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The Dielectric Constant of Ionic Solutions: A Field-Theory Approach ∗ Amir Levy, David Andelman Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel Henri Orland Institut de Physique Théorique, CE-Saclay, CEA, F-91191 Gif-sur-Yvette Cedex, France (Dated: Jan 23, 2012) 2 We study the variation of thedielectric response of a dielectric liquid (e.g. water) when a salt is 1 addedtothesolution. Employingfield-theoreticalmethodsweexpandtheGibbsfree-energytofirst 0 orderinaloop expansionand calculate self-consistently thedielectric constant. Wepredict analyt- 2 icallythedielectricdecrementwhichdependsontheionicstrengthinacomplexway. Furthermore, n aqualitativedescriptionofthehydrationshellisfoundandischaracterizedbyasinglelengthscale. a Our prediction fits rather well a large range of concentrations for different salts using only one fit J parameter related to thesize of ions and dipoles. 9 2 Electrostatic interactions in aqueous media between the dielectric constant of water, εw. Hence, in order ] dipoles and charged objects such as ions, colloidal par- to model the experimentally known dielectric decrement t f ticles and interfaces, play an important role in electro- phenomenon[7,15–17],oneneedstoemploymorerefined o chemistry, biology and materials science. The century- theories, which take into account ion-dipole correlations s . old Poisson-Boltzmann (PB) theory gives a simple and and fluctuations. The electric decrement stems from the t a powerful description for numerous systems, taking into factthatthelocalelectricfieldaroundeachionisgreater m account only Coulombic interactions on a mean-field than the external field, and orients the dipolar water - level, treating ions as point-like particles and the aque- molecules in its vicinity. This creates a hydration shell d ous solution as a continuous and homogeneous dielectric [18]thatencirclesthe ions. The totalresponseofdipoles n o mediumwith aconstantdielectricconstant, εw. The PB to the external field is thus smaller and leads to a re- c theorysucceededovertheyearsincapturingmuchofthe ductioninthe dielectricconstant. Indilute solutionsthe [ underlyingphysicsofelectrolytesolutions,and,inpartic- dielectric constant depends linearly on c , the salt con- s 1 ular, is successful when applied to monovalent ions and centration, ε(cs)=εw γcs, where γ is the coefficientof − v weak surface charges [1–4]. the linear term. The value of γ is ion specific and ranges 1 ThePBtheoryhasseverallimitations. Itdoesnottake from 8M−1 to 20M−1, for concentrations up to 1.5M. 8 0 intoaccountneitherthecorrelationsbetweenthecharges, At higher cs values, noticeable deviations from linearity 6 nor does it allow for any fluctuations beyond mean field. are observed and are due to ion-ion interactions [15–17]. . Thisleadstosignificantunaccountedcorrectionsincases In this Letter we go beyond the standard PB theory 1 of high charge density, especially near charged surfaces and investigate the microscopic origin of the dielectric 0 2 and interfaces [5]. In orderto improve upon the PB the- decrement. We construct the field theory for a grand- 1 ory, several extensions have been offered in recent years, canonical ensemble of point-like ions and dipoles. On : and effects of correlations and fluctuations for charged a mean-field level, we obtain an analytic description of v i interfaces were considered, for example, by using inte- the hydration shell around ions and calculate the aver- X gral equation theories [6]. In other approaches [7], the age dielectric constant in the shell. Then, using a field- r PB theory was modified in a simple and elegant way to theoreticalapproachweexaminethedielectricdecrement a include steric and other ionic-specific effects preventing by a one-loop calculation. A closed-formformula for the ionsfromaccumulatingnearchargedsurfaceforveryhigh dielectric constant is obtained and depends on a single ionic concentrations. Furthermore, in the well known length-scale,related to the typicalsize of ions and water Deryagin-Landau-Verwey-Overbeek (DLVO) theory [8], molecules. Our results show a substantial contribution van der Waals attractive interactions were added to the arising from the one-loop correction terms, and deviates electrostatic repulsion in order to explain charged col- from the linear relation, ε(c ) = ε γc . We fit the s w s − loidal stability. More recently, Monte-Carlo (MC) sim- calculated ε(c ) and compare it with the experimental s ulations and Molecular Dynamics (MD) were employed data for several monovalent ions. The results fit rather in order to study specific solvents and solutes and the well for a wide range of ionic concentrations,even in the interaction between them [9–13]. non-linear decrement regime. The PB theory as well as others primitive models of Although our formulation is very general, in this Let- ionic solutions [14] assumes that the ions are immersed ter we focus on the dielectric constantvariationfor ionic in a continuum dielectric background characterized by solutions. The solvent (water) is modeled as a liquid of 2 point-like permanent dipoles of dipolar moment p0 and tocalculateitsvariationasafunctionofthe saltconcen- thesymmetric1:1monovalentsaltasaliquidofpoint-like tration c . Following Eq. (7), we need only to calculate s charges, e. Using the standard Hubbard-Stratonovich the coefficient of the Ψ(r)2 terms in the Gibbs free- ± |∇ | transformation [19], the grand-canonical partition func- energy G[Ψ(r)]. tion Z and free energy F = β−1logZ[ρ(r)] in presence On the mean-field level the Gibbs free-energy for the − of an external charge distribution ρ(r) can be written as Dipolar PB system, GDPB, is a functional integralover the field φ(r) that is conjugate to ρ(r) βGDPB[Ψ]= βε0( Ψ)2 2cscosh(βeΨ) − 2 ∇ − c g(u), (8) Z[ρ(r)]= φ(r)exp β d3r f(φ(r)) − d D − Z (cid:18) Z with u = βp0 Ψ(r), u = u, and g(u) sinhu/u. The ∇ | | ≡ iβ d3rφ(r)ρ(r) (1) Dipolar Poisson-Boltzmann(DPB) equationis anexten- − sionofthestandardPBequation,andcanbederivedvia Z (cid:19) a variational principle from Eq. (8): with βf(φ(r))= βε0[ φ(r)]2 2λ cos[βeφ(r)] −ǫ0∇2Ψ=−2csesinh(βeΨ) s 2 ∇ − Ψ sin(βp0 φ(r)) +cdp0∇· ∇Ψ G(βp0|∇Ψ|) , (9) −λd βp0|∇|∇φ(r)| | , (2) where the function = g′(u(cid:20))|∇= c|oshu/u sin(cid:21)hu/u2 is where β = 1/kBT, and kBT is the thermal energy. The related to the LangeGvin function L(u)=co−th(u) 1/u. fugacities of the salt λs and water λd are determined by The dielectric constant ε is calculated by−substi- DPB solving the implicit equations tuting GDPB into Eq. (7). The result is the same as in λ ∂ Ref. [20]: s logZ =c , s λV ∂∂λs εDPB =ε0+βp20cd/3 . (10) d logZ =c , (3) V ∂λ d Ascanbe seenabove,the dielectricresponsedependson d thedipole(butnottheion)densityc and,hence,cannot where V is the volume and c and c are, respectively, d s d explain any dielectric decrement. At room temperature, the salt concentration and density of water molecules. The electrostatic potential Ψ(r) is given by T =300K,andforpurewaterwithdipolarmomentp0 = 1.8D, and density c = 55M, the obtained value of ε is d 1 δlogZ[ρ(r)] ε 11.1. Note that this value is much smaller than Ψ(r)=−β δρ(r) =ihφ(r)i0 , (4) thDePBm≃easuredwatervalueεw 80. Thisisnotsurprising ≃ sinceEq.(10)isadilutegasapproximationanddoesnot wherethebracket ... 0 denotesthethermalaveragewith capture the correlationeffects in the case of dense liquid h i the Boltzmann weight given in Eq. (1), for ρ=0. water as well as the finite-size of water molecules. The Gibbs free energy G is a function of Ψ(r) and is It is still possible, however, to circumvent this draw- definedastheLegendretransformofF[ρ(r)]withrespect back by solving the DPB equation, Eq. (9), around a to ρ(r) through the equations fixed point-like ion, and showing the existence of a hydration shell around it. Expanding in a Taylor series, G[Ψ(r)]=F[ρ(r)] d3r Ψ(r)ρ(r) , the dielectric constant is extracted Gas a function of the − Z distance r from the ion: δF[ρ(r)] Ψ(r)= , (5) ǫ δρ(r) ǫ(r) DPB , (11) ≃ 3h2(l /r)+1 h from which we deduce the Legendre relation where the function h(u) is obtained by solving a cubic δG[Ψ(r)] ρ(r)= , (6) equation: − δΨ(r) 1/3 Thedielectrictensorε isdefinedthroughtheFourier 1 u √u αβ h(u)= + + transform: "r27 4 2 # εαβ = ∂pα∂∂2pβ Z d3r eip·rδδΨ2(Gr[)ΨδΨ(r()0])(cid:12)(cid:12)Ψ=0,p=0 . (7) −"r217 + u4 − √2u#1/3. (12) (cid:12) In our present study, the dielectric tens(cid:12)(cid:12)or for isotropic The length, lh2 = lBd/√10 in Eq. (11), depends both aqueous solutions is diagonalεαβ =εδαβ, and our aim is on the dipole size d = p0/q and the Bjerrum length, 3 lB =e2/4πεkBT. It is the only length scale in the prob- sion [21, 22] of the Gibbs free-energy, G = GDPB+∆G. lem characterizing the hydration shell thickness as can To one-loop order the calculation of ∆G yields: be seen from the behavior of ǫ(r) (inset of Fig. 1). In 1 the vicinity of the ion (r ≤ lh), the dielectric response ∆G[Ψ(r)]= 2βTrlog −ε0∇2+2λsβe2cosh(βeΨ(r)) is very small and it smoothly rises to bulk values as the influence of the ion decreases, within distances of a few + λdβp20(∂α(cid:2)Γαγ(r)∂γ +Γαγ(r)∂α ∂γ) ,(13) · l . h with (cid:3) By averaging ε of Eq. (11) over a sphere of radius R around each ion, and equating R with the typical dis- g′(u) u u g′(u) tance between two ions, (2cs)−1/3/2, at concentration Γαγ(r)=δαγ u − αu2γ u −g′′(u) . (cid:18) (cid:19) c ,weobtainthe expressionof ε(c ) . Ascanbeseenin s h s i (14) Fig. 1, the non-linear dielectric decrement is reproduced and fits rather well the experimental data for RbCl and To get a consistent expression for the dielectric con- CsCl salts of Ref. [16]. For low salt concentration, the stant we need to pay extra attention in the loop expan- averaging of ε can be done analytically and results in sion to the expressions of the ion and dipole fugacities, γ ≃ 110ǫDPBlh3cs, which is proportional to the hydration λs andλd. While atmean-fieldlevelλs andλd areequal shell volume. With parametervalues as in Fig. 1,we get totheionanddipole(water)densities,c andc ,respec- s d γ 17. tively,their correctionsarederivedby expandingEq.(3) ≃ toone-looporder. Tothisorder,onlythecorrectiontoλ d 80 willaffectthe mean-fieldvalueof the dielectric constant, 80 ε , and is written as DPB 60 70 ε/ε040 λd =cd− 32aπ3εDPεBDP−Bε0 (cid:20)1− 4π32(κDa)2 20 3 2π εε〉 /0 60 00 2 4 6 8 + 8π3(κDa)3tan−1 κDa(cid:21) , (15) 〈 r[Å] whereaisamicroscopiccutofflengthandκD =√8πlBcs 50 is the inverse Debye length. The origin of the cutoff in our formulation is related to the fact that Coulombic interactions diverge at zero distance, while in reality such 40 0 1 2 3 4 5 6 a divergence is avoided because of steric repulsion. A c [M] s simple way to take this into account is through a cutoff length a, which is related to the minimal distance be- FIG. 1. (color online). The dielectric constant hεi averaged tween adjacent dipoles and charges, and thus indirectly inside a specific volume around a single ion (solid line) as func- also to the size of dipoles and ions. tion of ionic concentration, cs. The comparison is done with We can now calculate consistently the corrections to experimental values for RbCl (empty circles) and CsCl (full cir- the dielectric constantup to one-looporder. The correc- cles) [16]. In the inset, the exact (solid line) and approximated tion term can be split into water and salt contributions, (dashed, Eq. (11)) solutions of the DPB equation, Eq. (9) are ε ε =∆ε +∆ε : shown as function of the distance r from a point charge (ion). − DPB d s Cp0ho=os4i.n8gDas(inastfietadpaorfatmheetperhytshiecadlivpaoluleemp0om=e1n.t8oDf)w[2a0te],ratlolowbes ∆ε = (εDPB −ε0)2 4π us toobtain εDPB =80 and lh ≃1.5˚A. d εDPB 3cda3 ∆ε = (εDPB −ε0)2 κ2D 1 κDatan−1 2π So far in order to induce the necessary ion-dipole cor- s − εDPB πcda(cid:18) − 2π κDa(cid:19) relations, we had to rely on calculating the dielectric re- (16) sponse around a single fixed ion, which gives the dielectric decrement in an approximated way. Moreover, the Theterm∆ε representsthefluctuationeffectofthewa- d validity of the DPB theory can be justified only in the terdipoles beyondthe mean-fieldDPB level. Itvariesas dilute limit, where the dipole-dipole fluctuations are not 1/(c a3). This pure water fluctuation term essentially d ∼ important. adds a positive numerical prefactor of rather large mag- To overcome this limitation a more complete treat- nitude to the mean-field value of ε , meaning that the DPB ment of the statistical mechanics of ionic and dipolar one-loop correctionis important even for the pure water degrees of freedom (including their interactions) is de- case. veloped. We generalizethe Debye-Hu¨ckelapproximation The second term ∆ε has by itself two contributions. s to dipolar systems in presence of salt, via a loop expan- The leading term in the dilute solution limit, κDa 1, ≪ 4 depends linearly on the salt concentration, ∆ε = γc , Acknowledgements. We thank D. Ben-Yaakov, s s − with Y. Burak and X.-K. Man for useful discussions. This workwassupportedinpartbytheU.S.-IsraelBinational γ = (εDPB −ε0)2 8lB . (17) Science Foundation under Grant No. 2006/055 and the ε c a DPB d IsraelScience Foundationunder GrantNo. 231/08. One of us (HO) would like to thank the Raymond & Bev- Whenthe Debye lengthκ−1 is ofthe sameorderofmag- D erly Sackler Program for Senior Professors by Special nitudeasa(highsaltlimit),thelasttermin∆ε startsto s Appointment at Tel Aviv University. dominate and the dielectric decrement becomes smaller until eventually it will reverse the trend and may even cause a dielectric increment as seen in experiments [16] in the high salt limit. We compare our prediction for the dielectric constant [1] J. N. Israelachvili, Intermolecular and Surface Forces ε,Eq.(16),toexperimentalvalues[16]forsevendifferent (Academic Press, London, 2011). ionicsolutionsinconcentrationrangeof0–6M.Wesepa- [2] D. Andelman, in Handbook of Physics of Biological Sys- tems, edited by R. Lipowsky and E. Sackman, (Elsevier rate the seven salts into three subgroupaccordingto the Science, Amsterdam, 1995), Vol. I,Chap. 12. sizeofthealkalinecationsaspresentedinthethreeparts [3] P.Debye,PolarMolecules(ChemicalCatalog,NewYork, of Fig. 2. In each of the figure parts the a parameter is 1929). fittedseparately. Wetreataasafreeparameterandfind [4] L. Onsager, J. Am. Chem. Soc. 58, 1486 (1936). itsvaluebythebestfitofourprediction,Eq.(16),toex- [5] R.R.NetzandH.Orland,Eur.Phys.J.E1,203(2000). perimental data, while keeping the physical value of the [6] D. Henderson, L. Blum, and W. R. Smith, Chem. Phys. water dipolar moment, p0 = 1.8D. The best fit to the Let. 63, 381 (1979); Fundamentals of Inhomogeneous Fluids, edited by D. Henderson (Marcel Dekker, New data can be seen in 2(a) and corresponds to the largest ionic size of Cs+ and Rb+. In (b) the fit for K+ ions is York, 1992). [7] For a review see, e.g., D. Ben-Yaakov, D. Andelman, also quite good, while in 2(c) for the smallest ions, Na+ D. Harries, and R. Podgornik, J. Phys. Cond. Mat. 21 andLi+,thedeviationismorepronounced,especiallyfor 424106 (2009), and references therein. the larger values of cs 4M. [8] E.J.VerweyandJ.T.GOverbeek,TheoryoftheStabil- ≥ Note that our formula takes into account only in a ity of Lyophobic Colloids(Elsevier, Amsterdam, 1948). broad sense the finite size of ions (and the distance of [9] G. N. Patey and J. P. Valleau, J. Chem. Phys. 63, 2334 (1975). closest approach between them) via the a parameter. It [10] M. Sharma, R. Resta, and R. Car, Phys. Rev. Lett. 98, is beyond the level of the theory to give more specific 247401 (2007). ionic predictions. Hence, the obtainedvalue of a≃2.7˚A [11] I.KalcherandJ.Dzubiella,J.Chem.Phys.130,134507 isnotverysensitivetothe typeofsalt,butits maincon- (2009). tributioncomesfromthewaterdipolesthemselveswhose [12] S.ChowdhuriandA.Chandra,J.Chem.Phys.115,3732 diameter [23] is about 2.75˚A. On the other hand, as can (2001). beclearlyseenfromFig.2,importantcooperativeeffects [13] S. Zhu and G. W. Robinson, J. Chem. Phys. 97, 4336 (1992). ofionsanddipolesareaccountedforinournon-linearex- [14] K. S. Pitzer and D. R. Schreiber, Mol. Phys. 60, 1067 pressionforε(c ). Forsmallc ,thedashedlinerepresents s s (1987). the best linear fit and works well only when cs ≤ 1M, [15] Y. Wei and S.Sridhar, J. Chem. Phys.92, 923 (1990). while the non-linear prediction of Eq. (16) succeeds in [16] Y. Wei and S.Sridhar, J. Chem. Phys.96, 4569 (1992). fitting the large concentration range as well [24]. [17] R.Buchner,G.T.Hefter,andP.M.May,J.Phys.Chem. Inconclusion,we areable toreproduceratherwellthe A 103, 1 (1999). linearandnon-lineardielectricdecrementbehaviorovera [18] D. Ben-Yaakov, D. Andelman, and R. Podgornik, J. Chem. Phys.124, 7 (2010). largerangeofionicconcentrations,andtheobtainedval- [19] R.PodgornikandB.Zeks,J.Chem.Soc.84,611(1988). ues of ε are in quantitative agreement with the data for [20] A. Abrashkin,D. Andelman,and H.Orland,Phys. Rev. several types of monovalent salts. In addition, we found Lett. 99, 077801 (2007). a qualitative description of the hydration shell charac- [21] J. Schwinger, Proc. Natl. Acad. Sci. U.S.A 37, 452 terized by a single length scale, lh. Note that our model (1951). does not contain any significant ionic-specific effects. To [22] E. Brezin, J.C. Le Guillou, and J. Zinn-Justin, Phase improveonthelatter,wewouldneedtoincludetheionic Transitions and Critical Phenomena, Vol.VI (Academic Press, New York,1976). finite-sizeandspecificnon-electrostaticshort-rangeinter- [23] I. M. Svishchev and P. G. Kusalik, J. Chem. Phys. 99, actionsthataffectbothbulkpropertiesofionicsolutions 3049 (1993). such as dielectric constant and viscosity as well as their [24] Inthelowcs limit,weestimateγ ≃26fromEq.(17)and behavior at interfaces and, in particular, their surface for a=2.7˚A. tension. 5 80 80 80 RbCl KF LiCl 70 CsCl 70 KCl 70 NaI NaCl 60 60 60 0 0 0 ε 50 ε 50 ε 50 / / / ε ε ε 40 40 40 30 30 30 (a) (b) (c) 20 20 20 0 2 4 6 0 2 4 6 0 2 4 6 c [M] c [M] c [M] s s s FIG. 2. (color online). Comparison of the predicted dielectric constant, ε, from Eq. (16) with experimental data from Ref. [16], as function of ionic concentration, cs, for various salts. The theoretical prediction (solid line) was calculated using a as a fitting parameter. In(a)thefitisforRbClandCsClsaltswitha=2.66˚A;in(b)thefitisforKFandKClwitha=2.64˚A;whilein(c)the fit is for LiCl, NaIand NaClwith a=2.71˚A. Thedashedlines arethelinear fitto thedatain thelow cs ≤1Mrange. Theslope of the linear fit is γ =11.7M−1 in (a), 9.0M−1 in (b) and 13.8M−1 in (c). The value of γ for each salt varies by about 10-20% and thelinear fit should be takenas representative for thelow cs behavior.