Varying the charge of small cations in liquid water ... the charge of small cations in liquid water: Structural, transport, and thermodynamical ... effect of convection on the polarization

Varying the charge of small cations in liquid water: Structural, transport, andthermodynamical propertiesFausto Martelli, Rodolphe Vuilleumier, Jean-Pierre Simonin, and Riccardo Spezia Citation: The Journal of Chemical Physics 137, 164501 (2012); doi: 10.1063/1.4758452 View online: http://dx.doi.org/10.1063/1.4758452 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/137/16?ver=pdfcov Published by the AIP Publishing

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THE JOURNAL OF CHEMICAL PHYSICS 137, 164501 (2012)

Varying the charge of small cations in liquid water: Structural, transport,and thermodynamical properties

Fausto Martelli,1 Rodolphe Vuilleumier,2 Jean-Pierre Simonin,3 and Riccardo Spezia1,a)

1Université d’Evry Val d’Essonne, Laboratoire Analyse et Modélisation pour la Biologie et l’Environnement,CNRS, UMR 8587, Boulevard F. Mitterrand, 91025 Evry Cedex, France2Ecole Normale Supérieure, Département de Chimie, Paris, France and UPMC Université Paris 06, UMR8640 CNRS-ENS-UPMC, Paris, France3Laboratoire PECSA (UMR CNRS 7195), Université P.M. Curie, 4 Place Jussieu 75005 Paris, France

(Received 1 August 2012; accepted 26 September 2012; published online 22 October 2012)

In this work, we show how increasing the charge of small cations affects the structural, thermodynam-ical, and dynamical properties of these ions in liquid water. We have studied the case of lanthanoidand actinoid ions, for which we have recently developed accurate polarizable force fields, and theionic radius is in the 0.995–1.250 Å range, and explored the valency range from 0 to 4+. We foundthat the ion charge strongly structures the neighboring water molecules and that, in this range ofcharges, the hydration enthalpies exhibit a quadratic dependence with respect to the charge, in linewith the Born model. The diffusion process follows two main regimes: a hydrodynamical regime forneutral or low charges, and a dielectric friction regime for high charges in which the contraction ofthe ionic radius along the series of elements causes a decrease of the diffusion coefficient. This latterbehavior can be qualitatively described by theoretical models, such as the Zwanzig and the solvatedion models. However, these models need be modified in order to obtain agreement with the observedbehavior in the full charge range. We have thus modified the solvated ion model by introducing a de-pendence of the bare ion radius as a function of the ionic charge. Besides agreement between theoryand simulation this modification allows one to obtain an empirical unified model. Thus, by analyz-ing the contributions to the drag coefficient from the viscous and the dielectric terms, we are ableto explain the transition from a regime in which the effect of viscosity dominates to one in whichdielectric friction governs the motion of ions with radii of ca. 1 Å. © 2012 American Institute ofPhysics. [http://dx.doi.org/10.1063/1.4758452]

I. INTRODUCTION

Since the time of Arrhenius many efforts have been madeby physicists and chemists to understand ionic hydration. Thedetermination of hydration numbers of ions, ion-solvent equi-librium distances, ionic radii, rates of exchange of coordi-nated water molecules around ions, interaction energies be-tween ions and water molecules, and diffusion coefficients aresome of the properties that have been investigated at a theoret-ical and an experimental level. In the last decades, Ohtaki andRadnai,1 Helm and Merbach,2 and Marcus3, 4 have reviewedresults for these properties up to the last century through thewhole periodic table.

Although we have passed the centennial of the first theo-retical studies on molecular diffusion5 and many experimentaldata have been accumulated so far, the study of ionic diffu-sion in water is still a strongly active field, in particular withrespect to transport properties. Actually, this process is stillnot fully understood. Macroscopic Stokes’ law can be trans-ferred down to the atomic scale to describe the diffusion of abrownian particle, e.g., a neutral atom whose motion in wa-ter is determined by random collisions with water molecules,or when the solute size is much larger than the solvent size

a)Author to whom correspondence should be addressed. Electronic mail:[email protected].

(colloids). In these cases, the drag coefficient is linear withrespect to the solute radius, Ri, and to the solvent viscosity η,

ζv = cπηRi, (1)

with c = 4 or c = 6 in the case of slip or stick conditions,respectively. However, Stokes’ law can be wrong if the soluteis charged and is placed in a polar solvent. One may proposethat Stokes’ law be corrected by replacing the ionic radius,Ri, by an effective (larger) radius for the ion-solvent complex,that corresponds to a first hydration shell rigidly bound to theparticle. However, this assumption does not take into accountthe coupling between the motion of the ion and the relaxationof the dielectric medium around it.

The total friction is then the sum of two terms: the viscous(or hydrodynamic) friction, ζv , and the dielectric term, ζ D,

ζ = ζv + ζD. (2)

Models for dielectric friction were proposed by Born,6

Fuoss,7 Boyd,8 Zwanzig,9, 10 Hubbard and Onsager,11, 12

Adelman13 and Wolynes.14–16 A mode coupling theory hasbeen proposed by Chandra and Bagchi.17 Born6 model isbased on the fact that the rotation of a homogeneously po-larized dielectric should create no dissipation so that the re-laxation should depend on the local rate of change of thepolarization in a reference frame rotating with the liquid.Hubbard and Onsager11, 12 adopted a reference frame which

0021-9606/2012/137(16)/164501/11/$30.00 © 2012 American Institute of Physics137, 164501-1

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164501-2 Martelli et al. J. Chem. Phys. 137, 164501 (2012)

moves with the local velocity of the liquid, so that the rigidbody motion of a non-homogeneously polarized dielectricproduces no dissipation. Zwanzig9, 10 included this effect ofconvection on the polarization field considering its pertur-bation by local rotation. Adelman13 included a desolvationfunction which depends on the local structure and dynamicsof the solvent, studying solvation properties of ions in non-aqueous solutions. Wolynes and co-workers14–16 presented anon-analytical, numerically fitted procedure for ion solvationin water.

In the literature, one can find relevant data for lowcharged ions in water (Li+, Na+, Cs+, Zn2+, Hg2+, F−,I−3 , etc.): on ion mobilities, thermodynamical, structural andsolvent-induced symmetry making/breaking properties.4, 18–21

It has been observed that the faster diffusing alkali andhalide ions in water are Rb+ and Br−, respectively.19, 22 Inan attempt to explain why the conductivity of alkali ions inwater increases with their ionic radius when going from Li+

to Rb+, while it then decreases for the larger ion Cs+ (ortetramethyl ammonium), the role of dispersion interactionshas been investigated recently.23–25 Studies about ionic diffu-sion in molten salts by Madden and co-workers26, 27 pointedout that the effective hydrodynamic radius switches from abare value to that of the solvation shell, correlating this switchto the relative time scales of coordination shell relaxation andstress relaxation time.

However, lanthanoids (Ln) and actinoids (An) exist ashighly charged ions in aqueous solution and little has beenreported in order to explain their dynamical behavior in polarsolvents, besides an important role of dielectric friction.28 Inwater, Ln’s and An’s from Am to the end of the series form3+ cations, as also do some of the lighter An’s (U, Np, Pu) un-der certain conditions. Note that Th, U, and Pu can form 4+cations.29 They have been recently studied in our and othergroups in terms of geometrical, thermodynamical, and diffu-sive properties.2, 28, 30–35

An overview of the main theoretical and experimental re-sults about Ln(III) and An(III) properties is given in Figure 1and reported in the recent review of D’Angelo and Spezia.36

While some structural and thermodynamical properties werewell recovered in various simulations, a clear connection be-tween those and the role of (strong) electrostatic interactionsis still missing. In particular, one may wonder why the diffu-sion coefficients decrease across the Ln series while the ion-water distances decrease. This is contrary to what Stokes’ lawpredicts, and this behavior has been used in the literature toquestion the reliability of the structural properties.37, 38 Theradii of Ln ions in water decrease along the series (from 1.250to 0.995 Å39). Since these ions are small and highly charged,dielectric friction may have a big role in determining their dif-fusion properties.

With the purpose of clarifying this point and to under-stand the different contribution to the friction, we analyze inthis work the behavior of four Ln ions: Ce, Sm, Ho, and Yb,thus spanning the series from bigger to smaller ions.

In order to investigate the effect of the cation charge ondiffusion (and other properties) we have considered the inter-val of valency, q, from 0 to 4. This allows us to determinefor which charge values (in this ionic radius range) Stokes

2.25

2.4

2.55

2.7

d ion-

oxyg

en (A

)

8.18.48.7

9

CN

-3750

-3600

-3450

-3300

ΔH (

kcal

/mol

)

La Ce Pr Nd U

Pm Np

SmPu

EuAm

GdCm

TbBk

DyCf

HoEs

ErFm

TmMd

YbNo

LuLr

0.54

0.57

0.6

0.63

D (

10-5

cm

2 s-1

)

(a)

(b)

(c)

(d)

FIG. 1. Main literature results for hydration properties of Ln(III), as circles,and An(III), as triangles, ions in water : ion-oxygen distance (panel (a)), waterfirst shell coordination number (panel (b)); hydration enthalpy (panel (c));ionic diffusion coefficient (panel (d)). Simulations are as open symbols andexperimental as filled ones. Data are from Refs. 4, 28, 36, 37, 39 and 64.

regime is replaced by a dielectric friction diffusional regime.We have thus investigated different analytical models of theliterature and shown that solvated ion models can constitute agood model, provided that the size of the first hydration shellis taken as depending on the charge. We have also reported adetailed analysis of the structural and transport properties asa function of the cation charge in order to investigate the ef-fect of the latter on the structuring of the surrounding watermolecules and on the diffusion of both species. Moreover hy-dration enthalpies were analyzed as a function of the charge,showing that a simple parabolic behavior, thus following theBorn equation that does not take into account atomic level de-tails, is recovered by our results, strengthening the picture inwhich electrostatic interactions govern most of the propertiesof these small cations in water.

II. COMPUTATIONAL DETAILS

A. The force field

The total potential energy of our system is modeled as asum of three terms,

Vtot = Velec + VLJO−O + VM−O, (3)

where Velec is the electrostatic energy term composed of aCoulomb and a polarization term following Thole’s induceddipole model,40 VLJ

O−O is the 12-6 Lennard-Jones potential de-scribing the O–O interaction, and VM−O accounts for the non-electrostatic metal-oxygen interaction potential. Because ofthe explicit polarization introduced in the model, the originalTIP3P water41 was replaced by the TIP3P/P water model,42

i.e., the charges on O and H were rescaled to reproduce thedipole moment of liquid water. For VM−O, we have used apotential composed of a long range attractive part with a 1/r6

behavior and a short range repulsive part modeled via an ex-ponential function, dealing with the well-known Buckingham

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164501-3 Martelli et al. J. Chem. Phys. 137, 164501 (2012)

TABLE I. Radial distribution function (rdf) of Ce and Yb for differentcharge values. q is the charge of the metal cation (M); M–O is the oxygenfirst peak position in the rdf; M–H the hydrogen first peak position in the rdf.δO−H is the difference between the oxygen and hydrogen peaks positions.Results are reported in Å.

M–O M–H δO−H

q Ce Yb Ce Yb Ce Yb

0 3.05 2.88 2.85 2.65 0.20 0.230.5 3.00 2.80 2.89 2.70 0.11 0.101 2.90 2.70 2.92 2.97 –0.02 –0.271.5 2.79 2.59 3.34 3.15 –0.55 –0.562 2.65 2.49 3.27 3.11 –0.62 –0.622.5 2.58 2.40 3.20 3.15 –0.62 –0.653 2.50 2.34 3.17 2.99 –0.67 –0.653.5 2.44 2.25 3.10 3.05 –0.66 –0.654 2.35 2.27 3.00 2.92 –0.65 –0.65

potential

VM−O = AM−O exp (−BM−OrM−O) − CM−O

r6M−O

. (4)

This expression applied to Ln(III) and An(III) in wa-ter was shown to be able to correctly reproduce variousproperties.28, 30–32 This interaction potential, in conjunctionwith the TIP3P/P model for water, was able, for example,to accurately reproduce EXAFS spectra.39 With regards tothe potential between water molecules, many force fields arepresent in the literature and some are able to better describeliquid water properties.43, 44 However, this potential was cho-sen in the present case for its ability to very well describeLn(III) and An(III) experimental hydration properties. Fur-ther, the parameters of the Buckingham potential can be re-lated to ionic radii, as shown recently in our group32 and inprevious works of Madden and co-workers.45, 46 Details onthe meaning of the parameters for Ln(III) and An(III) usedin the present study are reported in Refs. 30–32, while param-eters used in the present study are reported in Table I of thesupplementary material.47

The Coulomb interaction was tuned by changing themetal cation valency from the original 3+ value of Ln(III)and An(III) ions to different values in the q ∈ [0; 4] interval,while keeping all other terms fixed. We have considered thefollowing q values: 0, +0.2, +0.5, +1, +1.5, +2, +2.5, +3,+3.5, +4.

B. Simulation details

Our simulation systems consist of one ion (or neutralatom when q = 0) immersed in cubic boxes composed by216, 500 or 1000 water molecules, with box edges of 18.64,24.84, and 31.05 Å, respectively. Periodic boundary condi-tions (PBC) were applied to each simulation box in orderto mimic bulk conditions. Long-range interactions were cal-culated by using the smooth particle mesh Ewald method.48

Simulations were performed using a velocity-Verlet-basedmultiple time step in the microcanonical NVE ensemble withour own developed CLMD code MDVRY.49 The extended La-grangian method to propagate induced dipoles in time50 and

Thole’s induced dipole model40 was used. Note that the dy-namics of the induced dipole degrees of freedom is fictitious.It only serves the purpose of keeping the induced dipoles closeto their values at minimum energy, that would be obtainedfrom the exact solution of the self-consistent equations at eachstep. Dipoles and temperature stability were verified along thepresent simulations (induced dipoles obtained with SCF arewithin the oscillations observed using the dipole dynamics)while a more detailed report of the performances of the ex-tended Lagrangian implementation is reported in Ref. 49.

The equations of motion were numerically integrated us-ing a 1 fs time step. The system was equilibrated at 298 K for2 ps. Production runs were subsequently collected for 3 ns.

Diffusion coefficients were calculated from simulationsvia the well-known Einstein relationship involving the meansquare displacement (MSD),

D = limt→∞

d

dt

〈|r(t) − r(0)|2〉6

, (5)

where D is the value obtained for each simulation box. Since,as shown by Hummer and co-workers,51 diffusion coefficientsdepend on simulation box size when PBC are employed, weextrapolated the values obtained for each ion at different boxdimension to obtain the D values at infinite dilution. The ex-trapolation procedure has been conducted by performing sim-ulations for each system with the three cubic box sizes. Forany 3 ns run, the trajectory was divided in 10 blocks, andeach block in three sub-blocks. The slope of the MSD wascalculated in the middle portion of each sub-block. In thisway, potential errors coming from the ballistic regime (thefirst part of the sub-blocks) and from numerical errors (thelast part of the sub-blocks) were minimized. Then, we havecalculated the diffusion coefficients by averaging the slopesof each MSD curve, similar to what was done by Kerisit andLiu.52 The uncertainty of the calculated diffusion coefficients,obtained from the standard deviation of the averaged values,resulted to be of about 0.2 × 10−6 cm2 s−1. Mean residencetimes (mrt) have been calculated using the Impey method53

with the value t* = 0.5 ps.

III. STRUCTURAL PROPERTIES

A. Radial distribution functions (RDFs)

We first describe how the effect of progressively charg-ing the Ln ions affects the structural properties in terms ofmetal-water RDFs. For the sake of clarity, we report resultsobtained for Ce and Yb–the metals with the biggest and thesmallest radii in the series we study here (see Table I of thesupplementary material47)–and with the polarizable potentialthat provides the best agreement for structural data in the caseof Ce(III) and Yb(III) in water.30, 31 The peak positions for themetal-oxygen and metal-hydrogen RDFs are presented in thefirst two columns of Table I. Switching on and progressivelyincreasing the valency of the metal cation has the effect ofdecreasing the metal-oxygen distance, while it has the oppo-site effect on the metal-hydrogen distance, as expected. Thisbehavior shows how the water molecules around the cationbehave when the ion charge is increased. Thus, the increase

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164501-4 Martelli et al. J. Chem. Phys. 137, 164501 (2012)

(a) (b)

(c) (d)

FIG. 2. Three-dimensional radial distribution functions and corresponding 2D maps for Ce-O (panels (a) and (c)) and Yb-O (panels (b) and (d)) as a functionof q (the metal ion valency).

of the metal-oxygen interaction (the oxygen bearing a localpartial negative charge) causes the water molecules to arrangethemselves neatly around the cation. Thus, we can use thedifference between metal-oxygen and metal-hydrogen RDFspeaks, δ = dM−O − dM−H, as a useful parameter carrying thisinformation. Values of δ for different cation charges are re-ported in the last column of Table I. Positive values for δ rep-resent geometries in which the hydrogen atoms are closer tothe ion than the oxygen atoms. They were obtained for small(or zero) values of q (the cation valency). This means thatthe metal ion positive charge does not interact so stronglywith the surrounding water as to induce a minimal structur-ing. On the other hand, negative values for δ represent ge-ometries in which the oxygen atoms are closer to the ion thanhydrogen atoms. This represents structures in which the pos-itive ion charge draws the partially negative oxygen chargecreating a local ordered structure. For Ce, a net inversion onδ from positive to negative values appears at q = 1.5, whilefor Yb it appears at q = 1. Further increase of the value ofthe charge leads to a limiting value of δ of ∼−0.65, whichis consistent with water molecules having the HOH bisec-tors aligned with the metal-oxygen line (see below). A bet-ter visualization of the water structuring around the metal canbe seen from metal-water RDFs as a function of the charge.Figure 2 shows these RDFs for Ce-O and Yb-O as a func-tion of q and the corresponding two-dimensional contour plot.

First, for Ce ions we observe that the intensity of the firstRDF peak increases with the charge and that this peak comescloser to the ion. Furthermore, a clear second shell structureemerges for q ≥ 1.5. As for the first shell peak, the secondshell peak grows in intensity and gets closer to the ion whenthe ion charge is increased. Finally, for q ≥ 2.0 we can ob-serve the creation of a small third peak. A similar behav-ior is observed for Yb. In this case, the third peak is moreintense and appears at lower q values, due to the strongermetal-oxygen attraction of Yb as compared to Ce for thesame q values. This behavior is due to the fact that the Ybionic radius is smaller than that of Ce (1.010 Å and 1.220 Å,respectively.39)

The effect of the ion charging on the reorientation of wa-ter molecules can be also observed, from a structural point ofview, from the metal-hydrogen RDFs. In Figure 3, we showthe three dimensional Ce-H and Yb-H RDFs and the corre-sponding contour plots. We note that, for q = 0 and q = 0.5,the RDFs are characterized by a broad first peak, a secondpeak not well separated from the first one, and a small broaderthird one. When the charge is increased, the first peak be-comes sharper and well separated from the second one–thiscan be also seen as a merging of the two peaks correspondingto the two hydrogen atoms of water molecules in the first hy-dration shell that are structured toward the outer sphere of theatom as the charge is increased.

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(a) (b)

(c) (d)

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3 3.5 4

q

0

1

2

3

4

5

6

7

8

9

r(A

ng)

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4

q

0

1

2

3

4

5

6

7

8

9

r(A

ng)

FIG. 3. Three-dimensional radial distribution functions and corresponding 2D maps for Ce-H (panels (a) and (c)) and Yb-H (panels (b) and (d)) as a functionof q (the metal ion valency).

B. Angular distribution functions (ADFs)

To better characterize the orientation of the watermolecules in the first hydration shell, we now report theoxygen-metal-oxygen ADF, for which the oxygen atoms arein the first shell of the metal. Panel (a) of Figure 4 clearlyshows the case of Ce where for q ≥1.5 a structuring ap-pears: two peaks at about 70◦ and 135◦, with a coordina-tion number (CN) of 9, corresponding to a tricapped prismstructure.30, 31, 54 In the case of Yb (see panel (b)) of Figure 4),we have up to q = 2.5 a situation similar to that of Ce (i.e.,two peaks at about the same angles and CN ∼ 9). When q isincreased a third peak appears at around 110◦ correspondingto CN ∼ 8. Note that for low charges, one does not observea defined structuring of water around the metal (for both Ceand Yb). Coordination numbers for each ion as a function ofq are reported in Table II.

It is possible to understand the flexibility of watermolecules in the first shell by analyzing the tilt angles be-tween the metal-oxygen vector and the plane defined by thewater molecule. Corresponding distribution functions are re-ported in Figure 5 where we also show the angles we consider.The shapes are symmetric and, as expected, centered around0◦. For neutral species and low charged ions, they present twosmall peaks at ±60◦ with an almost flat region in between.This results from the low (or zero) structuration of first shell

such that water molecules can move almost freely. Startingfrom q = 1.5 for Yb and q = 2 for Ce a unique peak ap-pears at 0◦ whose intensity increases with the charge, becauseof the increasing structuration of the first shell and loss ofwater mobility in the shell. As expected the tilt angle distribu-tions for Yb are sharper than for Ce. Further, these distribu-tions reach zero at values between ±90◦ and ±60◦ for neutralspecies and low charged ions (up to q = 2.5), correspond-ing to a high flexibility of the first hydration shell, while for

TABLE II. First (CN(1)) and second (CN(2)) solvation shell coordinationnumbers for Ce and Yb as a function of ion charge.

CN(1) CN(2)

q Ce Yb Ce Yb

0 14.6 12.7 32.4 32.50.5 13.8 12.1 31.3 19.21 12.6 10.3 29.0 32.61.5 10.2 8.7 21.8 19.32 9.3 8.4 21.7 21.52.5 9.1 8.2 21.7 20.83 9.0 8.1 21.0 18.23.5 9.0 8.0 19.2 17.54 9.0 8.0 18.5 17.3

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164501-6 Martelli et al. J. Chem. Phys. 137, 164501 (2012)

FIG. 4. Oxygen-metal ion-oxygen angular distribution function for the dif-ferent charge values. Ce in panel (a) and Yb in panel (b).

highly charged ions they reach zero from ±50◦ to ±40◦ indi-cating more rigid structures. Note that the charge value fromwhich a clear water structuring appears around the metal issimilar to what was pointed out in previous structural analy-sis. The concept of rigidity of the first shell can be related tothe residence time of the water molecules around the ion andto the hydration enthalpy. This further analysis is presented inSec. IV.

Based upon structural characterization, Migliorati et al.55

have recently observed that Zn2+ ions preserve the hydrogenbond pattern of bulk water, while ions with the same chargebut with larger ionic radii, like Hg2+, tend to destroy this pat-tern. Following the same structure-based spirit, we have stud-ied the role of the cation charge on the hydrogen bond pattern.As suggested by Migliorati et al.,55 the formation of an or-dered water structure around the ionic solute can be observedfrom the O1-O*-O2 angle distribution, O* being an oxygenon the second shell of the metal and O1 and O2 oxygens be-ing in the first shell of O*. Figure 6 shows the correspondingADF for Ce and Yb. Distributions are very similar to that ofpure water for q = 0 and q = 0.5 with the low value peak (atabout 0.4) smaller in intensity than the large and broad peakbetween 1 and 1.5. Increasing the valency, part of the broadpeak distribution is shifted toward lower values, such that forq = 1, 1.5, and 2 the distribution shows a large maximum inthe 0.5–0.7 values range. Then when the charge increases thelow value peak becomes dominant since the ion now decon-

FIG. 5. Angular distribution functions for the tilting angle at different chargevalues. Ce in panel (a) and Yb in panel (b).

structs the liquid water structure around, similar to what ob-served by Migliorati et al.55 The difference between Ce andYb is small, and it only results in higher values of the low peakfor q = 4. The size effect on the second shell is not seen prob-ably because for low charges the ionic radius difference is nothigh enough to produce such an effect, and for high chargesthe higher structuration of Yb due to electrostatics dominates.

IV. THERMODYNAMIC AND TRANSPORTPROPERTIES

A. Hydration enthalpies

The formation of ordered water structures around theion depends on the ion-water interaction strength and, re-versely, it has an effect on thermodynamic properties likehydration enthalpies. These, denoted �Hhyd, were calculatedfrom molecular dynamics simulations with the biggest box(containing 1000 water molecules) and proper correctionswere applied.56–58 The results are reported in Table III for Ceand Yb. The hydration enthalpies are very small for neutralspecies and the differences are small between low chargedions. They increase rapidly however while increasing thecharge, and we observe a quadratic dependence of the ab-solute value of �Hhyd with the charge. Note that we havein Ref. 28 determined hydration enthalpies for the wholelanthanoids(III) and actinoids(III) series, displaying good

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164501-7 Martelli et al. J. Chem. Phys. 137, 164501 (2012)

FIG. 6. Distribution functions of the O1-O*-O2 angles at different chargevalues. Ce in upper panel and Yb lower panel.

agreement between our results and experimental data,4 thusvalidating our force field and procedure. Among those, Ce3+

and Yb3+ are Ln cations for which experimental results areavailable.

This quadratic behavior of �Hhyd as a function of so-lute charge follows the Born-Bjerrum model,59 based on Born

TABLE III. Thermodynamic properties of Ce and Yb in water for differentq values. �Hhydr is hydration enthalpy in kJ mol−1, mrt is water first shellmean residence time in ps, and D is diffusion coefficient in 10−6 cm2 s−1.

�Hhydr mrt D

q Ce Yb Ce Yb Ce Yb

0 –184.4 –177.4 14.2 18.0 22.0 27.20.2 –188.1 –194.3 14.6 18.4 18.1 20.30.5 –321.7 –317.1 29.0 28.0 13.8 12.71 –505.2 –548.6 78.3 76.2 13.6 12.51.5 –916.7 –1035.4 86.0 215 11.8 11.02 –1617.9 –1608.1 167 269 10.8 10.42.5 –2364.7 –2555.2 530 383 7.6 7.33 –3382.3 –3721.3 891 300 6.8 6.33.5 –4521.4 –4990.9 1134 1020 6.5 6.54 –6133.8 –6793.2 2393 3696 6.2 5.5

FIG. 7. Hydration enthalpies as a function of the charge on Ce (circles),Sm (squares), Ho (diamonds), and Yb (triangles) ions. Corresponding fitsemploying the Born-Bjerrum equation are shown as full, dotted, dashed, anddotted-dashed lines, respectively. In the inset, we show the same hydrationenthalpies as a function of q2.

solvation model,6 according to which

�Hhyd = − 1

4πε0

(qe)2

2Ri

(1 − 1

ε− T

ε2

∂ε

∂T

), (6)

where ε0 is the permittivity of a vacuum, e the electron charge,Ri the ionic radius, T the temperature, and ε the solvent dielec-tric constant.

In Figure 7, we report our results for the 4 ions studiedand the fitted data using the Born solvation model.6 We ob-serve a very good agreement between our results and the be-havior predicted by the Born model.6 Born ionic radius, RB

i ,can be extrapolated from the fit of our numerical results withEq. (6). We obtain RB

i = 9.5 Å for Ce and RBi = 7.2 Å for

Yb. These values are far from the structural ionic radii ob-tained by EXAFS39 but also larger than those derived fromStokes’ law and experimental diffusion coefficients.37

A tentative explanation is that the rigidity of the watermolecules in the first shell around the ion leads to a local per-mittivity much smaller than in the bulk, such that the Born ra-dius, which characterizes the onset of the dielectric medium,is pushed further away. As expected, the Born model does nottake into account the discrete character of the solvent. It there-fore does not consider any local order produced by the ion andit is based on purely electrostatic considerations. Our simula-tions show that there is a clear change in the water structuringeffect when the valency is increased, with a transition fromdisorder to order for q values of ∼1–2 (depending on the ionicsize). It must be noticed that at the same time the solvation en-ergy seems to vary continuously.

This change in solvation structure must be associated toan energy contribution, analogous to a polarization energy ofthe first solvation shell. Indeed, we note that for q = 0 we donot have exactly �H = 0 since the first shell is polarizable(both atom and water molecules). Similarly, there should bea term linear in q to reflect the interaction of the frozen po-larization of the first shell with the central charge. Here, weare not devoted to improve the Born solvation model, but weonly want to see if this model works and which ionic radiusone should get if this popular model is applied.

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164501-8 Martelli et al. J. Chem. Phys. 137, 164501 (2012)

Note that Born radii exhibit a decrease across the series,as the EXAFS radii do, while those obtained from Stokes’law behave in the opposite way, i.e., they increase across theseries. We will discuss the implication of diffusion models inSec. V.

B. Water exchange dynamics and ionic diffusion

Another property of ions in solution are the exchangesoccurring between first and second hydration shells aroundmetal solute, as well as between second shell and bulk wa-ter. The mrt provides an idea of how long a water moleculeis hosted in a defined shell. Our results on mrt are reportedin Table III for Ce and Yb. As expected, the mrt increaseswith the charge because of the increasing attraction of theion. As shown above in Figure 5, we observe that the strongrigidity of the shell for q ≥ 2.5 corresponds to high residencetimes.

Furthermore, diffusion coefficients for Ce and Yb at dif-ferent charge values are shown in the same Table III. Asexpected, for neutral atoms, Yb diffuses faster than Ce, be-cause Yb is smaller than Ce in our model. The diffusion co-efficients decrease with q. Those for Ce become larger thanthe corresponding ones for Yb at a given charge value. Wehave discussed the case of q = 3+, corresponding to real ionsin water, where this behavior is observed experimentally37, 60

and our interaction potential and related simulations repro-duce this result.28 We have evoked the effect of dielectricfriction that depends on ionic radii in the opposite way withrespect to Stokes’ law. We discuss in more details the conse-quences of diffusion models on this behavior in Sec. V. Here,we first examine the relationship between solute diffusionand solvent exchange dynamics in the solute’s first solvationshell.

At room temperature and in the absence of external fields,water molecules diffuse at a speed about 3–4 times biggerthan that of the ion, as shown in Table IV. On the other hand,when water molecules encounter an ion they are slowed downby the ion, and water in the first shell has about the same dif-fusion coefficient as the ion that structures them (see sameTable IV). As recently shown by Masia and Rey,61 an esti-mate of the re-orientational time informs us about the timescale after which one would expect to observe similar diffu-sion coefficients for an ion and the water in the first hydrationshell. This time can be derived from the rotational version ofthe Stokes-Einstein relation62 generalized to the ion plus first

TABLE IV. Ratios of water-to-ion, and first hydration shell water-to-ion,diffusion coefficients (Dw/Dm and D1st/Dm, respectively) for Ce and Ybmetal ions.

Ce Yb

q Dw/Dm D1st/Dm Dw/Dm D1st/Dm

2.5 3.0 1.4 3.1 1.23 3.4 1.1 3.5 1.03.5 3.5 1.1 3.3 1.14 3.7 1.1 3.2 1.3

solvation shell complex as

τ = 8πη0R3

kBT, (7)

where kB is Boltzmann constant, η0 is the bulk solvent vis-cosity, and R is the radius of the ion-water complex. As longas the exchange time scale is much longer than that for rota-tional relaxation, the diffusion coefficient of the first solvationshell molecules should be taken equal to that of the ion. ForCe and Yb ions, we have determined τ to be around 190 psand 210 ps, respectively, which is of the same order of mag-nitude as the mrt (see Table III) for Ce2+ and Yb1.5+. We canthen explain by this simple interpretation why highly chargedions and first shell water molecules have similar diffusion co-efficients. In particular, the ratio between D of the ion and Dof the first shell water molecules is around one for Ce with qin the interval 2.5–4 and Yb with q in the interval 2–4 (seeTable IV), i.e., when the charge of the ion is such that orderedwater structures are created. Note that it has not been possi-ble to obtain this ratio for small q values since first hydrationshell is not well defined.

V. IONS DIFFUSION AND MACROSCOPIC MODELS

We now turn to the connection between the observed dif-fusion coefficients behavior across the series as a function ofcharge and the theoretical models for diffusion.

The diffusion coefficient is related to the drag (or friction)coefficient, ζ , through the well-known Einstein relation

D = kBT

ζ. (8)

As mentioned by Eq. (2) the drag coefficient is composed ofa viscous term (Eq. (1)) and a dielectric term.

Zwanzig,9, 10 based on assumption of slip flow field, ob-tained the dielectric friction term as

ζZwD = 3

4

q2e2

R3i

ε − ε∞ε(1 + 2ε)

τD, (9)

where Ri is the solute radius, ε∞ the high frequency dielec-tric constant of pure water, and τD the Debye relaxation time.Note that here and hereafter we skip the vacuum permittiv-ity, ε0, for the sake of simplicity and all quantities involvingcharges should be multiplied by a factor of 1

4πε0to recover

SI units. In Zwanzig model, the solvent is structureless (con-tinuum). The contribution of Eq. (9) becomes negligible forlarge radii, but it can be important for small Ri values.

Before looking at the results of this model, we must in-troduce another common model employed in the literature,often called the solvated ion (or solventberg) model.13, 61 Inthe latter, it is assumed that the ion is rigidly solvated, thatis, the solvent molecules form a stable and fixed structurearound the ion. The ionic radius is thus replaced by an ef-fective radius, σ ,

σ = Ri + 2Rs, (10)

where Rs is the radius of a solvent molecule. The viscous anddielectric terms are then

ζ siv = 4πη0σ, (11)

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164501-9 Martelli et al. J. Chem. Phys. 137, 164501 (2012)

FIG. 8. Walden products for q = 1 (panel (a)), q = 2 (panel (b)), q = 3(panel (c)), and q = 4 (panel (d)). Black line is for Stokes model, red line forZwanzig model, green line for solvated ion model, and blue line for effectivesolvated ion model developed in this work.

ζ siD = 3

4

q2e2

σ 3

ε − ε∞ε(1 + 2ε)

τD. (12)

A useful, and used, way of following diffusion behavioras a function of solute radius is to plot the so-called Waldenproduct, W ,

W = λ0η0 = η0 |q| e2NA

ζ(13)

with η0 the viscosity of pure solvent, λ0 the limiting ionicconductance, NA Avogadro’s number, and ζ the friction co-efficient obtained from one of the models described above(Stokes, Zwanzig or solvated ion). The expressions of theWalden products for each model are given in the supplemen-tary material.47

In Figure 8, we show the results for the Walden prod-ucts for Stokes, Zwanzig, and solvated ion models and vari-ous charges as a function of ionic radius. Note that we havehighlighted the radius range corresponding to structural ionicradii of Ce3+ and Yb3+ as reported by EXAFS.39 We shouldremind that in our model, the ionic radius enters into the def-inition of Buckingham potential parameters, and thus it is in-dependent of the charge. Surely, this would not be the caseif the species could be present with different charges in so-lution. Here, we modify the cation charge without modifyingthe ionic radius to point out the role of the charge, and thusdielectric friction, on diffusion properties. It is obvious thatin the case of Stokes’ behavior, the Walden product dependslinearly on R−1

i with different slopes just depending on dif-ferent q values. Following the Zwanzig model, the behaviorfollows Stokes’ regime for large radii, while it is inverted forsmaller ones, i.e., W decreases as radius decreases above acertain value. This is due to the effect of dielectric friction,which has an inverse cubic dependence on ionic radius. Ofcourse, the larger the ionic charge the larger the ionic radiusfrom which this behavior is inverted. For the ionic radii val-ues considered here, we obtain an inverted region in Waldenproducts for each charge value–as we will see in the follow-

ing, one should use q < 1 to obtain a Stokes’ regime for ionsof such small size. In the solvated ion model, the inclusionof an additional solvent radius decreases the role of dielectricfriction, since the effective radius is greater and ζ D has a R−3

dependence. Finally, this results in almost constant values ofW (and thus constant values of D) in the ionic radius rangeexamined here.

At this point we remark that the standard solvated ionmodel uses a constant σ value. In other words, the solventforms a rigid structure around the ion for each value of R andcharge, q. This is a crude approximation and, as observed re-cently about the possibility of considering a non-static sol-vent radius,63 we have introduced a dependence of σ withthe charge. We have thus taken an effective solvent radius de-pending on the metal charge as follows:

ReffS (q) = RS[1 − exp(−αq)], (14)

in which RS = 1.37 Å for the radius of water and α is a fittedparameter evaluated to be 7/8. Therefore, σ turns out to be afunction of the charge as

σ (q) = Ri + R′ + 2ReffS (q), (15)

where we have introduced R′ = 0.4 Å as a small additionalradius in order to exactly recover Stokes’ law for q = 0 in thecase of Ce. In our model, we have considered that at saturation(i.e., for q → ∞) a perfectly rigid solvation shell is created,corresponding to what is observed in the case of our smallions with large charges. Clearly, a greater number of increas-ingly rigid solvation shells should be expected in principle ascharge is increased, but this was not observed in our studyof these small ions. In fact, even for highly charged ions, thesecond shell turned out to be very mobile and it seems to be agood approximation to consider it as having the same proper-ties as bulk solvent. The definition of an effective radius de-pendent on ionic charge, here developed at an empirical levelthrough Eq. (14), corresponds to the following picture: for lowcharges the ion interacts weakly with the solvent, which is notstrongly structured around the ion (reflected in non-structuredradial and angular distribution functions and small mean res-idence times); as the charge is increased the ion-solvent in-teraction increases and thus the solvent is more structured.Saturation corresponds to the solventberg model, while inter-mediate Reff values describe a dynamical situation where thesolventberg species is not formed yet. Note that this pictureof continuous desolvation is similar to what was introducedby Adelman via the desolvation function.13 This latter modelrequires values for parameters that are not available for wa-ter. They should be obtained by fitting data for a given chargeover a large ionic radius range. Here, we have introduced acharge dependence of solvation structure that has a formallydifferent expression, even if similar in spirit.

The Walden products behavior using the effective sol-vated ion model, presented in the same Figure 8, exhibits abehavior in between the Zwanzig and solventberg models.To better compare the different models with our simulationresults, we plot in Figure 9 the diffusion coefficients as afunction of charge. For q = 0, we are in the Stokes’ regime,where the bigger solute (Ce) diffuses more slowly than thesmaller one (Yb). This regime holds for very small charges:

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164501-10 Martelli et al. J. Chem. Phys. 137, 164501 (2012)

FIG. 9. Diffusion coefficients for solvation radius depending on ion charge.Results for Ce in black and for Yb in red. Simulation results are shown as dotsand squares. Results from the models: original solventberg model (dottedlines), original Zwanzig model (dotted-dashed lines), effective solvated ionmodel (full lines). The inset shows the dependence of effective solvent radiuswith the solute charge.

modified solvated ion model, in which an effective solvent ra-dius defined by Eq. (14) is employed, provides an inversionpoint for q ∼ 2, while the original Zwanzig model locates thisinversion at smaller charges. Simulations show that alreadyabove q = 0.5 the Yb diffusion coefficient is smaller than thatof Ce, which is thus closer to Zwanzig results. On the otherhand, the effective radius solvated ion model better describeshigh charges tail. In fact, Zwanzig provides overdamped val-ues for high charges due to the high effect of dielectric fric-tion where any shield due to rigid, or partially rigid, solventmolecules is considered. Note also that for small q values,the Zwanzig model has zero slope behavior at q = 0, whilethe solvated ion model has a finite negative slope one at q= 0 that seems to be closer to simulation behavior also forsmall charges. In the inset of Figure 9, we show in detailsthe effective solvent radius as a function of the solute charge,corresponding to the fact that in our model the solventbergbehavior is observed only for highly charged ions.

We can thus explain why charged ions such as Ln3+

and An3+ in water exhibit an “anomalous” behavior: dielec-tric friction plays an important role and Zwanzig or mod-ified Zwanzig theory, where an effective solvent radius isconsidered, can describe this phenomenon. We show, seeFigure 10, the behavior of ζv and ζ D as a function of chargefor Ce and Yb (here we report only results for our effectiveradius solvated ion model). ζv is larger than ζ D up to q = 3and 3.5 for Yb and Ce, respectively. We have shown an in-version in D behavior for q = 2.5 in our model that corre-sponds to the charge for which δD = ∣∣ζCe

D − ζ YbD

∣∣ is equal toδv = ∣∣ζCe

v − ζ Ybv

∣∣. The ratio δv/δD is at the origin of the ob-served “anomalous” behavior: when it is greater than unity thehydrodynamic regime is observed, and in the reverse situation(when it is smaller than 1) the dielectric friction effect playsa great role and smaller ions diffuse slower than bigger ones.Note that, in the effective solvated ion model the viscous fric-tion has an indirect dependence on the charge, since the hy-drodynamic radius is now dependent on the charge: this corre-

FIG. 10. Viscous ζv and dielectric ζD contributions to the drag coefficient ζ

for Ce (black lines) and Yb (red lines) using the effective solvated ion model.

sponds to a picture in which as the ion charge is increased, thesolvent molecules structure themselves more firmly around it,finally reaching the limit of the solventberg picture for highcharges.

VI. CONCLUSIONS AND OUTLOOKS

In this work, we have studied the effect of varying thecharge on small ions of ca. 1 Å radius on their hydration struc-ture, and on their thermodynamic and transport properties inliquid water.

From a structural point of view, a transition from unde-fined to defined first hydration shell is observed for q ∼ 1.5− 2. This result was obtained by careful investigation of bothradial and angular distribution functions.

In particular, when q > 2 the first hydration shell be-comes rigid, as reflected also in the mean residence time ofwater molecules in the first hydration shell. Moreover, forq > 2 the ions destroy the water hydrogen bond patternformed by water molecules in the second shell. On the otherhand, from a thermodynamical point of view, the interactionbetween ions and water molecules in terms of hydration en-thalpy increases quadratically as the ion charge is increased.This observation is in perfect agreement with the Born modelof ion-solvent interactions even though the calculated Bornionic radii are much larger than the real ionic radii, which canagain be related to the rigidity of the solvation shell.

An analysis of the diffusion properties as a function ofionic charge leads one to identify two frictional regimes: Anhydrodynamic regime, in which the ions diffuse followingStokes’ law, and a dielectric friction regime, in which electro-static interactions between the ion and the solvent play a keyrole. The hydrodynamic regime occurs for neutral solute (asexpected) and ions bearing a small charge (i.e., q ≤ 0.2), whilethe dielectric friction regime is observed for highly chargedspecies. The popular analytical models, like Zwanzig and sol-vated ion models, are not able to fully describe this diffusionbehavior as a function of the charge. The Zwanzig model isable to provide the charge value for the transition betweenthe hydrodynamic and the dielectric friction regimes, but ittotally fails for highly charged ions. Further, the diffusion

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164501-11 Martelli et al. J. Chem. Phys. 137, 164501 (2012)

decay pattern at small charges seems to be of an exponentialform, while the Zwanzig theory indicates a Gaussian decay.The standard solvated ion model, based on a solventberg hy-dration picture, is not able to reproduce this behavior.

We have thus modified the latter model by introducingan empirical dependence of the ion radius on the ionic chargethat reflects the different degree of structuration. This mod-ification provides a model that smoothly changes from theZwanzig to the solvated ion picture. This model correctly fitsour simulation results even though some features are miss-ing. We were thus able to explain the nature of the transitionbetween hydrodynamic to dielectric friction regime: it occurswhen the difference between the viscous drag coefficients ofthe two ions (one smaller than the other) and the difference ofthe dielectric drag coefficients of the same two ions is equalto zero.

The present study has been developed at a phenomeno-logical level: we have introduced an effective solvent radiusinto the solventberg model with an empirical dependence onthe ionic charge. At the present time the switching betweentwo different diffusional regimes is not observed in mean res-idence time behavior that is related to the time scale of co-ordination shell relaxation,26, 27 but mrt should have some in-fluence. Then it would be tempting to connect mrt with somemeasure of “local” stress tensor which in Adelman picture13

is related to local density. However, we did not find yet anysimple connection: the mrt shows an exponential behavior butfaster than diffusion coefficient and the “local” stress tensor isnot uniquely defined (and this is also why local friction, whichis related to the stress tensor, is also an elusive quantity). Fu-ture developments should connect these findings with otherknown properties of ionic solutions (and in particular thoserelated to experiments, like mrt is) in order to provide a gen-eral framework valid for ions with any charge and radius inany solvent. Our research is currently going in that direction.

ACKNOWLEDGMENTS

We thank J. T. Hynes for useful discussions. This workwas supported by the ANR 2010 JCJC 080701 ACLASOLV(Actinoids and Lanthanoids Solvation).

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