ageneral formulation of the Debye-Hückel theory from themean electrostatic potential in the Appendix.

III. A MOLECULAR DEBYE-HüCKEL THEORY

In the Appendix, two phenomenological functions,namely, the dielectric function and the renormalized chargedensity in our approach are discussed at the microscopic level.In this section the average electric potential around a taggedion, which is the starting point of thermodynamic propertycalculations, will be obtained from these functions.

An inverse Fourier transform of �j (k)= ρ0

j (k)/[k2εl(k)], which provides a solution to Eq. (A15),leads to the average electric potential as

�j (r) = 1

8π3

∫∫∫dkeik·r�j (k)

= 1

2π2r

∫ ∞

0dk[�j (k)k] sin (kr). (17)

Suppose that the roots of εl(k) can be found as ikn, i.e.,

k2εl(k) = εs

∏n

(k2 + k2

n

)f (k)

, (18)

where f (k) is a function with no poles, then we can rewritethe electrical potential as

�j (k) = f (k)ρ0j (k)

εs

∏n

(k2 + k2

n

) . (19)

For an electrolyte solution, the traditional Debye-Hückeltheory is equivalent to use a dielectric function εl(k)= εs(1 + k2

DH/k2) and the charge density ρ0j (k) = qj , where

kDH =√

β∑

i niq2i /εs is the conventional inverse Debye

screening length.13

After some calculations, it can be shown that �j (r) hasthe following asymptotic behavior:

�j (r) ∼∑

n

qeff,jn

4πεeff,n

e−knr

r, (20)

where qeff,jn = ρ0j (ikn) is an effective charge, and εeff,n

= 1/2[kdεl(k)/dk]k=iknis an effective dielectric constant de-

termined from the bulk property εl(k), and ikn is the root withpositive imaginary part. It is interesting to note that differentions in the same solution will have the same set of screen pa-rameters kn, but with different effective charges.

The traditional DH theory has just one Yukawa term inthe electric potential outside the hard core with the conven-tional Debye screening length, bare charge of the tagged ion,and solvent dielectric constant εs . The multi-Yukawa termsin Eq. (20) motivate us to calculate the electric potential interms of a multi-parameter DH theory, i.e., the electric po-tential could be a linear combination of solutions from a DHtheory with different Debye length kn, effective charge qeff,jn

and effective dielectric constant εeff,n from molecular proper-ties of an electrolyte solution. A straightforward implemen-

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104104-4 T. Xiao and X. Song J. Chem. Phys. 135, 104104 (2011)

tation of this procedure as in the dressed ion theory13 willrequire many terms to achieve accurate thermodynamic prop-erties. Our approach is to use effective linear combinationcoefficients instead of qeff,jn/[4πεeff,n ], which are obtainedfrom some rigorous constraints, to calculate the average elec-tric potential accurately. This approach is called a molecularDH theory.

Consider a primitive model where ion j has a hard corewith diameter σ and the bare charge density is ρc

j (r) = qj δ(r).Denote φjn(r) as the solution of the DH equation with aDebye screening parameter kn,

−εs∇2φjn(r) = ρcj (r), r < σ,

∇2φjn(r) = k2nφjn(r), r > σ.

(21)

Here, φjn(r) is the nth basic mode of Eq. (20), which couldbe used to expand the electric potential, i.e.,

�j (r) =∑

n

Cjnφjn(r), (22)

where Cjns are the linear combination coefficients to bedetermined. A main advantage of this approach is the linearnature of the formulation as in the conventional DH theory,from which one can extend this approach to complex solutewith arbitrary shape and arbitrary charge distribution, andthe only price we need to pay is to find these coefficients Cjn

which contain nonlinear effect via the re-normalization of theeffective charge and effective dielectric constant. If appropri-ate Cjns are used, Eq. (22) would be an approximate solutionto �j (k) = ρ0

j (k)/[k2εl(k)]. Moment conditions related tothe dielectric function of the bulk system can be exploited todetermine these linear combination coefficients. Conceptu-ally, this approach is a resummation of the asymptotic seriesin Eq. (20) using moment conditions as constraints.

According to Eqs. (21) and (22), the mean electric poten-tial �(r) outside the core is a linear combination of multi-Yukawa potential,

�j (r) =∑

n

qjCjn

4πεs(1 + knσ )

e−kn(r−σ )

r, r > σ.

(23)Using the Poisson equation Eq. (A1), Eq. (23) is equivalent toassume that

∑i

niqihij (r) =∑

n

qjCjnk2n

4πεs(1 + knσ )

e−kn(r−σ )

r, r > σ.

(24)Inside the hard core, it is known that

�j (r) = qj

4πεsr+ B, r < σ. (25)

Continuous conditions of �j and εs ∂�j (r)/∂r at boundaryr = σ lead to the following restriction about B and Cjn,

qj

∑n Cjn/(1 + knσ )

4πεsσ= B + qj

4πεsσ,∑

n

Cjn = 1. (26)

It is note that the second restriction in Eq. (26) is equiva-lent to the charge neutrality condition. These two restrictions

lead to

B = qj (∑

n Cjn/(1 + knσ ) − 1)

4πεsσ= − qj

4πεs

∑n

Cjnkn

1 + knσ.

(27)As our purpose is to develop a molecular multi-parameter DHtheory given the bulk dielectric function of the electrolyte so-lution, it can be shown that both the effective Debye screen-ing parameter kn and the coefficient Cjn could be determinedfrom the bulk dielectric function.

Note that Szz(k) offers the connection between hij (r) andεl(k)−1, expansions of the functions εl(k)−1 and Szz(k) interms of k yield

εs

εl(k)=

∞∑n=1

Dnk2n (28)

and

1−nβe20

εsk2Szz(k)=1−nβe2

0

εsk2

(∑Z2

i xi+∞∑0

(−1)n+1)

(2n+1)!M2nk

2n

),

(29)where M2n = ∫

drr2n∑

i,j ZiZjxixjnhij (r) is the 2nth mo-ment. Using Eq. (A12) we have the following relations of mo-ments M2n,

−M0 =∑

Z2i xi,

− 1

3!

nβe2

εs

M2 = 1,

− 1

(2n + 1)!

nβe2

εs

M2n = (−1)nDn−1.(n ≥ 2). (30)

To derive Eq. (28), the perfect screening conditionlimk→0 εs/εl(k) = 0 is used. Note that the first two relationsin Eq. (30) are just the charge neutrality condition and theStillinger-Lovett relation,11 and the third equality is the re-striction about the high order moments.

For a restricted primitive model with element chargee = 1, we have xi = 1/2, qi = Zie = Zi, i = 1, 2, and Z1

= −Z2. By symmetry, we also have C1i = C2i = Ci . FromEq. (A11) and Eq. (24), we can relate Szz(k) to ki and Ci . Af-ter some calculations, we can find the moment M2n in termsof ki and Ci as

−M0 =∑

i

q2Ci,

− 1

3!

nβe2

εs

M2 =∑

i

Ci

k2DH

k2i

1 + kiσ + k2i σ

2/2 + k3i σ

3/6

1 + kiσ,

− 1

(2n + 1)!

nβe2

εs

M2n=∑

i

Ci

k2DH

k2ni

∑2n+1l=0 (kiσ )l/ l!

(1 + kiσ )(n ≥ 2).

(31)

Combine Eqs. (30) and (31), we have a set of linear equationsabout C = {Cj } , which reads

TC = Q, (32)

where T = {Tij } is a matrix with element Tij

= [k2DH/k

2j−2i ][

∑2j−1l=0 (kiσ )l/ l!/(1 + kiσ )], and Q = {Qj }

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104104-5 Molecular Debye-Hückel theory J. Chem. Phys. 135, 104104 (2011)

is a vector with element Q1 = k2DH,Q2 = 1,Qj

= (−1)j−1Dj−2, (j ≥ 3). From this linear equation, wecan find the solution C = T−1Q.

Now it is clear that we can find the electric potential giventhe dielectric function is known. Namely, one can first findthe screening parameters kl by finding the roots of εl(k) fromnumerical fitting. Second, Eq. (32) could be used to find thecoefficient Cl with these screening parameters kl . Finally, theelectric potential is given by Eq. (23). This electric potentialwill be used in Sec. IV to evaluate the electrostatic contribu-tion to thermodynamic properties.

IV. ELECTROSTATIC CONTRIBUTIONSTO THERMODYNAMIC PROPERTIES

As long as the mean electric potential �j (r) is deter-mined, one can calculate electrostatic contribution to the in-ternal energy according to the following procedure. Since theinduced electric potential ψj of ion j at origin is given by

ψj ≡ limr→0

[�j (r) − qj

4πεsr

], (33)

the electrostatic energy per particle reads

βU ele

N= 1

n

∑njβqj

∫ 1

0λψjdλ =

∑ βxjqjψj

2. (34)

It is possible to find ψj using the definition Eq. (33); how-ever, it is not convenient to calculate �j (r) by numerical cal-culations because of the singularity of �j (r) at the origin.Using the relation from inverse Fourier transform f (r = 0)= 1/8π3

∫ ∞0 dk4πk2f (k), it would be easier to evaluate ψj

by

ψj = 1

8π3

∫ ∞

0dk4πk2

[�j (k) − qj

εsk2

]= 1

2π2

∫ ∞

0dkf (k),

(35)where f (k) = k2[�j (k) − qj/εsk

2]. According to the Poissonequation, we find that f (k) = ∑

i niqihij (k)/εs . Using the lo-cal charge neutrality condition,

∑i qinihij (k = 0) = −qj , it

is easy to find that f (k = 0) = −qj/εs , which has no singu-larity.

In the context of our molecular DH theory, B is the in-duced potential ψj at the origin according to Eq. (33), thenthe electrostatic energy per particle becomes

βU ele

N= −

∑j

βnjq2j

8πεsn

∑i

Cjiki

1 + kiσ. (36)

For a restricted primitive model, the expression of the energyis further simplified

βU ele

N= −

∑j

βnjq2j

8πεsn

∑i

Ciki

1 + kiσ= − k2

DH

8πn

∑i

Ciki

1 + kiσ,

(37)which should be compared with the traditional DH theory

βU ele

N= − k2

DH

8πn

kDH

1 + kDHσ. (38)

In the traditional DH theory or mean sphere approximation(MSA), εl(k) depends on kDH only, and hence the electrostatic

energy depends only on the Debye parameter kDH. However,in general, this is not necessary true. For example, εl(k) couldbe a function of both kDH and β as indicated by hypernettedChain (HNC) calculations. One should note that generally thecoefficient Cjl depends on both subscripts “j” and “l” ratherthan only “l.” To a good approximation, we found that theapproximation Cjl ≈ Cl is always adequate when one try toevaluate the thermodynamic properties for electrolytes withsize or charge asymmetry.

Note Szz(k) is a linear function of hij (k), the electrostaticenergy per particle could also be evaluated as23

βU ele

N= 1

2n

∑i,j

ninjβ

∫ ∞

0

qiqj

4πεsr4πr2hij (r)dr

= βe20

4π2εs

∫ ∞

0

(Szz(k) −

∑i

Z2i xi

)dk

= 1

4π2n

∫ ∞

0

{[1 − εs

εl(k)

]k2 − k2

DH

}dk, (39)

which means that the electrostatic energy could be determinedgiven the static structure factor or equivalently the dielectricfunction. It is straightforward to prove that Eqs. (34) and (39)are equal.

With the electric potential, we can also evaluate electro-static contributions to other thermodynamic properties. Forexample, the excess chemical potential μex

i of species i couldbe evaluated from a Kirkwood coupling process,21

βμexi = βn

∑j

xj

∫ 1

0dξ

∫dr

∂uij (r; ξ )

∂ξgij (r; ξ ), (40)

where ξ is the coupling parameter to define uij (r; ξ ) which isthe pair interaction between a partially coupled ion of speciesj and another ion of species i, while the other particles interactfully with each other. Here, gij (r; ξ ) is the radial distributionfunction of species j around the partially coupled i ion. As dur-ing the coupling process, both the short-ranged and Coulombinteractions from the charges will change, this coupling couldbe realized in two stages. During the first stage, ξ1 is usedto specify the coupling strength of the short-ranged potential,us(r, ξ1). For example, if the short-ranged interaction is hardsphere interaction with diameter σ , then when ξ1 goes from0 to 1, the tagged hard sphere will grow from 0 to σ . Here,gs

ij (r; ξ1) is the radial distribution function between i and jparticles when their interaction is us(r, ξ1). During the secondstage, charge ξ2qj at the center of the sphere j grows linearlyfrom 0 to qj while keeping the full short-ranged interactions,and gij (r; ξ2) denotes the radial distribution function betweeni and j ions. Then the excess Gibbs free energy per particlecan be written as24

β�Gex

N= βn

∑i,j

xixj

∫ 1

0dξ1

∫dr

∂usij (r; ξ1)

∂ξ1gs

ij (r; ξ1)

+βn∑i,j

xixj

∫ 1

0dξ2

∫dr

qiqj

4πεsrgij (r; ξ2)

= β�Gs

N+ β�Gele

N. (41)

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104104-6 T. Xiao and X. Song J. Chem. Phys. 135, 104104 (2011)

For concreteness, the short-ranged interaction is taken as hardsphere with a diameter σ . Introducing the cavity functionys

ij (r; A) = gsij (r; A)eβus

ij (r), where A is the distance betweentwo hard spheres; one can evaluate the integral in the first termof Eq. (41),

β

∫dr

∂usij (r; A)

∂Ags

ij (r; A)

= β

∫dr

∂usij (r; A)

∂Ae−βus

ij (r)ysij (r; A)

= −∫

dr∂e−βus

ij (r;A)

∂Ays

ij (r; A)

=∫

drδ(r − A)ysij (r; A) = 4πA2ys

ij (A; A)

= 4πA2gsij (A; A), (42)

and then we have

β�Gs

N= βn

∑i,j

xixj

∫ σ

0dA

∫dr

∂usij (r; A)

∂Ags

ij (r; A)

= 4πn∑i,j

xixj

∫ σ

0dAA2gs

ij (A; A). (43)

Since in the first stage the particle i is neutral, the reversiblework performed to grow this hard sphere should be not rel-evant to the Coulomb interactions between the other parti-cles and thus can be evaluated using the Gibbs free energyof a hard sphere system with size A. Generally, this termcould be evaluated by the Canahan-Starling formulas.25 Thesecond term in Eq. (41) is the contribution from Coulombinteractions,

β�Gele

N= βn

εs

∑xixjqiqj

∫ 1

0dξ

∫ ∞

0rgij (r; ξ2)dr

= −∑

j

βxj

∫ 1

0dξ2

∫ ∞

0∇2�j (r; ξ2)dr, (44)

where �j (r; ξ2) is obtained from −εs∇2�j (r; ξ2)= qj ξ2δ(r) + ∑

qinigij (r; ξ2). During the charging pro-cess, only the charge of tagged jth particle is changed, andthus the bulk property εl(k) or ki in Eq. (23) can be used.Assuming that the coefficient Cl keeps constant duringthe charging process, then �j (r; ξ2) = ξ2�j (r). UsingEqs. (23) and (44), we have

β�Gele

N= β

εs

∑i,j

ninjqiqj

∫ 1

0dξ2

∫ ∞

0ξ2rhij (r)dr

= −∑

j

βnjq2j

8πεsn

∑i

Cjiki

1 + kiσ. (45)

In this case, we have β�Gele/N = βU ele/N , which is simi-lar to the case of traditional DH or MSA. Now the solvationenergy, which is defined as the excess Gibbs free energy perparticle (excess chemical potential), reads

Esol = β�Gs

N+ βU ele

N. (46)

Using the virial equation, we can find the pressure as

βP

n= 1 + 2πnσ 3

3

∑i,j

xixjgij (σ ) + βU ele

3N. (47)

From the thermodynamic relations, the excess Helmholtz freeenergy reads which is

β�Aex

N= β�Gex

N−

(βP

n− 1

)= β�Gs

N

− 2πnσ 3

3

∑i,j

xixjgij (σ ) + 2

3

βU ele

N. (48)

In general, the contact value gij (σ ) depends on Coulomb in-teraction and could be determined by other theories such asexponential approximation.26 When the electrostatic couplingis not strong, the contact value gij (σ ) could be approximatedby the contact value gs(σ ) of a reference hard sphere systemwith size σ , and then we have

1 + 2πnσ 3

3

∑i,j

xixjgij (σ ) 1 + 2πnσ 3

3

∑i,j

xixjgs(σ )

= 1 + 2πnσ 3

3gs(σ ) = βP s

n, (49)

where P s is the pressure of the reference hard sphere system.Then the pressure equation reads

βP

n βP s

n+ βU ele

3N, (50)

and the excess Helmholtz free energy is reduced to

β�Aex

N β�Gs

N− βP s

n+ 2

3

βU ele

N= β�As

N+ 2

3

βU ele

N,

(51)

where β�As/N = β�Gs/N − βP s/n is the Helmholtz freeenergy of the reference hard sphere system, which couldalso be evaluated by Carnahan-Starling formula. If theshort-ranged interaction is not hard sphere type, straightfor-ward generalization can be obtained for any short-rangedinteractions.

Therefore, all Coulomb contributions to the excess ther-modynamic properties could be found in terms of kl and Cl

and the short-ranged interaction contributions can be evalu-ated either from direct calculations, such as virial expansionsor perturbation theory.22 Since the matrix equation Eq. (32)has infinite elements, it is not very useful in practical cal-culations. On the other hand, a screened Coulomb potentialmode with large ki would make small contribution, i.e., whenki is very large, the coefficient Ci in Eq. (36) would be verysmall; the first few modes with small ki would dominate theelectrostatic contribution calculations. When the concentra-tion of the electrolyte solution is not large, k1 is a real numberthat close to but different from kDH. Taking C1 = 1 and ne-glect other Ci(i ≥ 2), then we recover the results of Attardand co-workers.9, 10 In Sec. V, we will show that it is possibleto obtain very accurate results using the first few terms in theexpansion of Eq. (36).

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104104-7 Molecular Debye-Hückel theory J. Chem. Phys. 135, 104104 (2011)

V. APPLICATIONS TO ELECTROLYTE SOLUTIONS

In this section, we will apply our theory to simplemodels of electrolyte solutions with various short-rangedinteractions. To illustrate these applications, we will calculatethe excess internal energy and the solvation energy of thesemodel systems in the context of MSA (Ref. 27) and HNC ap-proximation, where analytically or numerically exact resultsare known for these approximations. As noted in Sec. V,the electric potential is determined by multi-Yukawa modes.When kDH is not too large, typically for kDHσ < 2, the firsttwo Yukawa modes would make dominant contributions tothe electric potential, and the excess internal energy could bewell described by the first two Cis and kis. As kDH furtherincreases, more Yukawa parameters are required to obtainaccurate thermodynamic properties.

A. Tests against the mean spherical approximation

Using the mean spherical approximation,27 analytical re-sults for the dielectric function, excess energy and other ex-cess thermodynamic properties are available for primitivemodels, hence, a natural test case for the application of ourtheory.

For a 1:1 primitive model, the static charge-charge struc-ture factor is

Szz(k) = 1 + n[h11(k) − h12(k)] = k4

4�2F (k), (52)

where function F (k) = 2�2[1 − cos(kσ )] + 2k� sin(kσ )+ k2 cos(kσ ) + k4/(4�2) and 2σ� = √

1 + 2σkDH − 1. Thedielectric function reads

εs/εl(k) = 1 − k2DHSzz(k)

k2= 4�2F (k) − k2

DHk2

4�2F (k). (53)

The roots k = ikn of εl(k) = 0 are determined by equationF (ikn) = 0. Since the equation F (ikn) = 0 is highly non-linear and no analytical solutions are available, we need tosolve this equation numerically. For σ = 1, the first two rootsk1,2 are shown in Fig. 1, where Re(k1,2) denotes the realpart of k1,2, and Im(k1) denotes the imaginary part of k1.It is noted that k1,2 are real numbers when the coupling isweak, and then become complex conjugate after some crit-ical value kDHσ 1.23, which is related to the fact thatthe mean electric potential becomes oscillatory decay whenthe electric coupling is strong enough.28 Other roots (kn forn ≥ 3) appear in complex conjugates for all value of kDH(notshown).

The first few kn (n=1,2,...,L) are calculated via F (ikn)= 0, and then Cn is evaluated via Eq. (32), where the elementof Qj (j ≥ 3) or equivalently Dj−2 could be found byexpanding εs/εl(k) = ∑

j Djk2j . The first four elements of

Qj are

Q1 = k2DH, Q2 = 1, Q3 = D1

= −k2DH(−3 + 3�σ + 3�2σ 2 + �3σ 3)

48�4(1 + �σ )3, (54)

Q4 = −D2 = k2DH(15 − 30�σ − 15�2σ 2 + 20�3σ 3 + 15�4σ 4 + 6�5σ 5 + �6σ 6)

960�6(1 + �σ )4. (55)

Using these kn and Cn, we can calculate electrostaticcontribution to the excess energy βU ele/N via Eq. (37) andother thermodynamic properties. From these calculations, wefound that the electrostatic energy converges very rapidly asthe number L increases. As shown in Fig. 2, we take L = 2when kDH < 1.0 and take L = 4 for kDH ≥ 1.0. Our resultsare in very good agreements with the analytical MSA results.Even for very strong coupling cases such as kDH = 100, ourtheory prediction has only 5% error compared to MSA ana-lytical results, while the traditional DH theory has an error ofabout 42%.

An interesting observation is that we can also eval-uate Cjn in terms of effective charges and effective di-electric constants as prescribed in Sec. III. By comparingEqs. (20) and (23), we have

Cjn = qeff,jn/qj

εeff,n/εs

(1 + klσ )e−knσ . (56)

Under the context of MSA, kn can be determined numerically,and the effective charges and dielectric constants can also be

found

qeff,jn/qj = k2n/k2

DH,

εeff,n/εs = −2�2[(2�2σ + 2� − k2σ ) sin(kσ )/k

+ 2(1 + �σ ) cos(kσ ) + k2/�2]/k2DH|k=ikn

.

(57)

Equation (56) is then used to calculate Cjn, and electrostaticcontribution to the excess energy is determined by Eq. (36).Our calculation shows that this approximation could give usvery accurate results as long as we collect enough terms intothe summation. However, this strategy is not very satisfac-tory because of its slow convergency for high coupling cases.For inverse Debye length kDH = 1.0, the error is less than2% when we take the first two kn for evaluation; while forkDH = 3.0, the error is about 3% if the first eight kn are takeninto account. The strategy proposed in Sec. III to evaluate Cn

is equivalent to a resummation of the above series using mo-ment conditions, thus a faster convergent calculation of excessthermodynamic properties.

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104104-8 T. Xiao and X. Song J. Chem. Phys. 135, 104104 (2011)

FIG. 1. The first two roots from the MSA dielectric function (symbols). Thelines are guides to the eye.

B. Applications to binary electrolyte solutions

Consider a binary primitive model of electrolyte solu-tions with hard sphere diameter σ = 1, reduced inverse tem-perature β = 1, and dielectric constant εs = 1/4π . We willtest our theory using HNC approximation, which is knownto yield very accurate thermodynamic properties of primitivemodels.27 In this case, only numerical results rather than ana-lytical form of the dielectric function can be obtained. In orderto find the first few screened constants kn(n = 1, 2, ..., L), aPadé polynomial

P1(k)

P2(k)=

∑n≥1 ank

2n∑m≥0 bmk2m

(58)

is introduced to fit εl(k)−1, from which we can find kn as wellas the expansion coefficient Dn. Similar to the MSA case, thefirst two Debye parameters k1,2 under HNC are also real num-bers when the coupling is not strong, and then become com-plex conjugate after kcσ 1.3.10

As long as we have kn and Dn, we can solve Eq. (32)to find Cn and thermodynamic properties. L = 2 is used for

FIG. 2. Electrostatic contribution to the excess internal energies of a sym-metric 1:1 primitive model of an electrolyte solution from MSA (square), ourmolecular Debye Hückel approximation (circle), and traditional DH theory(triangle). The lines are guide to the eye.

kDH < 2, and L = 4 is used for kDH > 2. In this study, thefunctionχ (k) ≡ 1 − εs/εl(k) is fitted by

χ (k) = a0 + a1k2

a0 + k2 + a2k4for L = 2,

χ (k) = a0 + a1k2

2

[1

a0 + k2 + a2k3 + a3k4

+ 1

a0 + k2 − a2k3 + a3k4

]for L = 4.

(59)

The electrostatic contribution to excess energiesβU ele/N of the electrolyte solutions with (q1, q2)= (1,−1), (2,−2), (2,−1) have been calculated ac-cording to our molecular DH theory and are shown inFigs. 3(a)–3(c), respectively.

The number density n for kDH = 3.0 is about 0.72, 0.18,and 0.36 for the 1:1, 2:2, and 2:1 electrolyte solutions, respec-tively. Again, very good agreements between our theory andHNC calculations are found. For example, when kDH 3, theerror from our theory is 3%, while the traditional DH theoryis about 25%. We have also calculated the excess Gibbs freeenergy, where the hard sphere contribution to excess Gibbsfree energy is derived from the Carnahan-Starling formulas.22

As shown in Fig. 4, again very satisfactory results are found.Previous simulations show that HNC is accurate when com-pared with Monte Carlo simulations for n < 0.3,27 hence ourtheory should provide accurate thermodynamic properties aswell when compared with molecular simulation results.

Note that for the 1:1 electrolyte in high density region(e.g., kDH > 2.5 which roughly corresponding to n > 0.5 inFig. 4(a)), our prediction has significant difference when com-pared with HNC results. This is most probably due to the factthat HNC is not good enough at high densities to yield ac-curate hard sphere contribution to excess Gibss free energyβ�Gs/N . When β�Gs/N comes from HNC calculations,rather than the Carnahan-Starling formulas which is used inour molecular DH calculations, much better agreement be-tween our theory prediction and HNC is found, e.g., the erroris smaller than 2% for number densities up to n = 0.7.

In this study, our main focus is on the electrolyte so-lutions with size symmetry cases, i.e., all the ions have thesame size. However, one should note that we can extend ourtheory to a binary electrolyte solution with size asymmetrywithout difficulty, where the coefficients of multi-Yukawa po-tential will satisfy a similar linear equation as in Eq. (32), butwith the element of T and Q modified as functions of sizes.We have also calculated the excess thermodynamic propertiesfor a binary electrolyte system with size asymmetry, wheresatisfactory results are found as well (not shown).

C. Applications to multi-componentelectrolyte solutions

In this section, our molecular DH theory will be appliedto multi-component electrolyte solutions. By assuming thatthe coefficient Cjn is independent of j which is the index forions, i.e., Cjn = Cn, the application of our molecular DH

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104104-9 Molecular Debye-Hückel theory J. Chem. Phys. 135, 104104 (2011)

FIG. 3. Electrostatic contribution to excess energies of binary electrolyte solutions from HNC calculation (square), our molecular Debye Hückel approximation(circle), and traditional DH theory (triangle). The lines are guides to the eye.

FIG. 4. Solvation energy Esol of binary electrolyte solutions from direct HNC calculation (filled squares), our molecular Debye Hückel approximation (opensquares). The lines are guides to the eye.

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104104-10 T. Xiao and X. Song J. Chem. Phys. 135, 104104 (2011)

theory will be straightforward. Based on this assumption,which will be substantiated next from theoretical argumentsand calculations of model systems, it is interesting to notethat excess thermodynamic properties of the dilute solutecan be evaluated from properties of pure solvent (majoritycomponents of ions). For multi-component systems withouta minority component, the dielectric function of the wholesystem is needed to calculate the electrostatic contributionsto thermodynamic properties from straightforward general-izations of the method presented in Sec. III. As our interestsare for the systems with a clear minority component, suchas proteins in an electrolyte solution, this section is used toillustrate applications of our theory to such systems.

Consider a four-component system with qA : qA solventA and qB : qB solute B. The anion and cation of A has thesame size σA , while the anion and cation of B has the samesize σB . The molar fraction of B is xB = nB/n. In the dilutelimit xB → 0, this four-component system reduces to an ef-fective binary system. The electrostatic contribution to the in-ternal energy per solvent particle of A is

βU eleA

NA

= − βq2A

8πεs

∑n

CAnkn

1 + knσA

, (60)

where kn and CAn are evaluated from the pure solvent dielec-tric function. According to Eq. (36), electrostatic contributionto the internal energy per particle of solute B reads

βU eleB

NB

= − βq2B

8πεs

∑n

CBnkn

1 + knσAB

, (61)

where σAB = (σB + σA)/2 is the average size. When B is thesame as A, naturally we have CBn = CAn; but when B hasdifferent charge and size, it is still a good approximation toevaluate electrostatic energy using CBn CAn, which couldbe tested using the data from HNC or molecular simulations.

For the MSA case,27 we found that Cjn is not sensitiveto the charges and sizes. When MSA is applied to this four-component system, we can find that the electrostatic energyof the solute in the dilute limit reads

U eleB

NB

= − q2B�

4πεs(1 + σB�), (62)

where parameter � = (√

1 + 2σAkDH − 1)/σA depends only

on the inverse Debye length kDH =√

βnAq2A/εs and is the

property of the pure solvent. This analytical form is similarto that of the solvent

U eleA

NA

= − q2A�

4πεs(1 + σA�). (63)

This result implies that the analytical form for the electrostaticenergy of the solvent is applicable to the dilute solute whenthe charge qA and size σA is replaced by that of the solute. Itshould be noted that the electrostatic energy from MSA indi-cates that CBn should be close to CAn.

In the limit xB → 0, we can approximate CBn by CAn,and then we have

βUexB

NB

= − βq2B

8πεs

∑l

CAlkl

1 + klσAB

. (64)

FIG. 5. Electrostatic energies of solute B from HNC under different diameterσB , with xB = 0.1(square), xB = 0.01 (Circle), and our molecular Debye-Hückel approximation (triangle). The lines are guides to the eye.

From the electrostatic energy, we can evaluate other excessthermodynamic properties of the solute B following the ther-modynamic relations derived in Sec. IV.

Although in this report, our focus is on the ions withspherical symmetry, one should note that this molecular ap-proach could be extended to molecules with arbitrary shapesuch as biomolecules. The key difference is that the mean po-tentials in Eqs. (21) and (22) are obtained from solving thecorresponding linearized Poisson-Boltzmann equations withscreening length kn using appropriate boundary conditions;therefore, our approach may find broad applications in study-ing biomolecules solvation in electrolyte solutions.

To test Eq. (64), a four-component primitive model ofelectrolyte solutions with 1:1 solvent A and 1:1 solute B isconsidered. The diameter of A is fixed as σA = 1, while σB ,the diameter of B is used as a parameter. For a dilute solution,the molar fraction of solute B is small. As shown in Fig. 5, forn = 0.1 the electrostatic energy βU ele

B /NB for B is obtainedfrom HNC calculations. It is noted that βU ele

B /NB is not sensi-tive to the exact value of xB as long as it is small. For example,the results for xB = 0.1 and xB = 0.01 only have a differenceup to 3% for σB = 3.5 from the electrostatic energy of so-lute B calculated from Eq. (64). Furthermore, our theory isalso in very good agreement with the excess thermodynamicproperties directly calculated from HNC. We have also cal-culated βU ele

B /NB for other values of number density n andsize σB , where good agreements between our theoretical pre-dictions and direct HNC calculations are found (not shown).These results indicate that excess thermodynamic propertiesof the solute can be obtained from the pure solvent dielectricproperties in dilute electrolyte solutions.

D. Applications to electrolyte solutions beyondprimitive models

As models of an electrolyte solution with soft short-ranged interactions are more realistic than a primitive modelextension of our approach beyond primitive models will beuseful. Consider a simple binary 1:1 soft electrolyte modelwith pair additive interaction uij (r) = us(r) + qiqj /εsr .

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104104-11 Molecular Debye-Hückel theory J. Chem. Phys. 135, 104104 (2011)

Similar to the case of primitive models, the solvation energy(excess chemical potential) of soft electrolyte solution canbe evaluated from a summation of the Gibbs free energyβ�Gs/N of a simple fluid with pair interaction us(r) and theelectrostatic energy βU ele/N evaluated from the dielectricfunction of the soft electrolyte model, i.e.,

Esol = β�Gs

N+ βU ele

N. (65)

Then the problem reduces to find Gibbs free energy of theshort-ranged system and the Coulomb contribution to excessinternal energy.

In general, perturbation theory can be used to calcu-late the thermodynamic properties of a simple fluid withshort-ranged interactions such as Weeks-Chandler-Anderson(WCA) theory,29–31 where the effective size of the referencehard sphere system is determined by the short-ranged repul-sive part of the interaction. With the effective size and the di-electric function of the soft system, we can evaluate βU ele/N

in the same way as for primitive models accurately. In thissense, our approach essentially extends the perturbation the-ory to systems of long-ranged interactions with the same ac-curacy as the successful perturbation theory widely used forshort-ranged systems.

Namely, using the concept of effective size, the tradi-tional perturbation theory is extended to soft electrolyte mod-els if an effective primitive model is used as a reference sys-tem of the soft electrolyte model. The dielectric function ofthe soft system can be approximated by that of the referenceprimitive model, and βU ele/N is evaluated from the electro-static energy βU ele

eff /N of the effective primitive model, i.e.,

βU ele

N∼= βU ele

eff

N. (66)

As in the traditional perturbation theory for short-ranged sys-tems, the evaluation of other thermodynamic properties of thesoft electrolyte model would be similar.

Validity of Eq. (65) is tested for a binary electrolytewith various soft interactions once an effective size ofthe soft ion can be found. For one test, the short-rangedinteraction for the soft ion is taken as an inverse 12 powerpotential, i.e., βus(r) = βε(σ/r)12 and the parameters

TABLE I. Coulomb excess internal energy βU ele/N for 1:1 soft electrolytemodel with r−12 potential, where parameters used are β = 1, ε = 1, εs

= 1/4π, σ = 1.

n Soft SD BH WCA0.01 −0.1326 −0.1342 −0.1324 −0.13190.02 −0.1707 −0.1731 −0.1704 −0.16960.05 −0.2315 −0.2350 −0.2306 −0.22940.1 −0.2855 −0.2896 −0.2836 −0.28220.2 −0.3477 −0.3514 −0.3437 −0.34130.3 −0.3891 −0.3915 −0.3831 −0.38150.4 −0.4216 −0.4223 −0.4135 −0.41330.5 −0.4491 −0.4479 −0.4389 −0.43860.6 −0.4732 −0.4700 −0.4609 −0.46270.7 −0.4951 −0.4897 −0.4806 −0.4828

β = 1, ε = 1, q1 = 1, q2 = −1, εs = 1/4π, σ = 1 are usedin the calculations. As the choice of an effective size is notunique from various versions of perturbation theories, for ther−12 potential, the effective size is determined by the widelyused WCA prescription31 in our calculations. The effectivesize decreases from about 1.07 to 1.04 when the number den-sity increases from 0.1 to 0.7. The set of Cn, kn are directlycalculated using the dielectric function εl(k) for soft elec-trolyte model from HNC calculation or the effective primitivemodel. As a comparison, the effective size is also used in thetraditional DH theory to evaluate the electrostatic energy. Asshown in Fig. 6(a), our theory works much better than thetraditional DH in the high coupling region. The excess Gibbsfree energy β�Gs/N of short-ranged system is evaluatedfrom the WCA theory and the result for solvation energy isshown in Fig. 6(b). Again satisfactory agreements betweenour theoretical model and HNC calculation are found.

Even though the choice of reference hard sphere di-ameter is crucial for the perturbation theory calculationof short-ranged systems, for the effective primitive modelused in the Columbic systems the hard sphere diameter isnot that sensitive. Hence the effective parameters Cn, kn

can be obtained easily from the effective primitive modelusing molecular simulations or accurate integral equationssuch as HNC. To demonstrate this, Eq. (66) is tested fortwo soft electrolyte models, one with short-ranged r−12

potential and the other with Leonard-Jones (LJ) potential

FIG. 6. Comparisons for electrostatic energies and solvation energies of soft electrolyte solution model with 1/r12 potential.

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104104-12 T. Xiao and X. Song J. Chem. Phys. 135, 104104 (2011)

TABLE II. Coulomb excess internal energy βUexl /N for 1:1 soft elec-

trolyte model with Leonard-Jones potential, where the parameters used areβ = 0.5, ε = 1, εs = 1/4π, σ = 1.

n Soft SD BH WCA0.01 −0.0531 −0.0507 −0.0508 −0.05070.05 −0.0989 −0.0935 −0.0938 −0.09360.1 −0.1244 −0.1180 −0.1184 −0.11820.2 −0.1528 −0.1466 −0.1472 −0.14700.3 −0.1707 −0.1656 −0.1663 −0.16610.4 −0.1843 −0.1804 −0.1811 −0.18100.5 −0.1959 −0.1928 −0.1935 −0.19350.6 −0.2062 −0.2036 −0.2043 −0.20440.7 −0.2157 −0.2133 −0.2140 −0.21420.8 −0.2247 −0.2221 −0.2228 −0.2232

us(r) = 4ε[(σ/r)12 − (σ/r)6]. Three kinds of effective sizeare used for comparison. The first one is simply taken from thesoft sphere diameter as an effective one, i.e., σeff = σ (denotedby SD). The second one takes the Barker-Henderson (BH)prescription σeff = ∫ ∞

0 (1 − e−βVs (r))dr (denoted by BH); andthe third one is calculated from WCA perturbation theory(denoted by WCA). The short-ranged strong repulsive partof us(r), Vs(r), is used for both BH diameter and WCAdiameter calculation. For r−12 potential, Vs(r) = ε(σ/r)12;while for LJ potential, Vs(r) = 4ε[(σ/r)12 − (σ/r)6] + ε

for r < rm = 21/6σ and Vs(r) = 0 for r > rm. These threekinds of effective hard sphere diameters are used as input ofthe effective primitive model. Electrostatic energy βU ele/N

for the same soft electrolyte models directly calculated fromHNC (denoted by Soft) with r−12 and LJ core are also shownin Tables I and II. It is found that the electrostatic energyfrom these three kinds of reference systems does not havesignificant difference from that of the soft electrolyte models.This implies that Eq. (66) is a good approximation even ifthe effective diameter is not very accurate. This property isdifferent from that of short-ranged systems, i.e., when the per-turbation theory is applied to a short-ranged system, the freeenergy is sensitive to the small difference in effective sizes.23

It should be noted that the above results may not be uni-versal, especially for the cases where the charge is off-centerand close to the dielectric boundary. Further studies should beperformed to test the best strategy to determine the dielectricboundary for a solute with general geometry.

VI. CONCLUDING REMARKS

In conclusion, a molecular Debye-Hückel theory forionic fluids is developed. Starting from the macroscopicMaxwell equation for bulk systems, an application of the dis-persion relations to electrolyte solutions leads to a molecu-lar Debye-Hückel theory which is related to the dressed iontheory in the static case. The two phenomenological func-tions, namely, the dielectric function and the effective chargedensity are discussed at the microscopic level. Generally, thedielectric function has many poles, which leads to multi-Yukawa forms for the electric potential around a single ion.Parameters of these multi-Yukawa potentials, such as the lin-ear combination coefficients and effective Debye lengths, can

be obtained from general relations such as charge neutralitycondition, Stillinger-Lovett relation and other high order mo-ment restrictions. Namely, these parameters can be obtainedas long as we know the bulk dielectric function and the hardsphere diameter of primitive models. Using these parameters,the excess thermodynamic properties of primitive models caneasily be evaluated accurately. For electrolyte solution modelswith soft short-ranged interactions, it is shown that the tradi-tional perturbation theory can be extended to ionic fluids suc-cessfully just as the perturbation theory has been successfullyused for short-ranged systems.

ACKNOWLEDGMENTS

We thank Qiang Zhang for discussions and contributionsat the early stage of this work. The authors are grateful tothe financial support from a Petroleum Research FoundationGrant No. 46451AC5 administrated by the American Chem-ical Society, and from a National Science Foundation (NSF)Grant No. CHE-0809431.

APPENDIX: THE DEBYE-HÜCKEL THEORY FROMTHE MEAN ELECTRIC POTENTIAL

Consider a simple model of isotropic electrolytesolutions, where the solvent is treated as a continuummedium with dielectric constant εs and a solute particle,say “j,” is treated as a sphere with a point charge qj

at the center. The pair potential between i and j readsuij (r) = us

ij (r) + ulij (r), where us

ij (r) is a short-ranged po-tential, while ul

ij (r) = qiqj /εsr is the long-ranged Coulombpotential. The average electric potential �j (r) around ion j ,satisfies the following Poisson equation:21

− εs∇2�j (r) = ρcj (r) + ρind

j (r) = ρtotj (r), (A1)

where ρcj (r) = qj δ(r) is the bare charge density of the

particle j, ni is the particle number density of species i,ρind

j (r) = ∑niqigij (r) = ∑

niqihij (r) is the induced chargedensity around j; gij (r) is the radial distribution functionbetween i and j, hij (r) = gij (r) − 1 is the correlation func-tion, and ρtot

j (r) is the total charge density. Note that theinformation of dielectric response has been included inthe induced charge density, one can derive the traditionalDebye-Hückel theory from this Poisson equation after amean field approximation of the induced charge density.21

In an isotropic equilibrium fluid, the total correlationfunction hij (r) and the direct correlation function cij (r) sat-isfies the conventional Ornstein-Zenike (OZ) equation

hij (r) = cij (r) +∑

l

∫dr′cil(|r − r′|)nlhlj (r ′), (A2)

where r = (x, y, z) is the three-dimensional coordinate of theparticle and r = |r| is the distance from the origin.

As noted in the dressed ion theory,12 a renormalizationof charge density could be realized by splitting the functionf (r) = h(r), c(r), ρ(r) into a well-defined long-ranged partf l(r) and a short-ranged part f 0(r). Note that the asymp-totic behavior of the direct correlation function behaves as

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104104-13 Molecular Debye-Hückel theory J. Chem. Phys. 135, 104104 (2011)

c(r) ∼ −βqiqj /εsr , we can define

clij (r) = −βqiqj

εsr(A3)

and

c0ij (r) = cij (r) + βqiqj

εsr. (A4)

The short-ranged part h0ij (r) of hij (r) is determined via the

OZ equation

h0ij (r) = c0

ij (r) +∑

l

∫dr′c0

il(|r − r′|)nlh0lj (r ′), (A5)

and then the short-ranged part of induced charge reads

ρind,0j (r) =

∑i

niqih0ij (r). (A6)

It is noted that the long-ranged part of correlation functionand charge density, defined via hl

ij (r) = hij (r) − h0ij (r) and

ρind,lj (r) = ρind

j (r) − ρind,0(r), are linearly related to the av-eraged electric potential,12 i.e.,

hlij (r) = −β

∫dr′ρ0

i (|r − r′|)�j (r ′), (A7)

ρind,lj (r) =

∑i

niqihlij (r) = −

∫dr′α(|r − r′|)�j (r ′),

(A8)where α(r) = β

∑i niqiρ

0i (r). Introduce an effective charge

density as ρ0j (r) = ρc

j (r) + ρind,0j (r), then from Eq. (A1), we

have

− εs∇2�j (r) = ρ0j (r) −

∫dr′α(|r − r′|)�j (r ′). (A9)

In the Fourier space, Eq. (A9) reduces to the following equa-tion for potential �j (k),

k2

[εs + α(k)

k2

]�j (k) = ρ0

j (k), (A10)

where α(k) = ∫dreik·rα(r) = ∑

i qiniρ0ij (k) is a screened

response function, since the potential is a response to a localcharge density ρ0

j (r) rather than an external charge densityqj δ(r).22 When we apply this equation to a multi-componentsystem with solvent and solute, note α(k) = ∑

i qiniρ0ij (k)

and∑

i qinih0ij (r) are proportional to the particle density ni ,

then in the dilute limit of solute, i.e., nj → 0 for all the solutespecies, only the solvent will contribute to these two quanti-ties. This means one can use the dielectric function of the puresolvent to evaluate the properties of a dilute solute, while theeffective charge is still related to the cross correlation functionh0

ij (k) between solute and solvent species.When we compare Eq. (A10) with Eq. (16), similar struc-

tures are apparent. Introduce the static charge structure factorSzz(k) as

Szz(k) =∑

i

Z2i xi + n

∑i,j

ZiZjxixjhij (k), (A11)

where n = ∑i ni is the total number density, xi = ni/n is the

molar fraction of ith type particle, e is the element charge, and

Zj = qj/e is the charge number of the jth ion. Then accordingto the linear response theory in ionic fluids,22 �l(k) could bereplaced by the bulk dielectric function εl(k), which is relatedto the Szz(k) by the relation

εl(k)−1 = ε−1s

[1 − βne2

εsk2Szz(k)

]. (A12)

Note that the screened response function α(k) defined inthe dressed ion theory is also related to the charge structurefactor,14 i.e.,

Szz(k) = εsk2α(k)

βne2[εsk2 + α(k)], (A13)

and it is straightforward to verify that

εl(k) = εs + α(k)

k2. (A14)

For a macroscopic system where the macroscopic Maxwellequations are used, the charge density ρ in Eq. (16) is al-ways interpreted as free charge density, which is a macro-scopic quantity.18 However, we should be careful when weapply this equation to the microscopic level, e.g., to a singleion. For a macroscopic system of interest, the linear responseis applicable, since the scale we care about is much larger thanthe molecular scale, e.g., typically order of 1 mm, where theexternal field is always weak enough. However, when we con-sider the potential of a single ion, we are interested in the scalethat is comparable with molecular size, where the Coulombinteraction may be not weak enough, then linear responsewould not be satisfactory, and the nonlinear effect should betaken into account. So at the molecular level, it would be es-sential to interpret the charge density in Eq. (16) as an ef-fective one, which should be derived from statistical mechan-ics calculations of a microscopic model as the derivation ofEq. (16).21 As noted by Kjellander,13 the nonlinear responsecan be included in the effective charge density ρ0

j (k). Whenthe dielectric function is replaced by the term from linear re-sponse, i.e., �l(k) = εl(k), the charge density is interpretedas ρ(k) = ρ0

j (k), and then we have a rigorous electric poten-tial equation that is applicable to the ionic fluids at molecularlevel,

k2εl(k)�j (k) = ρ0j (k). (A15)

Therefore, a rigorous statistical mechanics derivation of thePoisson equation with a renormalized charge density is ob-tained for a molecular model with pair additive interactions.The solution of the above equation can be obtained as a su-perposition of linearized PB equations with effective screen-ing lengths and charges, which forms the starting point ofour molecular DH theory presented in Sec. III. Dynamicalextension of the above result can be formulated in a similarfashion, namely, a rigorous statistical mechanics derivation ofthe macroscopic Maxwell Eqs. (14) and (15) from the micro-scopic ones is needed just as how Eq. (16) is derived. Sincea time-dependent Ornstein-Zernike equation has not been for-mulated rigorously thus a rigorous dynamical extension of theDH theory is still an open question.

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104104-14 T. Xiao and X. Song J. Chem. Phys. 135, 104104 (2011)

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