Journal of Molecular Liquids 174 (2012) 135–142

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Journal of Molecular Liquids

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Internal structure of water around cations

S.J. Suresh a,⁎, Kriti Kapoor b, Sanchit Talwar c, Abhishek Rastogi a

a Unilever R&D Bangalore, 64 Main Road, Whitefield, Bangalore 560066, Indiab Department of Chemical Engineering, Indian Institute of Technology, Powai, Mumbai 400076, Indiac Department of Chemical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India

⁎ Corresponding author. Tel.: +91 9008030941; fax:E-mail address: sj.suresh@unilever.com (S.J. Suresh)

0167-7322/$ – see front matter © 2012 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.molliq.2012.07.021

a b s t r a c t

a r t i c l e i n f o

Article history:Received 27 March 2012Received in revised form 16 May 2012Accepted 16 July 2012Available online 28 July 2012

Keywords:Hofmeister seriesHydration shellWater structureHydrogen bondsDipolar alignment

In the present paper, we use statistical mechanics to probe into the changes induced by cations (Al3+, Mg2+,Ca2+, Li+, Na+, K+, Rb+, Cs+) on the structure of water. The theory aims to find the minimum free energystate, taking into account the hydrogen bonding interactions between water molecules, electrostatic interac-tions between water and the ion, and thermal energy. Water molecules in the first shell of Na+ ions are foundto largely retain their structure; the average number of H-bonds (bnHB>) and the average dipolar alignment(bθ>) of a water molecule are only marginally different from the corresponding values of bulk water. This ismade possible by the “caging” of the Na+ ions by water molecules. The magnitudes of bnHB> and bθ> are,however, found to decrease for ions on either side of Na+ ions in the Hofmeister Series. Water moleculesaround small ions with high charge density (e.g. Al3+) are found to strongly align their dipoles in the direc-tion of field, despite the reduction in the number of H-bonds per molecule. Those around large ions with lowcharge density (e.g. Cs+) are oriented such that one of their H-bond axes involving a lone pair of electrons isdirectly facing the ion, thereby maximizing their H-bond interactions at the other three bonding sites. Beyondthe first shell of all the ions studied, the degree of hydrogen bonding is similar to that of bulk water mole-cules. Changes in the molecular orientation and non-linear polarization effects, however, persist up to ~3–4 hydration shells in the case of salting-in ions, and ~7–9 shells in the case of salting-out ions.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The Hofmeister series is an arrangement of ions in the order oftheir ability to alter the solubility of various proteins in water. It isgenerally thought to be related to the ability of ions to alter the struc-ture of water. Gurney [1], for example, considered the effect of ions onthe viscosity of water. The Jones–Dole equation [2] for the viscosity ofaqueous salt solutions (up to concentrations of 1 M) states thatη−ηo� �

=ηo ¼ Affiffiffic

p þ Bc: Where c is the concentration of ion, η is theviscosity of the solution, ηo is the viscosity of pure water, and A andB are constants. The two terms on the right side are attributed tothe ion–ion interactions and the influence of ions on the H-bond net-work of water, respectively. “Structure-breakers” have large B coeffi-cients, while “structure-makers” have negative B coefficients. This isin line with the expectation that structured water should be more vis-cous than unstructured water. The lower mobilities of small ions withhigh charge densities, as compared to large ions with low charge den-sities, are also attributed to the ability of ions to either “make” or“break” the structure of water. Studies on the vibrational spectra ofsolutions containing high charge density ions have reported a slightred-shift, indicative of enhanced hydrogen bonding [3]. Employing

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classical molecular dynamics simulation techniques, Chandra [4]found a reduction in the number of H-bonds with increasing concen-tration of ions (NaCl and KCl), indicating that water molecules are sig-nificantly influenced by the presence of ions. Chandra et al. [5] furtherdeduced that in the presence of ions, pressure had a weaker influenceon the number of H-bonds per water molecule and the strength ofH-bond.

The above view has, however, been recently challenged. Gurau etal. [6] concluded, based on their studies using vibrational sum fre-quency spectroscopy, that the Hofmeister effect is primarily due tothe ions' ability to penetrate the head-group region of surfactantmonolayers thereby disrupting the hydrocarbon packing, and notdue to any long-range changes in the water structure. Batchelor etal. [7] used a new technique, pressure perturbation calorimetry, andfound no correlation between the solute's impact on water structureand its effect on protein stability. Kropman and Bakker [8] probedthe structure of water using femtosecond pump–probe spectroscopy,and found that the presence of ions (both salting-out and salting-in)does not lead to an enhancement or a breakdown of the hydrogenbond network outside nearest vicinity (first shell). Naslund et al. [9]showed that the H-bond network in bulk water, in terms of formingand breaking H-bonds as detected by XAS/XRS, remains unchanged,and only water molecules in the close vicinity to the ions are affected.Cappa et al. [10] found, using XA spectroscopy, that monovalent cat-ions induce no long-range changes to the H-bond network of water.

Fig. 1. Our model for water. d1 and d2 denote the two donor sites (protons). e1 and e2denote the two acceptor sites (lone pairs of electrons). μ

→and E

→are vectors

corresponding to the dipole moment of water and electric field due to the ion. θ isthe angle made by the dipole moment of water and the electric field due to ion. Φ isthe angle of rotation of the molecule about the dipole moment axis. ω is the angle ofrotation of the molecule about the H-bond axis.

136 S.J. Suresh et al. / Journal of Molecular Liquids 174 (2012) 135–142

Obst and Bradaczek [11] performedMD simulations and found no sig-nificant differences between H-bond structure of water molecules be-yond the second hydration shell of various cations.

One explanation for the apparent contradictions of the above ex-periments and the conventional view that the influence of ions isnot just restricted to the first hydration shell is that the H-bondsnear ions may be distorted, but not necessarily broken (analogousto what happens to pure water in high pressure); however, ionsmay perturb the orientational ordering of neighboring water mole-cules, and that this perturbation could have a knock-down effect onwater molecules in the second and subsequent hydration shells[12,13]. This explanation has also been supported by Monte Carlosimulations with revised ab initio based potential [14]; the referencevalue of the energy of water molecules in bulk was found to be recov-ered only at a distance of 10 Å away from the cations.

The present study aims to remove some of the above-mentionedcontradictions using statistical mechanics. The approach we followis similar to the one described earlier in the context of water mole-cules near a planar electrode surface [15], except that curvatureeffects due to the small size of the ions are additionally accounted for.

2. Theory

2.1. Statistical mechanics

The theory presented here is an extension of that proposed earlierfor bulk water [16] (hereafter referred to as Paper-I), and for waternear charged electrode surfaces [15] (hereafter referred to asPaper-II).

Each water molecule is assumed to be spherical in shape, and isassigned two proton donor sites (referred to as d-sites) and two pro-ton acceptor (lone pair of electrons) sites (referred to as e-sites). Di-ameter of a water molecule was taken to be the same as reported inRefs. [17,18]. Ionic radii were taken from the literature [19]. The tetra-hedral arrangement of the donor and acceptor sites, the H-bond, andthe permanent dipole moment of a water molecule is schematicallyshown in Fig. 1. The energy of a water molecule near an ion is as-sumed to depend on two angles: the dipolar angle (θ) w.r.t theE-field axis, and the angle of rotation (Φ) of the molecule about thedipole moment axis (see Fig. 1). The former angle governs the torqueexperienced by the molecule due to E-field, and hence determines thedipole energy. The later angle, at a given θ, governs the angular orien-tation of the four H-bonding axis of the molecule w.r.t the E-field axis,and hence governs its H-bond interaction. Our goal now is to evaluatethe number distribution of water molecules {N(θ,Φ,r)} in orientation (θ,Φ), and with its center at a distance r from the center of the ion.

The key assumptions related to H-bonds are: a) they can form onlybetween a donor and an acceptor site belonging to neighboring mol-ecules, b) they are not permanent; they continually break and formunder the influence of thermal energy, c) the O-d and e-O axes ofthe two molecules must be nearly collinear (within a deviation of~0.5°), d) formation of a H-bond is accompanied by the release of en-ergy (exothermic), and e) bonding at any site on a given water mol-ecule is independent of whether other sites on the same moleculeare bonded or not.

The system under consideration comprises N∘ water moleculesnear an ion. The free energy (F) of this system, in excess of thatcontaining non-polar, water-like hard-spheres, is written as:

F ¼ U−TS ¼ U−kT: ln Ωð Þ ð1Þ

where U, S, T andΩ are the internal energy, entropy, temperature andthe number of distinguishable ways of arranging molecules in thesystem. U is assumed to be composed of three terms: a) H-bond inter-actions between water molecules, b) electrostatic interactions be-tween water molecules and the ion, and c) all other non-specific

intermolecular interactions (e.g. dispersion) other than electrostaticsand H-bonding. They are termed as “physical” interactions.

U ¼ Uhb þ Uel þ Uph ð2Þ

Similarly, the term Ω in Eq. (1) is assumed to be composed of twoterms: a) Electrostatic interactions between water molecules and theion, and b) H-bond interactions between water molecules:

Ω ¼ Ωel⋅Ωhb: ð3Þ

The “physical” interactions do not contribute to Ω as they are as-sumed to be independent of molecular orientations. Combining Eqs.(1)–(3), we obtain:

F ¼ Uel−kT ln Ωelð Þf g þ Uhb−kT ln Ωhbð Þf g þ Uph: ð4Þ

We will now derive expressions for each term of the Eq. (4), usingstatistical mechanics, and find the minimum energy state of thesystem.

We first consider the electrostatic term, Uel, of Eq. (4). Our expres-sion is the same as that derived earlier by Onsager [20]:

Uel ¼ −X∞

r¼ dionþð Þ=2⋅X2πΦ¼0

⋅Xπθ¼0

Er⋅μeffr ⋅N θ;Φ;rð Þ⋅ cos θð Þ

h i; where μeff

r

¼εr⋅ R2 þ 2� �2εr þ R2� � ⋅μ ð5Þ

where dion is the diameter of the ion, σ is the diameter of water mol-ecule, R and εr are the refractive index and dielectric constant of liquidwater, μ is the permanent dipole moment of a water molecule invapor phase, and μreff is the “effective” dipole moment of a water mol-ecule in the liquid phase. εr and μreff depend on the radial position (r)of the water molecule (w.r.t the center of the ion) because they in-clude contributions from dipole moments induced by the cavity and

137S.J. Suresh et al. / Journal of Molecular Liquids 174 (2012) 135–142

reaction fields [20], which in turn originate from the E-field of the ion

Er ¼ 14πεεr

zer2

� �. e is the electronic charge and z is the valency of the ion.

Next, we consider the combinatorial term, Ωel, of Eq. (4). It ac-counts for constant disruption of dipolar orientation of water mole-cules by thermal energy, and is derived by counting the number ofdistinguishable arrangements of the set {N(θ,Φ,r)} of water dipoles inthe system [20,21]:

Ωel ¼N∘!

∏∞

r¼ dionþð Þ=2⋅ ∏

2π

Φ¼0⋅∏

π

θ¼0N θ;Φ;rð Þ!

8>>>><>>>>:

9>>>>=>>>>;

� ∏∞

r¼ dionþð Þ=2⋅ ∏

2π

Φ¼0⋅∏

π

θ¼02π⋅sinθ⋅Δθ⋅ΔΦ⋅Δrð ÞN θ;Φ;rð Þ

( ):

ð6Þ

The term (2π⋅sinθ⋅Δθ⋅ΔΦ⋅Δr) accounts for degeneracies arisingfrom the rotation of water molecule about the E-field axis (holding θconstant).

Before we consider the H-bond terms of Eq. (4), we define the rel-evant H-bond parameters. Let n(d,β,r) be the number of donor sites be-longing to water molecules whose center lies at a distance r from thecenter of the ion, and whose O−d axis lies at an angle β w.r.t theE-field axis. By mass balance, we have:

n d;β;rð Þ ¼X2πω¼0

N θc ;Φc ;rð Þ⋅Δω ð7Þ

where (θc,Φc) are the molecular orientations corresponding to the ex-tent of rotation (ω) of the molecule about the fixed constrained O−daxis (see molecule ‘1’ in Fig. 1). The subscript c implies that thecorresponding angle is constrained (in this case, by β and ω), and can-not be arbitrarily assigned. Ofn(d,β,r) donor sites, let n d;β;rð Þ be unbondedat any given instant of time, and the remainingm(d,β,r),(e,β ',r ') donor sitesbe bonded to acceptor sites of molecules whose center lies at a distancer ' from the center of the ion, and whose e−O axis lies at an angle β 'w.r.t the E-field axis (see Fig. 1). Similarly, let n(e,β ',r ') be the total num-ber of acceptor sites belonging towatermolecules whose center lies at adistance r ' from the center of the ion, and whose e−O axis lies at anangle β ' w.r.t the E-field axis. Of them, let n e;β0 ;r0ð Þ acceptor sites beunbonded. By mass balance, we have:

m d;β;rð Þ; e;β0 ;r0ð Þ ¼ n d;β;rð Þ− n d;β;rð Þ ¼ n e;β0 ;r0ð Þ− n e;β0 ;r0ð Þ: ð8Þ

The coordinates (r,β) and (r ',β ') of the two molecules shown inFig. 1 are related (through geometry) as follows:

β0 ¼ β−sin−1 σ⋅sin βð Þr0

� �ð9Þ

r0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2 þ r2 þ 2⋅σ⋅r⋅cos βð Þ

q: ð10Þ

We now consider the two H-bond terms of Eq. (4). The energeticterm, Uhb, is given by the energy released upon the formation of oneH-bond, uhb, times the total number of H-bonds formed:

Uhb ¼ uhb⋅X∞

r¼ dionþσð Þ=2⋅Xπβ¼0

m d;β;rð Þ; e;β0 ;r0ð Þ

0@

1A: ð11Þ

The combinatorial term, Ωhb, of Eq. (4) arises from the continualbreaking and formation of H-bonds under the influence of thermalenergy. It is found by counting the number of ways of distributingthe set {m(d,β,r),(e,β ',r ')} of H-bonds in the system using the methodoriginally proposed by Veytsman [22] in the context of bulk,

H-bonding fluids (and later extended to treat water molecules nearsolid surfaces [23,24] and in uniform E-fields [16]):

Ωhb ¼ ∏∞

r¼ dionþσð Þ=2⋅∏

π

β¼0

n d;β;rð Þ!n d;β;rð Þ!⋅m d;β;rð Þ; e;β0 ;r0ð Þ!

" #⋅ ∏

∞

r0¼ dionþσð Þ=2⋅ ∏

π

β0¼0

n e;β0 ;r0ð Þ!n e;β0 ;r0ð Þ!⋅m d;β;rð Þ; e;β0 ;r0ð Þ!

" #

∏∞

r¼ dionþσð Þ=2⋅∏

π

β¼0m d;β;rð Þ; e;β0 ;r0ð Þ!

" #⋅

∏∞

r¼ dionþσð Þ=2⋅∏

π

β¼0

eshb=k

V d;β;rð Þ; e;β0 ;r0ð Þ⋅ 12π⋅ sin βð Þ⋅dβ

!m d;β;rð Þ; e;β0 ;r0ð Þ" #

:

ð12Þ

The first term represents the number of ways of picking the set{m(d,β,r),(e,β ',r ')} of donor sites from the set {n(d,β,r)}. The second termrepresents the number of ways of picking the set {m(d,β,r),(e,β ',r ')} of ac-ceptor sites from the set {n(e,β ',r ')}. The third term represents thenumber of ways that the selected donor and acceptor sites can pairwith each other to form H-bonds. The term (1/V(d,β,r),(e,β ',r ')), whereV(d,β,r),(e,β ',r ') is the volume of the spherical shell within which bond-ing is possible, accounts for the probability that a selected donor–ac-ceptor pair satisfies the proximity requirement for bond-formation.shb represents the entropy loss due to changes in the frequency ofO\H covalent bond accompanying H-bond formation. The last term,

12π⋅ sin βð Þ⋅dβ, accounts for the probability that the e−O and O−d axes

(belonging to two different molecules) are collinear. The collinearitycondition may not be satisfied due to independent rotation of thetwo molecules (see Fig. 1) about the H-bond axis.

At equilibrium, all N(θ,Φ,r)* andm(d,β,r),(e,β ',r ')* have such values thatthe free energy as a function of {N(θ,Φ,r)*} and {m(d,β,r),(e,β ',r ')*} has theminimal value. i.e.

dF=kTdN�

θ;Φ;rð Þ

" #T ; mf g

¼ 0 ð13Þ

dF=kTdm�

d;β;rð Þ; e;β0 ;r0ð Þ

" #T; Nf g; all m except m�

d;β;rð Þ; e;β0 ;r0ð Þ

n o ¼ 0: ð14Þ

Combining Eqs. (1)–(12) with first criterion (see Eq. (13)) yieldsthe following expression for the equilibrium distribution of molecularorientations:

x�θ;Φ;rð Þ:Y�d;χc ;rð Þ⋅Y

�d;λc ;rð Þ⋅Y

�e;γc ;rð Þ⋅Y

�e;ψc ;rð Þ

¼ Cr⋅eμeffr ⋅Er ⋅ cos θð Þ

kT ⋅2π⋅sin θð Þ⋅Δθ⋅ΔΦ⋅Δr: ð15Þ

x(θ,Φ,r)* is the probability of a water molecule to be in (θ,Φ,r) coordi-nates. Y is the fraction of unbonded sites of the type denoted by thecorresponding subscripts. Cr arises from the presence of the physicalinteractions Uph, and hence depends only on the radial distance ‘r’ be-tween the ion and the water molecule. The angles (χc,λc,γc,ψc) madeby the four H-bond axes (w.r.t the E-field axis) are constrained(through geometry) by the angles (θ,Φ) of the same moleculethrough the following equations:

χc ¼ cos−1 − sin θð Þ sin 54:73o� �cos Φð Þ þ cos θð Þcos 54:73o� � ð16:aÞ

λc ¼ cos−1 sin θð Þ sin 54:73o� �cos Φð Þ þ cos θð Þ cos 54:73o� � ð16:bÞ

γc ¼ cos−1 − sin θð Þ sin 54:73o� �sin Φð Þ þ cos θð Þ cos 54:73o� � ð16:cÞ

ψc ¼ cos−1 sin θð Þ sin 54:73o� �sin Φð Þ þ cos θð Þ cos 54:73o� �

: ð16:dÞ

The exact form of Uph, and hence that of Cr in Eq. (15), isunimportant for the purpose of the present paper as Cr can be

138 S.J. Suresh et al. / Journal of Molecular Liquids 174 (2012) 135–142

alternatively deduced using the following space-filling constraint foreach r:

Xπθ¼0

X2πΦ¼0

x�θ;Φ;rð Þ ¼ 1: ð17Þ

Combining Eqs. (1)–(12) with the second criterion (see Eq. (14))yields the following expression for the equilibrium distribution ofH-bonds in the system:

m�d;β;rð Þ; e;β0 ;r0ð Þ

n�d;β;rð Þ⋅ n�

e;β0 ;r0ð Þ¼ e −f hb=kTð Þ

V d;β;rð Þ; e;β0 ;r0ð Þ⋅2π⋅ sin βð Þ⋅dβ ð18Þ

where fhb=uHB−Tshb. Combining Eqs. (7), (8), (15), and (18), andrecognizing that:

n�d;β;rð Þ

2π⋅ sin βð Þ⋅dβ ¼X2πω¼0

N�θc ;Φc ;rð Þ

2π⋅ sin θcð Þ⋅dθc⋅dΦc⋅dr⋅Δω: ð19Þ

We derive the following expression for the fraction of unbondeddonor sites of the type denoted by the corresponding subscripts:

Y�d;β;rð Þ ¼

n�d;β;rð Þ

n�d;β;rð Þ

¼ 1

1þ 2⋅ρo⋅e−f hb=kT⋅X2πω¼0

eμeffr ⋅Er0 ⋅ cos θcð Þ

kT

Y�d;χc ;r

0ð Þ⋅Y�d;λc ;r

0ð Þ⋅Y�e;γc ;r

0ð Þ

0B@

1CA⋅Δω:

ð20Þ

Where χc,λc,γc, and θc are the angles made by O−d1, O−d2, e2−O and dipole moment axes of molecule ‘2’ (shown in Fig. 1 ) w.r.tthe E-field axis. They are geometrically constrained by the twoangles(β,ω) of the same molecule through the following equations:

χc ¼ cos−1 sin βð Þ⋅ sin 109:47o� �⋅sin ω þ 30ð Þ− cos βð Þ cos 109:47o� �� � ð21:aÞ

λc ¼ cos−1 − sin βð Þ sin ω−30o� �sin 109:47o� �

− cos βð Þ cos 109:47o� �� �ð21:bÞγc ¼ cos−1 sin βð Þ⋅ sin 109:47o� �

⋅ cos ωð Þ þ cos βð Þ cos 109:47o� �� � ð21:cÞ

θc ¼ cos−1 sin βð Þ sin 54:73o� �cos ωð Þ þ cos βð Þ cos 54:73o� �� �

: ð21:dÞ

The expression, analogous to Eq. (20), for the fraction of unbondedacceptor sites is:

Y�e;β0 ;r0ð Þ ¼

n�e;β0 ;r0ð Þ

n�e;β0 ;r0ð Þ

¼ 1

1þ 2⋅ρo⋅e−f hb=kT⋅X2πω¼0

eμeffr ⋅Er ⋅ cos θcð Þ

kT

Y�d;χc ;rð Þ⋅Y�

e;λc ;rð Þ⋅Y�e;γc ;rð Þ

⋅Δω

0B@

1CA:

ð22Þ

Where χc,λc,γc, and θc are given by Eqs. (21.a)–(21.d).Knowing the equilibrium distributions, {Y(e,β ',r ')*}, {Y(d,β,r)*} and

{x(θ,Φ,r)*}, we can calculate two properties that can be considered torepresent the “average structure” of the liquid: the radial distributionof a) average number of H-bonds per molecule (bnHB> r) and b) aver-age dipolar angle of a molecule (bcos(θ)> r):

bnHB>r ¼

Xπθ¼0

X2πΦ¼0

1−Y�d;χc ;rð Þ

� �þ 1−Y�

d;λc ;rð Þ� �

þ 1−Y�e;γc ;rð Þ

� �þ 1−Y�

e;ψc ;rð Þ� �

24

35⋅x�θ;Φ;rð Þ⋅dθ⋅dΦ ð23Þ

b cos θð Þ>r ¼Xπθ¼0

X2πΦ¼0

x�θ;Φ;rð Þ⋅ cos θð Þ⋅dθ⋅dΦ: ð24Þ

where angles (χc,λc,γc,ψc) are given by Eqs. (16.a)–(16.d).

Finally, the mean moment of the molecule, parallel to the E-fieldof the ion, is calculated to be [16,20]:

b cos θð Þ>r⋅ 2εr þ 1ð Þ⋅μeffr

3εrþσ3⋅ε⋅εr⋅Er⋅ R2−1

� �2εr þ R2� �

0@

1A

where ε is permittivity in vacuum. Hence, polarization Pr, which ismoment per unit volume at distance r from the center of the ion, be-comes:

Pr ¼ε4π

⋅ εr−1ð Þ⋅Er

¼ ρo⋅ b cos θð Þ>r⋅ 2εr þ 1ð Þ⋅μeffr

3εrþσ3⋅ε⋅εrEr⋅ R2−1

� �2εr þ R2� �

0@

1A: ð25Þ

Eq. (25) relates the dielectric constant (εr) of the liquid and E-field(Er) at distance r from the center of the ion.

2.2. Physical interpretation of expressions

Eq. (15) has a very simple physical interpretation; the two sidesrepresent the fraction of monomeric water molecules with coordi-nates (θ,Φ,r). The left side calculates this fraction by finding the prod-uct of: a), and b) probability, Y(d,χc,r)*.Y(d,λc,r)*.Y(e,γc,r)*.Y(e,ψc,r)*, that allits four association sites are simultaneously unbonded. The right sidecalculates this fraction using Boltzmann's concepts. The orientationalpreference of a monomeric molecule (with no H-bonds) is governedprimarily by its dipole energy [−μreff⋅Er⋅cos(θ)]; hence, the fractionof monomeric molecules with coordinates (θ,Φ, r) is proportional to

eμeffr ⋅Er ⋅ cos θð Þ

kT . The term 2⋅π⋅sin(θ) accounts for degeneracies arisingfrom the dipole moment axis lying anywhere on the surface of acone of half-angle θ caused by the rotation of the molecule aboutthe E-field axis.

Eq. (18) can be identified as the equilibrium criteria for the reac-tion of the form

eþ d↔e…d ð26Þ

wherein, the forward and backward reactions represent formationand rupture of H-bonds, respectively, and the equilibrium constantis given by Khb=e(−fhb/kT). This equivalence between the so-called“Reaction Equilibrium” approach, which formed the basis of theoriesdeveloped by Wertheim [25], Stell and Zhou [26] and Heidemannand Prausnitz [27], and the currently adopted Bond-Counting ap-proach, was established earlier also, although in the context ofH-bonding fluids in bulk and near interfaces [28].

2.3. Limiting forms of expressions

Far away from the ion, Eqs. (20) and (22) for the non-bondingprobability become similar in form to that developed earlier byWertheim [25] and Veytsman [22] for bulk water. The application ofthis expression for representing H-bonding interactions has, overthe past two decades, been implemented by several workers in thecontext of different problems related to structuring and thermody-namic properties of water and aqueous mixtures. These include:

a) Favorable comparison of results of the theory for average numberof H-bonds per molecule vis a vis those extracted from neutrondiffraction, NMR, IR and Monte Carlo simulation studies [29].

b) Prediction of multi-phase equilibria [30], surface properties [23,31],dielectric constant [29,32] and heats of mixing [33] of aqueoussolutions.

139S.J. Suresh et al. / Journal of Molecular Liquids 174 (2012) 135–142

In the limit of water molecules experiencing a uniform E-field,Eqs. (20) and (22) reduce to those derived in Paper-I. The resultsfor dielectric constant of water in E-fields ranging from 0 to4×1010 V/m [16] compare well with molecular simulation data[34,35].

In the limit of the size of the ion being much larger in comparisonto that of a water molecule, and the water molecule being locatedclose to the ion, Eqs. (20) and (22) reduce to those derived inPaper-II for water molecules near a charged, planar electrode surface.The key result is that with increase in the strength of E-field, the av-erage number of H-bonds per molecule located adjacent to thecharged electrode surface decreases, while that for a bulk moleculeincreases. Similar trends were experimentally observed earlier byToney et al. [36].

2.4. Justification of the model

The question one might pose is: how can a simplistic model, suchas that proposed here, provide a reasonable picture of how watermolecules structure around ions? The problem here is two-fold,both of which are relatively less understood. The first one is relatedto the H-bond interactions. Till date, there is no universal definitionof “what constitutes a H-bond”? Some definitions of the H-bond arebased on a single operational criterion, such as infrared spectralshift, others on a listing of suitable functional groups, and still otherson a mathematical or physical picture of H-bond formation. Theseambiguities have made experimental investigations difficult. For ex-ample, the so-called “experimental data” for H-bond properties areoften extracted from spectroscopic measurements using molecularmodels, and hence are prone to model deficiencies. The second prob-lem relates to the characterization of H-bonded fluids in inhomoge-neous force fields (e.g. presence of a surface, inhomogeneousE-fields, etc.). Development of techniques for such studies has, in gen-eral, lagged behind that for homogeneous, H-bonded fluids. For ex-ample, experimental studies on the fraction of unbonded protonsfor bulk water were reported [37] as early as in 1967; however, a sim-ilar experimental study on the spatial variation of H-bond propertiesnear an ion has not yet been performed.

The understanding of complex systems, such as the one underconsideration, is not just limited by experimental difficulties. Molecu-lar modeling relies on intermolecular potential functions, which areoften not known from first principles. Hence, their predictive abilitiesare limited in scope.

Our approach, in the above context, has been to focus on two as-pects of intermolecular interactions which we believe are most signif-icant in the context of the problem at hand; a) H-bond interactionsbetween water molecules, and b) electrostatic interactions betweenthe water molecule and the ion. Clearly, the implicit assumption isthat other types of interactions, such as higher multipole effects, areof secondary importance. Rather than developing a theory startingfrom first principles, we have opted to build on past theoretical devel-opments related to the internal H-bond structure of bulk water mol-ecules [22] and its interaction with a uniform E-field [15,16]. Theprincipal advantage of this approach is that all results of the originaltheory, as well as those of extended theories for treating surface ef-fects [23,24] and molecular polarization in uniform E-fields [15,16],are valid. It gives us confidence that the theory reproduces both theH-bond structure and thermodynamic properties of water for awide range of experimental conditions (temperature, pressure,E-field strength, etc.). The disadvantage, of-course, is that it is subjectto all assumptions inherent in the original theory [22], the mostimportant one being that local density fluctuations are neglected (al-though composition fluctuations are accounted for through the treat-ment of H-bonded clusters). This limitation can be easily removed byintegrating the proposed approach of dealing with inhomogeneousfields and surface effects within existing Density Functional Theories.

However, past work in the context of bulk water and water near in-terfaces has shown that the final expressions for H-bond componentof free energy, as derived using these two different approaches, aresimilar in form [28]. Hence, it is more likely that adopting the densityfunctional approach, in the context of the current problem also, willnot significantly influence our ability to treat H-bond interactions.However, we will be unable to capture several other phenomena aris-ing out of density fluctuations, such as the “piling-up” of hard-spheresnear a surface [31]. Whether these effects will significantly alter ourresults is debatable. Our current view is that they will play a second-ary role, at least in relation to the H-bond network and inhomoge-neous field effects.

3. Results and discussions

All molecular parameters used in the present paper are the sameas those given in Papers I and II.

Fig. 2a shows the distribution of molecules in different orienta-tions around an ion with high charge density (Al3+). The fieldstrength in the first hydration shell is ~4.6×1010 V/m. Consequently,most molecules in the first shell are found with their dipoles highlyaligned (θ~6°) in the direction of E-field. The two donor sites of mol-ecules in the most probable conformation (shown in Fig. 2a) are ex-tensively H-bonded to molecules located beyond the first shell, witha bonding probability ~0.92. The two acceptor sites, on the otherhand, are not as well H-bonded (bonding probability ~0.23), due tothe absence of complementary donor sites pointing towards the ion.As a result, the average number of H-bonds per molecule in the firsthydration shell is ~2.2, a number that is significantly lower than thecorresponding bulk value of ~3.8 (Fig. 3). Dipolar alignment in thefirst shell has hence taken place at the cost of H-bond interactions.This result is to be expected since the dipole energy of a fully aligneddipole at this field is very high (~100kT), in comparison with theH-bond energy (~6kT). The situation in the second and subsequentshells is, however, quite different since the E-field decays rapidly;e.g. in the second shell, E~3.4×108 V/m, and the electrostatic energyof a fully aligned dipole is only ~0.9kT. Consequently, molecules in thesecond and subsequent layers orient themselves such that they canmaximize their H-bond interactions. Hence, disruptions to the extentof H-bonding are largely restricted to the first shell (Fig. 3), in agree-ment with the results of XAS/XRS study by Naslund et al. [9]. Howev-er, we do see structural perturbations in terms of disruptions to theaverage dipolar alignment of water molecules to extend up to ~9 hy-dration shells (Fig. 4).

The twomost favorable orientations of water molecules in the firstshell of Na+ are shown in Fig. 2b; their respective dipole angles (w.r.tE-field axis) are ~63° and 117°. Both orientations are equally likely.This result is surprising at first sight. The E-field in the first hydrationshell of Na+ is ~5×108 V/m, which is far lower that near that nearAl3+ ion; however, the dipole energy (~1.4kT) even at this weakfield is not negligible. One would hence expect the E-field to stabilizemolecules more in orientation “A” (with lower dipole angle) than in“B (with higher dipole angle). The answer lies in Fig. 3; the extentof H-bonding in the first shell is quite high, and only slightly lessthan that of bulk water. This is possible only if there is roughlyequal number of donor and acceptor sites available in the near vicin-ity of the ion for bonding. The presence of molecules in both orienta-tions “A” and “B” allows for this possibility (as seen in the schematicof the two orientations in Fig. 2b). Another way of looking at this re-sult is that the first-shell molecules in orientation “B” (with the unfa-vorably high dipole angle) are principally stabilized by those inorientation “A” (with the low dipole angle) through H-bonding inter-actions. Consequently, the average dipolar angle of a molecule in thefirst shell (~80°) is only slightly less than that of a bulk water mole-cule (Fig. 4). In contrast, recent computational studies with theTIP4P model [38], revised ab initio based potentials taking into

A

B

C

A BA B+

+

++

+

+

+

a) Al3+ b) Na+

c) Cs+

Fig. 2. Orientation of water molecules in the first hydration shell of: a) Al3+; b) Na+; c) Cs+ ions. Schematic of orientation of water molecules corresponding to peaks in the mo-lecular distribution is drawn. The grey, black and white circles denote the oxygen atom, proton acceptor and proton donor sites, respectively. The shaded grey circle with the + signdenotes the ion. The black arrow denotes the direction of dipole moment of water molecule. Refer Fig. 1 for definition of θ and Φ.

140 S.J. Suresh et al. / Journal of Molecular Liquids 174 (2012) 135–142

account many-body non-interactive effects [14] and ab initio quan-tum mechanical/molecular mechanical potentials [39] showed onlya single peak at ~48°, ~30° and ~60°, respectively, possibly suggestingthe dominance of electrostatic interactions with the ion (over H-bondinteractions with other water molecules) in determining the orienta-tion of first shell molecules.

At first instance, considering that disruptions to the average mo-lecular orientation and average number of H-bonds per molecule inthe first shell of Na+ ion are minimal, one might expect thecorresponding changes in the second and subsequent shells also to

Fig. 3. The average number of hydrogen bonds per water molecule at different dis-tances from ions.

be minimal. This is, however, not so. While Fig. 3 shows negligible in-fluence of ion on the H-bond network of water beyond the first hy-dration shell, in agreement with the X-ray spectroscopy study [9]and molecular dynamics study of Obst and Bradaczek [10], Fig. 4shows that disruptions to molecular orientations extend up to threehydration shells. The reasoning for this observation is as follows.First, the average dipolar angle in the first shell being only slightlyless than that in bulk does not imply minimal disruption to molecularorientations in the first shell; it is more due to the dipolar angle of thetwo preferred orientations (shown as “A” and “B” in Fig. 2b) being

Fig. 4. The average dipolar angle of water molecules in the different hydration shells ofions.

Fig. 5. The average number of hydrogen bonds per water molecules in the first hydra-tion shell of different ions.

Fig. 6. The average dipolar angle of water molecules in the first shell of different ions.

141S.J. Suresh et al. / Journal of Molecular Liquids 174 (2012) 135–142

roughly equally spaced out on either side of 90°. Second, these pre-ferred orientations in the first-shell influence the orientation ofsecond-shell molecules through H-bonding interactions, which, inturn, influence those of third-shell molecules, and so on. Theseknock-down effects persist up to the third hydration shell, beyondwhich, bulk structure is attained. In comparison, Carrillo-Trip [14]reported, based on a Monte Carlo study, that the reference energy ofthe energy of water molecule is recovered after ~10 Å from a Na+ ion.

Fig. 2c shows the orientation of water molecules around a relative-ly larger ion with lower charge density (Cs+). The E-field in the first

+ +

“A”

“A

“B”

b) Na) Al3+

Fig. 7. A schematic of structuring of water in the first shell of ions. The symbols are the samement, thus leading to poor availability of proton donor sites for bonding with the two accepinate over the electrostatic interactions with the ion, resulting in two preferred dipolar orien(denoted as “B”) with high dipolar angle. Consequently, the extent of bonding at all four sitesize does not permit all four sites of water in its first hydration shell to participate in H-bopoints towards the ion, thereby maximizing its H-bond interactions at its other three sites.

shell is ~2.3×108 V/m, which is weaker than that near Na+ ion dueto the larger size of Cs+ ion. Unlike in the case with Na+ ions, the mo-lecular distribution is largely skewed towards orientation “C”(Fig. 2c); the dipolar angle is ~57°, and one of the H-bond axes, asso-ciated with a lone pair of electrons, directly faces the ion. The reasonfor this difference lies in the geometric constraint on H-bonding im-posed by size of the ions and the tetrahedral arrangement of thefour H-bonding sites on a water molecule. This constraint can be bet-ter understood by considering the two extreme limits; a point charge,and an extremely large ion that presents a more-or-less planar sur-face to the water molecule. In the former limit of point charge, thepresence of the ion would impose no geometric barrier for a watermolecule to be associated with four H-bonds. In the later limit of aplanar surface, a water molecule can be involved with a maximumof only three H-bonds; the fourth H-bond axis would be necessarilypointing towards the ion (due to the tetrahedral arrangement ofsites), and hence cannot hydrogen bond. Na+ and Cs+ lie betweenthese two extreme limits; however, Cs+, being larger than Na+,would impose more geometric constraint on a water molecule inthe first shell to be involved with four H-bonds. This constraint, inturn, affects bonding patterns in the second shell, and so on. Conse-quently, the water molecules orient themselves, as shown in Fig. 2c.H-bonding interactions at three sites (two protons and one lonepair of electrons) in the first shell are maximized, as reflected bybnHB> being ~3 in Fig. 3. The fourth site, involving the second lonepair of electrons, points towards the ion, and hence does not hydro-gen bond. Similar structuring effects have been seen earlier also[40,41].

Figs. 5 and 6 summarize the influence of various cations on thestructure of water in the first shell. Na+ can be considered to be a“neutral” ion as its presence induces least disruption to the extentof H-bonding and orientation of water molecules. Conversely, theNa+ ion can hence be considered to be “caged” within the H-bondnetwork, as shown in Fig. 7b. To the left of Na+ ions lie theso-called “salting-in” cations; they are larger in size and are associat-ed with lower charge densities. At the extreme left is the Cs+ ion,which disturbs the structure of water in the first shell primarilythrough the so-called “hydrophobic” effect. The water moleculesaround them are structured as shown in Fig. 7c; one lone pair of elec-trons of a water molecule directly faces the ion and does not partici-pate in H-bonding, while the other three sites point away from theion and are extensively H-bonded. The structure of water in the firstshell of other ions, such as K+ and Rb+, that lie between Na+ andCs+ in Figs. 5 and 6, has signatures of both caging and hydrophobiceffects. To the right of Na+ ions lie the “salting-out” cations; theyare smaller in size and are associated with higher charge densities.At the extreme right is the Al3+ ion, which disturbs the structure of

+

”

“B”

c) Cs+a+

as used in Figure 2. Near the Al3+ ion, the high electric field drives strong dipolar align-tor sites (lone electron pair) of water. Near the Na+ ion, the H-bond interactions dom-tations, one (denoted as “A”) with low dipolar angle w.r.t the E-field axis and the others of water molecules in the first hydration shell is high. In the case of Cs+ ion, its largending; instead, the water molecule orients such that one of the lone-pair site of water

+

Fig. 8. A schematic of structuring of water in the first few shells of Na+ ion. The sym-bols are the same as used in Fig. 7. Molecular orientations in the first shell (which isthe same as that shown in Fig. 7b) are predominantly governed by the interplay be-tween H-bonding, E-field due to the ion and steric factors. These disruptions thenhave a knock-down effect on molecular orientations of farther water molecules, medi-ated through H-bonding interactions.

142 S.J. Suresh et al. / Journal of Molecular Liquids 174 (2012) 135–142

water in the first shell primarily through ion-induced dipolar align-ment (Fig. 7a). The strong electric field around the ion causes watermolecules to align such that the lone pairs of electrons are directedtowards the cation, although this alignment does not favor formationof H-bonds between water molecules in the first shell. The proton ac-ceptor sites (lone electron pairs) are weakly H-bonded due to thescarcity of proton donor sites immediately adjacent to the ions,while the proton donor are well bonded to water molecules fartheraway from the ion. The structure of water in the first shell of otherions, such as Ca2+ and Mg2+, that lie between Na+ and Al3+ inFigs. 5 and 6, has signatures of both caging and dipole orderingeffects.

4. Conclusions

Our key conclusions are as follows:

a) The long-range influence of cations on the H-bond interactions ofwater is negligible. A similar conclusion was arrived at in somepast studies in the case of Mg2+ ions [9,42,43], Al3+ ions [9],and K+ ions [9–11]. The molecular dynamics study of Chandra[4,5], however, does not agree with this conclusion; the averagenumber of H-bonds per water molecule was found to decreasewith increasing ion concentration, suggesting a long-range influ-ence of ions on the H-bond structure of water.

b) Disruptions to the orientational ordering of water molecules arelong-ranged; 3–4 hydration shells in the case of salting-in cations,and 7–9 shells in the case of salting-out cations. This observation isconsistent with the deductions by Leberman and Soper [12] andMancilleni et al. [13], based on neutron diffraction data, that theprincipal effect of ions is to perturb the orientational distributionof neighboring water molecules, which then has a knock-on effectin the second and subsequent shells.

The physical explanation for the above conclusions in our work isas follows. H-bonding of water molecules in the first hydration shellof cations is disrupted either due to the steric hindrance of the ionor due to the high dipole ordering by the ionic field. In either case,

the direct influence of ions significantly diminishes in the secondshell and beyond. For example, even in the case of a highlysalting-out cation such as Al3+, the field in the first hydration shellis of the order of 1010 V/m while that in the second shell is only ofthe order of 108 V/m (corresponding to a dipole energy of ~kT,which is small compared to the H-bond energy of ~6kT). Hence, mol-ecules in the second shell and beyond adjust their orientations inorder to maximize their H-bond interactions. However, since theO−d and the e−O axes of two H-bonded molecules must satisfythe angular criteria, it takes several shells for the orientational pertur-bation, induced by the first shell molecules, to fade away (as shownschematically in Fig. 8).

References

[1] R.W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.[2] G. Jones, M. Dole, Journal of the American Chemical Society 51 (1929)

2950–2964.[3] F. Franks, in: Water: A Comprehensive Treatise, Vol. 3, Plenum, London, 1973.[4] A. Chandra, Physical Review Letters 85 (2000) 768–771.[5] A. Chandra, S. Chowdhuri, The Journal of Physical Chemistry. B 106 (2002)

6779–6783.[6] M.C. Gurau, S.M. Lim, E.T. Castellana, F. Albertorio, S. Kataoka, P.S. Cremer, Journal

of the American Chemical Society 126 (2004) 10522–10523.[7] J.D. Batchelor, A. Olteanu, A. Tripathy, G.J. Pielak, Journal of the American Chemi-

cal Society 126 (2004) 1958–1961.[8] M.F. Kropman, H.J. Bakker, Science 291 (2001) 2118–2120.[9] L.A. Naslund, D.C. Edwards, P. Wernet, U. Bergmann, H. Ogasawara, L.G.M.

Pettersson, S. Myneni, A. Nilsson, The Journal of Physical Chemistry. A 109(2005) 5995–6002.

[10] C.D. Cappa, J.D. Smith, B.M. Messer, R.C. Cohen, R.J. Saykally, The Journal of Phys-ical Chemistry. B 110 (2006) 5301–5309.

[11] S. Obst, H. Bradaczek, The Journal of Physical Chemistry 100 (1996)15677–15687.

[12] R. Leberman, A.K. Soper, Nature 378 (1995) 364–366.[13] R. Mancinelli, A. Botti, F. Bruni, M.A. Ricci, A.K. Soper, Physical Chemistry Chemical

Physics 9 (2007) 2959–2967.[14] M. Carrillo-Tripp, H. Saint-Martin, I. Ortega-Blake, Journal of Chemical Physics

118 (2003) 7062–7073.[15] S.J. Suresh, Journal of Chemical Physics 126 (2007) 204705.[16] S.J. Suresh, A.V. Satish, A. Choudhary, Journal of Chemical Physics 124 (2006)

074506.[17] P.G. Kusalik, I.M. Svishchev, Science 265 (1994) 1219–1221.[18] M.W. Mahoney, W.L. Jorgensen, Journal of Chemical Physics 112 (2000)

8910–8922.[19] Y. Marcus, Ion Solvation, Wiley-Interscience, New York, 1985.[20] L. Onsager, Journal of the American Chemical Society 58 (1936) 1486–1493.[21] S.J. Suresh, Journal of Chemical Physics 122 (2005) 134502.[22] B.A. Veytsman, The Journal of Physical Chemistry 94 (1990) 8499–8500.[23] S.J. Suresh, V.M. Naik, Journal of Chemical Physics 109 (1998) 6021–6042.[24] S.J. Suresh, V.M. Naik, Journal of Chemical Physics 111 (1999) 10389–10390.[25] M.S. Wertheim, Journal of Statistical Physics 42 (1986) 477–492.[26] G. Stell, Y.Q. Zhou, Journal of Chemical Physics 91 (1989) 3618–3623.[27] R.A. Heidemann, J.M. Prausnitz, Proceedings of the National Academy of Sciences

73 (1976) 1773–1776.[28] S.J. Suresh, in: Encyclopedia of Surface and Colloid Science, Marcel Dekker, New

York, 2002, p. 2455.[29] S.J. Suresh, V.M. Naik, Journal of Chemical Physics 113 (2000) 9727–9732.[30] M.N. Garcia-Lisbona, A. Galindo, G. Jackson, A.N. Burgess, Journal of the American

Chemical Society 120 (1998) 4191–4199.[31] C.J. Segura, W.G. Chapman, K.P. Shukla, Molecular Physics 90 (1997) 759–772.[32] S.J. Suresh, The Journal of Physical Chemistry. B 108 (2004) 715–720.[33] C. Panayiotou, I.C. Sanchez, The Journal of Physical Chemistry 95 (1991)

10090–10097.[34] I.C. Yeh, M.L. Berkowitz, Journal of Chemical Physics 110 (1999) 7935–7942.[35] G. Sutmann, Journal of Electroanalytical Chemistry 450 (1998) 289–302.[36] M.F. Toney, J.N. Howard, J. Richer, G.L. Borges, J.G. Gordon, O.R. Melroy, D.G.

Wiesler, D. Yee, L.B. Sorensen, Nature 368 (1994) 444–446.[37] W.A.P. Luck, Discussions of the Faraday Society 43 (1967) 115–127.[38] L. Degreve, S.M. Vechi, C. Quintale, Biochimica et Biophysica Acta 1274 (1996)

149–156.[39] A. Tongraar, K.R. Leidl, B.M. Rode, The Journal of Physical Chemistry 102 (1998)

10340–10347.[40] C.Y. Lee, J.A. McCammon, P.J. Rossky, Journal of Chemical Physics 80 (1984)

4448–4455.[41] G.M. Torrie, P.G. Kusalik, G.N. Patey, Journal of Chemical Physics 88 (1988)

7826–7840.[42] A.W. Omta, M.F. Kropman, S. Woutersen, H.J. Bakker, Journal of Chemical Physics

119 (2003) 12457–12461.[43] A.W. Omta, M.F. Kropman, S. Woutersen, H.J. Bakker, Science 301 (2003)

347–349.