JOURNAL OF CHEMICAL PHYSICS VOLUME 111, NUMBER 11 15 SEPTEMBER 1999

Large ionic clusters in concentrated aqueous NaCl solutionLeo Degrevea)

Grupo de Simulac¸ao Molecular, Depto. de Quı´mica, Faculdade de Filosofia, Cieˆncias e Letras de Ribeira˜oPreto, Universidade de Sa˜o Paulo, 14040–901 Ribeirao Preto, SP, Brasil

Fernando Luıs B. da SilvaGrupo de Simulac¸ao Molecular, Depto. de Quı´mica, Faculdade de Filosofia, Cieˆncias e Letras de Ribeira˜oPreto, Universidade de Sa˜o Paulo, 14040–901 Ribeirao Preto, SP, Brasil, and Physical Chemistry II—P.O. Box 124—Lund University, S-221 00 Lund, Sweden

~Received 29 May 1998; accepted 11 June 1999!

The stability of the local structure of aqueous 1.0 M NaCl solution at 293 K was investigated bymolecular dynamics simulations. The mean and maximum life-times of the ion pairs weredetermined to be 0.13, 0.14 and 0.27 ps for negatively charged, neutral and positively charged ionspairs, respectively. The stability of non-neutral ion pairs was studied from the structure of theanion–cation radial distribution function and other structural functions. We found that non-neutralions pairs are stabilized by at least one counter-ion forming in this way large ionic clusters thatinclude the hydration molecules. About 25% of the ions are included in neutral clusters formed bya minimum of four ions. The existence of these large clusters obfuscates the commonly acceptedchemical reaction scheme for the interconvention of ion pairs in aqueous solutions. ©1999American Institute of Physics.@S0021-9606~99!50134-7#

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I. INTRODUCTION

The behavior of aqueous solutions in the bulk or atinterfaces has receiving considerable attention for myears. It is clear that the understanding of these processea wide range of technological applications and theoretdevelopments in liquid purification, homogeneous and nhomogeneous catalysis, transport, biophysical proceschemical technology, petrochemistry, theory of chemical aphysical processes.1–9 One particular aspect of great intereis the interconversion of ion pairs in solution due to its direrelation with important chemical reactions.10,11 However,this mechanism remains poorly understood.10 The reactionscheme that is more accepted isA11B2 ~free ions!⇔A1iB2 ~solvent separated ion pair—SSIP! ⇔A1uB2

~contact ion pair—CIP! ⇔AB ~reaction product!, where thespecific role and nature of these ion pairs are suncertain.10,12 The so-called computer experiments, or simlation techniques, can by-pass experimental and theoredifficulties and furnish a molecular level characterizationthese intermediate states and barriers separating them.3,10,12

They can also provide detailed information on the local ccentration in the neighborhood of molecules or macmolecules at nonhomogeneous interfaces13–16 or aroundions in solutions using either the usual radial distributifunction ~rdf!17,18 and/or multi-variable distributionfunctions.13–16,19–22

Recently, we presented an analysis of the structure oelectrolyte solution at finite concentration, precisely aque1.0 M NaCl solution.3 One important point in that study wathat the analysis focused on the ion pairs that were identiby the fact that two ions share at least two hydration m

a!Author to whom correspondence should be addressed; electronicleo@obelix.ffclrp.usp.br

5150021-9606/99/111(11)/5150/7/$15.00

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ecules. The main idea was based on the fact that the numof water molecules in the first solvation shells of these ioare quite stable and close to six, i.e., 6.0060.27 for Na1 and5.7360.99 for Cl2.22 The fluctuations in these hydrationumbers are small because ion–water energies are lothan the bulk energy.22 Another observation is that only fewexchanges between hydration molecules and the bulkobserved.23 These stability conditions suggest that ion asciation must occur keeping as far as possible the solvashells. At least two cases of ion pairing can be identified. Tsimplest one is the case where there is the direct conbetween the ions~this is the commonly accepted definition oion association24–26!. The second one is detected when tsolvated water molecules are shared by two ions keepthem close or making them approach each other. Thismaintain the stability of the structure.3 The consequence ithe stabilization of the clusters formed by two ions altogetwith all their solvation molecules. In all ion association, tions must share some water molecules in their solvashells: their number cannot be less than two excludingcases of single nonassociating collisions. Therefore, the sing of the minimum of two water molecules by two ions cabe a good criterion for the identification of an ioassociation.3 As a result, it was observed3 that the fractionsof associated ions were unexpectedly high, e.g., 0.46 forcations and 0.32 for the anions. In particular, associatibetween identical ions were largely observed. For instanroughly 13% of the anions and 16% of the cations are asciated in non-neutral pairs.3 These pairs were studied befoby other authors in different ways mainly on primitivemodeled solutions.27–29 However, apparently no informationon the stabilization process of charged pairs was obtainStudies on nonprimitive models for electrolyte solutiowere also able to predict the stability of some ion pairs froil:

0 © 1999 American Institute of Physics

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5151J. Chem. Phys., Vol. 111, No. 11, 15 September 1999 Large ionic clusters in aqueous solution

mthe mean force potential, such as the Cl2–Cl2 pair,30 whilethe stability of Na1–Na1 pairs was not predicted. Consequently, the present question is to determine what is thebilization mechanism of highly repulsive charged pairs ato determine if the stabilization process is mainly explainby the solvent or ions properties.

II. METHOD

Systems constituted by 2117 water molecules andNaCl were submitted at 293 K to molecular dynamsimulation.18,31,37The volume was adjusted to corresponda density equal to 1.1402 kg/l so that the electrolyte conctration was 0.993 M. These conditions correspond to Nsolutions under atmospheric pressure. Solvent molecwere described by the TIP4P water model33 and their inter-actions with the ions by a 6–12 plus Coulomb interactpotential.34 Ion–ion interactions were modeled by thGROMOS force-field.38 Periodic boundary conditions anminimum image convention were adopted. The long-ranelectrostatic energies were included through a PoissBoltzmann generalized reaction field32 with a spherical cut-off at 1.5 nm. The Newton movement equations were ingrated by a time step of 1.0 fs and the total simulation tiwas 0.2 ns. Quaternions36 and the Gear predictor–correctomethod37 were used.

Ion–ion rdf gion,ion8(r ), whereion could be both the anion and the cation andr is the interionic distance, were caculated in the usual way.18 As it will be seen, thegNa1,Cl2(r )function presents two very well defined peaks at short dtances. The first peakP1 is included in the range ofr 50.0 tor 5r min,1 and the second peakP2 in the range ofr min,1 tor min,2, wherer min,1 and r min,2 correspond to the positions othe first and secondgNa1,Cl2(r ) minima, respectively~it willbe noted thatr 5r min,150.34 nm andr min,250.58 nm). Theionic structures that are responsible for these peaks wilresolved from the fraction of ions present inP1 andP2 . Thisentity will denoteSion(n1

k ,n2k), where the subscription is the

reference anion, or cation, at the origin. The numberscounter-ions of typek in P1 andP2 are given byn1

k andn2k ,

respectively. For instance, assuming that the Na1 is on the

center,SNa1(n1Cl2 ,n2

Cl2) records the frequency that the num

ber of counter-ions Cl2 presents on peakP1(n1Cl2) and, si-

multaneously, inP2 (n2Cl2), appeared during the observatio

time. TheSion(n1k ,n2

k) functions were not calculated from thradial distribution functions. Instead, they were calculateddirect counting of the respective occurrence during the simlation process, after the proper equilibration of the system

The counter-ions present inn1k andn2

k are possibly dis-tributed in space according to the local energy minima,that we can define three partial radial distribution functiogion

11 (r ), gion12 (r ) andgion

22 (r ), where the exponents 11, 12 an22 refer to the simultaneous presence of counter-ionspeaksP1 and P1 , P1 and P2 , and in P2 and P2 , respec-tively. The distancer is the separation distance between tcounter-ions in the respective cases. It is convenient asto determine the distribution functions of the angleC formedby one counter-ion, the central ion and the second counion: C ion

11 , C ion12 and C ion

22 . For example,CNa111 refers to the

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angle formed by one Cl2, a central Na1 and a second Cl2,when both Cl2’s are inP1 . Analogously, if the anions are indifferent peaks, the corresponding function isCNa1

12 . Thepartial radial distribution functionsgion

11 (r ), gion12 (r ) and

gion22 (r ), together with the anglesC allow us to determine

accurately the most frequent interionic distances, and hewill allow us to associate the respective configurations wthe features of the ion–ion radial distribution functions.

III. RESULTS

The mean total configurational energy was found to217.460.1 kcal/mol. The configurational energy for the iteractions between the water molecules was27.04760.07 kcal/mol, which is slightly above the configurationenergy of the pure TIP4P water bulk phase (210.07kcal/mol30!. Ion–water energy was equal to24.2 kcal/mol.These results are similar to the Monte Carlo data formeobtained.3 The diffusion coefficients for Na1, Cl2 and watermolecules were determined as being equal to 1.23, 2.54.6031029 m2 s21, respectively. These numbers are in goagreement with the results obtained by the same ions moand rigid SPC water model.35 Some difference between ouresult for the diffusion coefficient and the value 2.931029 m2 s21 for pure TIP4P water38 is observed while theconcordance with more recent data39 is good.

ThegCl2,Na1(r ) distribution function is displayed in Fig1. It presents a first very well defined peak of 32.15 ar50.25 nm and a smooth second one of 2.5 atr 50.48 nm, atypical trait of strong long-range electrostatic attraction anshort-range soft core repulsion. This data is similar to resobtained with SPC water and the same ion–ion interacpotential reported in the literature.35 In Fig. 1, the number of

FIG. 1. The ionic radial distribution functions as a function of the separatdistancer between the ions:gCl2,Cl2(r ) ~dots!, gNa1,Cl2(r ) ~solid line! andgNa1,Na1(r ) ~dashed line!. The running integrals of these curves are alrepresented.

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5152 J. Chem. Phys., Vol. 111, No. 11, 15 September 1999 L. Degreve and F. L. B. da Silva

cations–anions neighbors is also plotted. This graphic shthat the firstgCl2,Na1(r ) peak corresponds to 0.41 ions sthat the fraction of ions in contact and consequently assated in neutral pairs is 0.41. The secondgCl2,Na1(r ) peak andthe number of neighbors profile indicate the existence ostructure in ther 50.49 nm region too. The other rdfsgCl2,Cl2(r ) andgNa1,Na1(r ), are also plotted in Fig. 1. Thesfunctions are badly defined such that the comparison wpublished data35 is not fruitful. However, in all cases thesfunctions present a first, but not so well defined peak cosponding to a mean of a 0.1 Cl2 ion near each Cl2 and 0.06in the Na1 case. More data are summarized in Table I.

The analysis of the stabilities or mean life-times begwith the ion–ion rdfs. The characteristics of the various ioion rdf are fundamentally different; see Fig. 1. If we compaeithergCl2,Na1(r ) with gCl2,Cl2(r ) or with gNa1,Na1(r ), onlythe gCl2,Na1(r ) function presents two very well definepeaks. This might indicate that a relatively long-range strture of counter-ions exists around the central cation or anFrom data of Fig. 1, the values ofr min,1 and r min,2 are iden-tified as equal to 0.34 and 0.58 nm, respectively. The cospondingSion(n1

k ,n2k) data are reported in Table II. Both fre

quencies SCl2(n1Na1

,n2Na1

) and SNa1(n1Cl2 ,n2

Cl2) are verysimilar showing roughly that 30% of the ions have no neigbor in eitherP1 or P2 , 44% one inP1 or P2 so that about

TABLE I. Positions (nm) and intensities of the first two extrema of thradial distribution functions and the corresponding value of the coordinanumber calculated until the corresponding minimum,n(r min).

r max g(r max) r min g(r min) n(r min)

gCl2,Cl2(r )

0.43 1.15 0.47 0.83 0.100.51 1.41 0.63 0.78 0.47

gNa1,Cl2(r )

0.25 32.15 0.34 0.83 0.410.48 2.50 0.58 0.95 1.05

gNa1Na1(r )

0.35 1.39 0.39 0.59 0.060.44 0.88 0.49 0.32 0.15

TABLE II. The frequenciesSCl2(n1Na1

,n2Na1

) andSNa1(n1Cl2,n2

Cl2). See thetext for details.

SCl2(n1Na1

,n2Na1

)

n2Na1

50 n2Na1

51 n2Na1

52 n2Na1

53

n1Na1

50 0.302 0.255 0.079 0.006

n1Na1

51 0.179 0.114 0.015 0.002

n1Na1

52 0.046 0.004 0.000 0.000

SNa1(n1Cl2 ,n2

Cl2)

n2Cl250 n2

Cl251 n2Cl252 n2

Cl253 n2Cl254

n1Cl250 0.296 0.257 0.069 0.009 0.001

n1Cl251 0.203 0.109 0.015 0.003 0.000

n1Cl252 0.004 0.004 0.000 0.000 0.000

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25% of the ions are embedded in clusters of three or mions. The frequency of these clusters is so large that westudy their structure by means of the partial radial distribtion functionsgion

11 (r ), gion12 (r ) and gion

22 (r ) @see Figs. 2~a!–~c!#, and from the anglesC ion

11 , C ion12 and C ion

22 @see Figs.3~a!–~c!#. In Fig. 2~a!, one sharp peak is present ingNa1

11 (r )and two peaks ingCl2

11 (r ) at the same distance as those fouat thegCl2,Cl2(r ) andgNa1,Na1(r ) peaks~see Fig. 1!. Despitesome fluctuations, a similar behavior is observed in Fig. 2~b!.Figure 2~c! shows a double behavior taking into account telectrostatic repulsion at a large distance and the formaof ion pairs at short distances. Comparing the positions ofpeaks in Fig. 1 and in Figs. 2~a!–~c!, the identification of thenature of the peaks, or bumps, comes out with no effoThe first peak ofgCl2,Cl2(r ) at 0.43 nm appears at least patially as a consequence of the first peaks presented atthree partial distributions,gNa1

11 (r ), gNa112 (r ) and gNa1

22 (r ),while the second peak is related to the second one foungNa1

12 (r ) and gNa122 (r ). Similar identifications can be mad

betweengNa1,Na1(r ) andgCl211 (r ), gCl2

12 (r ) andgCl222 (r ). The

correlations between the peaks and their positions are exaone to one. This is an obvious influence of the solvent aquite important factor on the determination of the peak psitions, i.e., the structure of the medium is largely definedthe solvation structures of the ion pairs. The remark incluthe cation–anion pairs that are the starting point ofpresent analyses. A comparison between the three partiafor the anion and for the cation reveals one characteridifference. In the anion case as the reference ion, the catcation distribution presents two peaks that must be assated with the high stability of the cationic pairs. This eplains the high stability of the first cationic hydration sheby the presence of one anion turning the cluster energeticstable.

The same structural properties are obtained for theCfunctions as well; see Figs. 3~a!–~c!. These figures also showthe same tendency to increase the interionic distance betwco-ions implying definite angles that correspond to the peobserved in Figs. 2~a!–~c!. A detailed analysis of the correlations between the anglesC and the partial and total radiadistribution functions can now be made. The position of tfirst gCl2,Na1(r ) maxima in Fig. 1 arer max,150.25 nm andr max,250.48 nm, while the peaks ingCl2,Cl2(r ) are observedat 0.43 and 0.51 nm; see Fig. 1. Yet ingNa1,Na1(r ), the peaksare found at 0.35, 0.44 and 0.61 nm. From these sets of pdistances and together with the coordinates of the obsepeaks in Fig. 2~a!–~c!, the values of the maxima ofC ion

11 ,C ion

12 andC ion22 observed in the Figs. 3~a!–~c! are recovered.

A consistency check together with a yet more complcomparison between the partial and the full ion–ion rdfs cnow be made comparing the sums of the partial rdf withfull rdf. In Fig. 4~a!, gNa1,Na1(r ) is compared withgCl2

11 (r )1gCl2

12 (r )1gCl222 (r ) and in Fig. 4~b! gCl2,Cl2(r ) with

gNa111 (r )1gNa1

12 (r )1gNa122 (r ). The gNa1,Na1(r ) is extremely

well reproduced mainly at short distances so that allNa12Na1 pairs are associated with counter-ions. At shdistances, i.e., when both counter-ions pertain toP1 and/or

n

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5153J. Chem. Phys., Vol. 111, No. 11, 15 September 1999 Large ionic clusters in aqueous solution

FIG. 2. Partial radial distribution functions as a function of the interionseparation distancer . See the text for details.~a! gNa1

11 (r ) ~solid curve! andgCl2

11 (r ) ~dots!; ~b! gNa112 (r ) ~solid curve! and gCl2

12 (r ) ~dots!; ~c! gNa122 (r )

~solid curve! andgCl222 (r ) ~dots!.

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totheare; seeear

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a-re

P2 , the full rdf are always smaller than the sum of the parrdf. This is evidence that the same two ions give more thone contribution to the partial rdf. This is possible only if thsame two counter-ions are found inP1 and/or P2 of twodifferent central ions. Consequently not only the Na12Na1

pairs are associated with counter-ions but the Cl22Cl2 pairsare too. The fact that the full rdf is always smaller than tsum of the partial rdf at short distance is the consequencthe existence of clusters formed by four ions, two pairscounter-ions. The quantity of these large clusters can beduced from the running integrals of the full rdf and from tsums of the partial rdf shown in Fig. 5. Running integracalculated until the same minima positions as the minimagNa1,Na1(r ) and gCl2,Cl2(r ) are found to be 0.09 and 0.15respectively. These values are exactly 50% larger than

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values in Table I, i.e., half of the ion pairs are associatedtwo counter-ions. The ratios of the running integrals onsums of the partial and on the full rdf show peaks thatconsequence of structural fluctuations at short distancesFig. 6. More important is the stabilization of these ratios n2.0 for the Na12Na1 and 3.5 for Cl22Cl2 pairs in bothcases for interionic distances at about 0.03 nm, beforefirst peaks of their respective distributions. These numbindicate the existence of larger clusters.

The mean life-times of the ion pairs were determindirectly from the generated trajectory, see Table III, whethey are listed together with the longest observed life-timThey were determined ast f2t i . The initial timet i is definedby the moment of the first identification of the pair formtion, when the first solvation shells of two ions begin to sha

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5154 J. Chem. Phys., Vol. 111, No. 11, 15 September 1999 L. Degreve and F. L. B. da Silva

FIG. 3. Distribution functions of the angle formed by one counter-ioncentral ion and a second counter-ion,C, in arbitrary units.C is given indegrees. See the text for details.~a! CNa1

11 ~solid curve! andCCl211 ~dots!; ~b!

CNa112 ~solid curve! andCCl2

12 ~dots!; ~c! CNa122 ~solid curve! andCCl2

22 ~dots!.

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two water molecules. The final timet f is the last time imme-diately before the same two ions discontinue the sharingany two water molecules in their first solvation shells. Nsort of time correlation function or ‘‘survival function’’40

was adopted. Consequently, fluctuations in the sharingsolvent molecules, resulting in the discontinuity of the extence of the ion pair, imply the limit of its life-time. It shoulbe noticed from the data in Table III that there is a raisethe mean life and longest life-times from the Cl22Cl2 pairsto the Na12Cl2 and Na12Na1 pairs. The positive pairs aruncommonly stable comparing with the neutral pairs. Athough they have both a mean life-time and a maximlife-time of almost twice the neutral pairs, their relative frquency ~number of detected pairs! is ca. 10% of what isfound for the neutral pairs. The last ones are sur

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predominant.3 The numbers of detected pairs are of the saorder as the numbers of ions in the first rdf peaks.

IV. DISCUSSION

In a former study,3 the existence of ion pairs was confirmed. The pairs can be charged or not. The structurethese pairs is determined by the solvent because the hytion energies are so attractive that the pairs were able toformed and to maintain some degree of stability. An imptant point was that the ions in these pairs must share shydration molecules. In the case of the non-neutral pairs,highly repulsive electrostatic interactions must prohibit tapproach of the ions, at first sight. This is not the case, sithe ions can approach one another; see Fig. 1. Together

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5155J. Chem. Phys., Vol. 111, No. 11, 15 September 1999 Large ionic clusters in aqueous solution

this fact, it is relevant to point out that these pairs wedetected too by direct and independent inspection intotrajectories. The present results furnish the simple explation that ion pairs can exist and that they exist in the prence of at least one counter-ion and probably two counions. The resulting electrostatic energy is clearly attractMany consequences are immediately identified, e.g., ionaqueous solution are not only associated in pairs, bucharged triplets and larger clusters. The running integralthe full and sums of the partial rdf suggest strongly thmany neutral clusters are formed by two anions andcations. A rough estimation of the fraction of ions associa

FIG. 4. A comparison of the full and partial sums of rdf:~a! gNa1,Na1(r )~solid line!, gCl2

11 (r )1gCl212 (r )1gCl2

22 (r ) ~dashed line!; ~b! gCl2,Cl2(r ) ~solidline!, gNa1

11 (r )1gNa112 (r )1gNa1

22 (r ) ~dashed line!.

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in these clusters, calculated from the fraction of non-neupairs, results in more than 0.25. The second consequenthat an important component of the electrolyte diffusion aself-diffusion is determined by the diffusion of large clustecontaining at least three ions and a high number of solvmolecules~speculatively about 12 to 14!. In order to deter-mine this physical property, it is probably advisable to stusuch aggregates center of mass displacement. A third co

FIG. 5. Running integrals ofgCl2,Cl2(r ) ~circle!, gCl211 (r )1gCl2

12 (r )1gCl2

22 (r ) ~dashed line!, gNa1,Na1(r ) ~cross! gNa111 (r )1gNa1

12 (r )1gNa122 (r )

~solid line!.

FIG. 6. Ratio of the running integral ongCl2,Cl2(r ) divided by the runningintegral ongNa1

11 (r )1gNa112 (r )1gNa1

22 (r ) ~solid curve!. The same plot for theNa1,Na1 pairs is shown as dashed lines.

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5156 J. Chem. Phys., Vol. 111, No. 11, 15 September 1999 L. Degreve and F. L. B. da Silva

quence is that no structure involving only ions can bepected. This is because their relative positions in the thdimensional space are governed by the scheme of ionicdration. The slow diffusion of the ions always observedexperiments41 and in simulations of aqueous electrolyte sytems is easily explained by the existence of these large cters.

Some question remains yet unanswered such as thetension, the kinetics of formation/destruction of the clustewhat is the fraction of the ions associated in these large cters? What are the consequences of the existence of theters on the kinetics and mechanisms of the ionic reactioHow is the behavior of the clusters in external fields? Thiscurrently under investigation and our results will be psented in a forthcoming paper.

V. CONCLUSION

The structure and properties of the aqueous NaCl is gerned by the solvent properties. The fact that the tetrahestructure of water is maintained~due to the high energeticost to be broken! together with the fact that the interioniinteractions are not able to by-pass this cost, result inextremely good solvent properties of water regarding iosolutes. These observations are direct and indirect coquences of the high water polarity. The 1 M NaCl solution isvery similar to a coloidal dispersion of entities formedclusters of solvated and associated ions. The neutralcharged ionic clusters are energetically stable and contaleast three ions. Clusters formed by two anions and twoions are frequent and include a quarter of the all the ioThis explains the ambiguity between the number of idefied paired ions and free ions. The existence of large solvaion clusters obfuscates the apparently simplicity of the mtioned chemical reaction by the increasing of a new ion~thecounter-ion! in the picture of SSIP and CIP pair schemes

ACKNOWLEDGMENTS

This work was supported in part by the Conselho Nacnal de Desenvolvimento Cientı´fico e Tecnolo´gico and by theFundac¸ao de Amparo a` Pesquisa do Estado de Sa˜o Paulo.

TABLE III. Life-times of the ion pairs (ps).

Mean life-time Relative frequency Maximum life-time

Cl2/Cl2 0.13 0.193 1.41Na1/Cl2 0.14 0.728 3.83Na1/Na1 0.27 0.079 6.63

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