Chemical Physics Letters 491 (2010) 54–58

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Chemical Physics Letters

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Hydrophobic interaction of two large plates: An analysisof salting-in/salting-out effects

Giuseppe Graziano *

Dipartimento di Scienze Biologiche ed Ambientali, Università del Sannio, Via Port’Arsa 11, 82100 Benevento, Italy

a r t i c l e i n f o

Article history:Received 8 March 2010In final form 30 March 2010Available online 2 April 2010

0009-2614/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.cplett.2010.03.092

* Fax: +39 0824 23013.E-mail address: graziano@unisannio.it

a b s t r a c t

The decrease in water accessible surface area drives the association of two large hydrophobic plates inboth water and aqueous salt solutions: the contact configuration of the two plates possesses the mini-mum Gibbs energy in all cases. High charge density ions, having strong electrostatic attractions withwaters, render more costly the process of cavity creation, as determined by classic scaled particle theorycalculations, and strengthen pairwise hydrophobic interaction. Low-charge density ions, being preferen-tially bound to plate surfaces due to dispersion interactions, have to be removed to arrive at the contactconfiguration and weaken pairwise hydrophobic interaction.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

The hydrophobic effect, or better hydrophobic interaction, HI,plays a fundamental role for the stability of micelles, double-layermembranes and the folded conformation of globular proteins[1–3]. It is also well known that the addition of salts to watercan markedly affect the magnitude of the hydrophobic effect, caus-ing either salting-in or salting-out [4,5]; the famous Hofmeisterseries was proposed in 1888 [6]. However, a molecular-level ratio-nalization of such salt effects has not yet been achieved, and sev-eral computer simulation studies have been performed to try toapproach the matter in a more controlled way [7,8]. In particular,Zangi et al. [9], ZHB, studied, at 300 K and 1 atm, the associationthermodynamics of two large hydrophobic plates: each one had adisk-like shape with a diameter of about 21 Å and consisted of 31carbon atoms arranged in a triangular lattice with a bond-length of3.2 Å, and each atom had a 4 Å diameter and a Lennard-Jonesparameter e/k = 60 K. Molecular dynamics, MD, simulations wereperformed in 1090 SPC/E water molecules [10], the positions ofthe plate atoms were held fixed, interactions between atoms onthe same plate were excluded, and the orientation of the two plateswith respect to each other was parallel and in-registry. MD simu-lations were done at different values of the inter-plate distance,d, ranging from 3.6 Å to 14.4 Å. The calculated potential of meanforce, PMF, showed a pronounced minimum at d = 4.1 Å, corre-sponding to the associated configuration of the two plates [9]. Byconsidering the completely dissociated configuration at d = 14.4 Åas the initial state, and the associated configuration at d = 4.1 Åas the final state, the Gibbs energy change DG(HI) = �178 kJ mol�1.

ll rights reserved.

This number demonstrates that the associated configuration isstrongly favoured.

To study the effect of salts with different ionic charge density onHI between the two plates, ZHB performed MD simulations, at300 K and 1 atm, in 30 anions, 30 cations and 1030 SPC/E watermolecules [9], by keeping fixed the Lennard-Jones diameter of allthe ions, but changing the charge magnitude in lock-steps, from0.5e up to 1.4e (i.e., the charge assumed non-integer values, eventhough this cannot happen in nature). ZHB found that salts with|q| < 0.9e cause a weakening of HI between the two plates with re-spect to the situation in SPC/E water (i.e., salting-in), whereas saltswith |q| P 0.9e cause a strengthening of HI between the two plates(i.e., salting-out). In particular, the change in the association Gibbsenergy caused by salt addition to SPC/E water was:DDG(HI) = +42.5 kJ mol�1 for |q| = 0.5e, +12.0 kJ mol�1 for|q| = 0.7e, �12.5 kJ mol�1 for |q| = 1.2e, and �20.0 kJ mol�1 for|q| = 1.4e. These numbers emphasize that the salt effects are largeand strongly dependent on the ion charge density. ZHB found agood correlation between the HI weakening/strengthening andthe preferential binding/exclusion of ions from plate surface [9].Snapshots of MD trajectories showed unequivocally the preferen-tial binding to the hydrophobic plate surface of ions with|q| = 0.5e, and the preferential exclusion of ions with |q| = 1.4e.Low-charge density ions interact preferentially with plate surfacesrather than with water molecules; the reverse holds for highcharge density ions.

Zangi, in a subsequent investigation [11], studied several struc-tural properties of the same aqueous salt solutions (i.e., 30 anions,30 cations and 1030 SPC/E water molecules, at 300 K and 1 atm)considered by ZHB, with the aim to clarify if one of such structuralproperties would correlate, in a quantitative way, with the salting-in/salting-out ability of the different salts. The average number ofH-bonds per water molecule proved to be 3.47 in SPC/E water,

G. Graziano / Chemical Physics Letters 491 (2010) 54–58 55

and decreased in a continuous manner in aqueous salt solutions,from 3.29 for |q| = 0.5e to 3.04 for |q| = 1.4e. These numbers wouldsuggest a breaking effect of salts on the water H-bond network,regardless of the charge density. On the other hand, the height ofthe first peak in the oxygen–oxygen radial distribution function,O–O rdf, of water molecules, always centred at 2.74 Å, was 3.09in SPC/E water, and in aqueous salt solutions it amounted to 3.59for |q| = 0.5e, 3.38 for |q| = 0.9e, and 3.00 for |q| = 1.4e. The lattervalues would suggest an inverse proportionality between thestructure of the water H-bond network (as measured by the heightof the O–O rdf peak) and the ion charge density increase. Similarly,the average potential energy between two waters amounted to�16.5 kJ mol�1 in SPC/E water, whereas in aqueous salt solutionsit was �17.1 kJ mol�1 for |q| = 0.5e, �16.7 kJ mol�1 for |q| = 0.9e,and �15.1 kJ mol�1 for |q| = 1.4e. In addition, Zangi found that theheight of the first peak in the water oxygen-ion rdf increased withthe ion charge density as a consequence of the strengthening ofion–water electrostatic interactions (these height values are listedin the third column of Table 1). On the basis of such results, the ionclassification as kosmotropes (i.e., structure-makers) or chaotropes(i.e., structure-breakers) should be considered at least misleadingand not useful to provide insight [11,12]. In particular, there isno connection between the structural features of aqueous saltsolutions determined by Zangi and the ability of such salts toweaken or strengthen HI between the two plates.

The PMF between the two hydrophobic plates, calculated byZHB, showed a pronounced minimum at d = 4.1 Å, correspondingto the contact configuration, and no indication of a desolvation bar-rier in both SPC/E water and aqueous salt solutions [9]. These pro-files are in line with the idea that the decrease in the wateraccessible surface area [13], WASA, upon dimerization is the rightvariable to describe the process [14–16]. Two plates occupy thesame van der Waals volume both when they are far apart andwhen they are in contact; however, the contact configuration is fa-voured because WASA (i.e., more correctly the solvent-excludedvolume) decreases significantly upon association, leading to a largegain of translational entropy of water molecules [16]. The approachI developed along these lines, performing classic scaled particletheory calculations, proved able to describe the temperature

Table 1Values of the location, Rmax, and height, g(Rmax), of the first maximum in the wateroxygen-ion rdf, the effective diameter of the various ions, reff(ion), the total volume ofthe simulation box, Vtot, for the different aqueous salt solutions, the correspondingvolume packing density g, and average effective diameter hri, at 300 K and 1 atm; allsimulation data come from [11]. See text for further details.

Rmax

(Å)g(Rmax) reff(ion)

(Å)Vtot

(Å3)g hri

(Å)

Ow-M(+0.5e) 4.10 1.759 5.46 38 137 0.4041 2.872Ow-X(�0.5e) 3.78 1.328 4.82Ow-M(+0.6e) 4.00 1.893 5.26 37 763 0.3964 2.862Ow-X(�0.6e) 3.70 1.587 4.66Ow-M(+0.7e) 3.92 1.989 5.10 37 456 0.3911 2.854Ow-X(�0.7e) 3.64 1.907 4.54Ow-M(+0.8e) 3.90 2.100 5.06 37 095 0.3925 2.852Ow-X(�0.8e) 3.62 2.263 4.50Ow-M(+0.9e) 3.80 2.247 4.86 36 671 0.3877 2.843Ow-X(�0.9e) 3.56 2.685 4.38Ow-M(+1.0e) 3.76 2.410 4.78 36 183 0.3905 2.841Ow-X(�1.0e) 3.56 3.071 4.38Ow-M(+1.1e) 3.70 2.582 4.66 35 655 0.3908 2.836Ow-X(�1.1e) 3.52 3.536 4.30Ow-M(+1.2e) 3.68 2.801 4.62 35 101 0.3948 2.833Ow-X(�1.2e) 3.50 3.955 4.26Ow-M(+1.3e) 3.62 3.036 4.50 34 541 0.3968 2.829Ow-X(�1.3e) 3.48 4.395 4.22Ow-M(+1.4e) 3.60 3.296 4.46 34 005 0.4010 2.827Ow-X(�1.4e) 3.46 4.821 4.18

dependence of the association thermodynamics of the two largehydrophobic plates [16,17]. The same approach is used in this Let-ter to shed light on the role played by the solvent-excluded volumeeffect in causing salting-in/salting-out phenomena.

2. Theoretical description

The process of bringing two hydrophobic plates from a fixed po-sition at infinite separation to a fixed position at contact distance inwater or aqueous solution, at constant temperature and pressure,can be treated as a case of pairwise HI. The associated Gibbs energychange can be rigorously separated in two terms [18]:

DGðHIÞ ¼ Eaðm—mÞ þ dGðHIÞ; ð1Þ

where Ea(m–m) is the direct monomer–monomer (i.e., plate–plate)interaction energy that does not depend on the presence of the sol-vent and its nature; dG(HI) is the indirect part of the reversible workto carry out the process, and accounts for the specific features of thesolvent in which association occurs. The dependence of DG(HI) onthe distance between the two plates produces the PMF. A generalrelationship connects dG(HI) to the Ben-Naim standard hydrationGibbs energy of dimer and monomer [18]:

dGðHIÞ ¼ DG�ðdimerÞ � 2 � DG�ðmonomerÞ; ð2Þ

where DG�(dimer) represents the Gibbs energy change associatedwith the transfer of the two plates locked in the contact configura-tion from a fixed position in the ideal gas phase to a fixed position inwater, at constant temperature and pressure [19]; DG�(monomer)has the same meaning for a single plate. A rigorous application ofstatistical mechanics allows the exact splitting of DG� in two contri-butions [20,21]:

DG� ¼ DGc þ DGa; ð3Þ

where DGc is the reversible work to create at a fixed position inwater a cavity suitable to host the solute molecule, and DGa is thereversible work to turn on the attractive interactions between thesolute molecule inserted in the cavity and all the surrounding watermolecules. Turning on attractive solute–solvent interactions is con-ditional to the creation of a suitable cavity: there is not the simpleadditivity of independent contributions. The process of water–water H-bond reorganization upon nonpolar solute insertion doesnot affect DG� because it is characterized by an almost complete en-thalpy–entropy compensation [22,23]. This enthalpy–entropy com-pensation should depend on the size and curvature of thehydrophobic surface [3]; the plates considered by ZHB have a sur-face roughness that should not cause problems for the water–waterH-bond reorganization. By using Eq. (3) in the definition of dG(HI),one obtains:

dGðHIÞ ¼ ½DGcðdimerÞ � 2 � DGcðmonomerÞ�þ ½DGaðdimerÞ � 2 � DGaðmonomerÞ�¼ dGcðHIÞ þ dGaðHIÞ; ð4Þ

where dGc(HI) and dGa(HI) correspond to the two square brackets inthe upper line, respectively. The dGc(HI) term, having fixed the sizeand shape of the desired cavities, depends solely on the propertiesof water and is of purely entropic origin [24]; in contrast, thedGa(HI) term accounts for the partial loss of plate–water attractiveinteractions upon association, due to WASA decrease, and is mainlyof enthalpic origin [21].

I have demonstrated, by means of classic scaled particle theory,SPT [25], that: (a) for a fixed cavity volume, the DGc magnitude de-pends on the cavity shape and proves to be roughly proportional tothe water accessible surface area of the cavity, WASAc; (b) the va-lue of the DGc/WASAc ratio calculated for spherical cavities can beused, to a good approximation, also for non-spherical cavities [16].

56 G. Graziano / Chemical Physics Letters 491 (2010) 54–58

Therefore, the dGc(HI) contribution can be calculated from theknowledge of WASA buried upon association by means of the fol-lowing relationship:

dGcðHIÞ ¼ ðDGc=WASAcÞ � DWASAðdimerizationÞ: ð5Þ

Since the WASA buried upon dimer formation is a negativequantity, while the DGc/WASAc ratio is always positive, the dGc(HI)contribution provides a negative Gibbs energy change favourableto dimerization [16].

In order to estimate the effect of salts on the dimerizationstrength, I have simply extended the validity of Eq. (5) to aqueoussalt solutions [26], obtaining:

DdGcðHIÞ ¼ f½DGcðsaltÞ�DGcðwaterÞ�=WASAcg �DWASAðdimerizationÞ:ð6Þ

This relationship is a measure of the change in the solvent-excludedvolume effect contribution caused by salt addition to water. Clearly,according to Eqs. (1)–(4), the overall Gibbs energy change caused bysalt addition to water is exactly given by:

DDGðHIÞ ¼ DdGcðHIÞ þ DdGaðHIÞ: ð7Þ

This relationship can be used to obtain reliable estimates of DdGa

(HI) by comparing the DdGc(HI) numbers calculated by means ofEq. (6) with the DDG(HI) values determined via MD simulationsby ZHB.

Table 2Values of DGc to create a spherical cavity of 10 and 15 Å diameter, first and secondnumber in the second column, in SPC/E water and the various aqueous salt solutions,calculated by means of classic SPT at 300 K and 1 atm; estimates of dGc(HI) andDdGc(HI), calculated by means of Eqs. (5) and (6); values of DDG(HI) from [9], andDdGa(HI) obtained by algebraically subtracting the numbers in the fourth column tothose in the fifth column. The total number of ions released upon plate dimerization[9] is listed in the last column. See text for further details.

|q| DGc

(kJ mol�1)dGc(HI)(kJ mol�1)

DdGc(HI)(kJ mol�1)

DDG(HI)(kJ mol�1)

DdGa(HI)(kJ mol�1)

Dm

H2O 117.0–250.6 �214.8 – – – –0.5e 123.1–263.3 �225.8 �11.0 42.5 53.5 �16.70.6e 121.0–258.8 �222.0 �7.2 17.5 24.7 �13.40.7e 119.7–256.0 �219.6 �4.8 12.0 16.8 �9.90.8e 121.4–259.7 �222.7 �7.9 7.5 15.4 �8.50.9e 120.7–258.1 �221.4 �6.6 0 6.6 �7.51.0e 123.4–264.1 �226.5 �11.7 �5.0 6.7 �6.91.1e 125.2–268.1 �229.8 �15.0 �6.0 9.0 �6.31.2e 128.9–276.0 �236.5 �21.7 �12.5 9.2 �6.21.3e 131.8–282.4 �241.9 �27.1 �14.0 13.1 �6.11.4e 135.7–291.0 �249.2 �34.4 �20.0 14.4 �6.0

3. Results

To perform classic SPT calculations in SPC/E water and aqueoussalt solutions, it is necessary to have estimates for the effective sizeof water molecules and all the ions [27]. I have obtained estimatesof the effective diameter of the various ions from the location ofthe first peak in the water oxygen-ion rdfs calculated by Zangi[11]. The location of this maximum is given by Rmax = [reff(H2O) +reff(ion)]/2, because the size of a water molecule corresponds inpractice to that of the oxygen atom. Since the first maximum inthe O–O rdf of SPC/E water occurs at 2.74 Å, this is the value as-signed to reff(H2O). The Rmax and g(Rmax) values are listed in thesecond and third columns of Table 1, and the obtained estimatesof reff(ion) are listed in the fourth column. A detailed analysis ofsuch estimates is in order. All the ions considered by ZHB havethe same Lennard-Jones diameter, rLJ(ion) = 5.0 Å, and the sameLennard-Jones energy parameter, eLJ(ion)/k = 120.3 K [9]; however,as emphasized by the numbers listed in the fourth column of Table1, the effective diameter of the ions depends markedly on theircharge. On increasing the ion charge, the strength of the electro-static interactions with SPC/E waters increases [i.e., the g(Rmax) val-ues raise strongly; see the third column of Table1], causing abunching up effect manifested in a significant decrease of the effec-tive diameter [note that a similar effect is operative in the case ofwater molecules [28–30]: the first peak in O–O rdf occurs at 2.74 Åin SPC/E water, even though rLJ(oxygen) = 3.165 Å]. For the cations,reff = 5.46 Å for q = 0.5e, 4.86 Å for q = 0.9e, 4.62 Å for q = 1.2e, and4.26 Å for q = 1.4e; for the anions, reff = 4.82 Å for q = �0.5e, 4.38 Åfor q = �0.9e, 4.26 Å for q = �1.2e, and 4.18 Å for q = �1.4e. It isevident that: (a) the anions are smaller in size than the cationspossessing the same absolute charge; (b) for all anions, the effec-tive diameter is smaller than the Lennard-Jones diameter,reff < rLJ = 5.0 Å; (c) for cations with q < 0.8e, the effective diameteris larger than rLJ = 5.0 Å (for more on this point, see the Appendix).The difference in charge density between the various ions is evenlarger than that expected by construction.

Having assigned an effective diameter to both SPC/E waters andthe various ions, and possessing the values of the volume of the

simulation box (listed in the fifth column of Table 1), it is straight-forward to calculate the volume packing density g of the differentaqueous salt solutions. The g values, listed in the sixth column ofTable 1, cover a narrow range from 0.388 up to 0.404, because,even though the volume of the simulation box markedly decreaseson increasing the ion charge, the same occurs for the effectivediameter of the ions, so that the volume fraction really occupiedby ions and water molecules is little affected.

Using classic SPT, I calculated DGc values to create in SPC/Ewater and in all the aqueous salt solutions, at 300 K and 1 atm,spherical cavities of 10 Å and 15 Å diameter (i.e., the sphere corre-sponding to the plate van der Waals volume has a diameter ofabout 13.8 Å, and so two cavity diameters bridging such numberwere selected [16]). By looking at such DGc values, listed in the sec-ond column of Table 2, it proves that the DGc magnitude in all theaqueous salt solutions is larger than in SPC/E water, simply be-cause the latter has a smaller volume packing density: g(SPC/Ewater) = 0.356 (i.e., the volume of 1090 SPC/E waters was32 985 Å3). I divided the DGc numbers by WASAc, using for watersa radius of 1.4 Å as is usually done [13,14]. Then, I multiplied theDGc/WASAc ratios times the WASA decrease associated with platedimerization, DWASA = �890 Å2 [16], to obtain average valuesfor dGc(HI). The latter are listed in the third column of Table 2and point out that the contact configuration of the two plates is lar-gely favoured with respect to the configuration in which the twoplates are far apart, in both water and aqueous salt solutions,due to the large WASA decrease.

Finally, I calculated DdGc(HI) by means of Eq. (6); the corre-sponding numbers, reported in the fourth column of Table 2, indi-cate that dimerization is always more favoured in aqueous saltsolutions than in SPC/E water, regardless of the ion charge. Themagnitude of this salting-out effect depends on the ion charge:DdGc(HI) = �11.0 kJ mol�1 for |q| = 0.5e, �6.6 kJ mol�1 for|q| = 0.9e, �21.7 kJ mol�1 for |q| = 1.2e, and �34.4 kJ mol�1 for|q| = 1.4e. This happens because, as emphasized by the analysis ofclassic SPT relationship [27,31], the DGc magnitude increases: (a)on increasing g (i.e., on reducing the fraction of unoccupied vol-ume), keeping the effective size fixed; (b) on decreasing the effec-tive size of liquid molecules or, in the case of solutions, the averageeffective size hri =

Pv(j) � reff(j), where v(j) is the molar fraction of

the species j in solution (i.e., on rendering smaller the packets ofunoccupied volume), keeping g fixed. For the aqueous salt solutionwith |q| = 0.5e, hri = 2.872 Å, g = 0.404 and DGc(rc = 15 Å) =263.3 kJ mol�1; for the aqueous salt solution with |q| = 1.4e, hri =2.827 Å, g = 0.401 and DGc(rc = 15 Å) = 291.0 kJ mol�1.

G. Graziano / Chemical Physics Letters 491 (2010) 54–58 57

However, direct MD simulations by ZHB showed that salt addi-tion causes salting-in with respect to the situation in SPC/E waterfor |q| 6 0.9e, and salting-out for |q| > 0.9e. Specifically, DDG(HI) =+42.5 kJ mol�1 for |q| = 0.5e, and �20.0 kJ mol�1 for |q| = 1.4e; allthe DDG(HI) values obtained by ZHB are reported in the fifth col-umn of Table 2. By taking into account solely the contribution ofWASA decrease associated with dimerization (i.e., the solvent-ex-cluded volume effect), it results that all aqueous salt solutions causesalting-out in contrast with direct MD simulation data. This meansthat it is necessary to account for the direct interaction of ions withthe hydrophobic surface of the two plates by means of dispersionforces [32,33]. Such a direct dispersion interaction is much moreeffective and stronger on lowering the ion charge (i.e., by weaken-ing the electrostatic interactions with water molecules), as con-firmed by the number of ions released upon plate dimerizationobtained by ZHB [9], and listed in the last column of Table 2. Theneed to remove ions from plate surfaces contrasts dimerization,affording a positive DdGa(HI) term, whose magnitude has to be solarge to produce salting-in for ions with |q| 6 0.9e. Estimates ofDdGa(HI) have been obtained by means of Eq. (7), subtracting theSPT-DdGc(HI) numbers to the DDG(HI) values of ZHB, and are listedin the sixth column of Table 2. In particular, DdGa(HI) = 53.5kJ mol�1 for |q| = 0.5e, 24.7 kJ mol�1 for |q| = 0.6e, 9.2 kJ mol�1 for|q| = 1.2e, and 14.4 kJ mol�1 for |q| = 1.4e. The latter numbers arein line with the energy changes obtained by ZHB [9]: DDE(HI) = 57kJ mol�1 for |q| = 0.5e, 22 kJ mol�1 for |q| = 0.6e, 5 kJ mol�1 for|q| = 1.2e, and 0 for |q| = 1.4e.

4. Discussion

The results obtained by ZHB are very interesting because, onchanging the ion charge of the salt added to SPC/E water, bothweakening and strengthening of HI between the two platesemerged [9]. No structural property of such aqueous salt solutionscan reliably rationalize the ZHB results [11]. The latter show a goodcorrelation only with the number of ions preferentially bound orexcluded from the plate surface [note that the Dm(pref) quantitycalculated by ZHB, in the case of an association process, assumespositive values when there is preferential binding of water mole-cules to the hydrophobic surface and preferential exclusion of ions,and vice versa for negative values]. Such a correlation, however,cannot be considered entirely satisfactory to achieve a quantitativeunderstanding. For instance, the Dm(pref) quantity slightly chan-ged from 0.5 up to 1.0 for salts with 1.0e 6 |q| 6 1.4e, even thoughDDG(HI) passed from �5.0 kJ mol�1 up to �20 kJ mol�1, and themolecular origin of the salting-out proves to be absolutely nottransparent.

I have approached the matter from an entirely different point ofview: the WASA decrease upon association is the central quantityto arrive at an explanation [16,26]. Cavity creation at a fixed posi-tion in a liquid, keeping the number of particles, temperature andpressure constant, causes an increase in the ensemble average vol-ume of the system by a quantity equal to the cavity volume. How-ever, one has to be careful in considering that, upon cavity creation,a region around the cavity van der Waals surface becomes inacces-sible to solvent molecules in all liquids. Specifically, the centre ofsolvent molecules cannot enter the shell region between the cavityvan der Waals surface and the solvent-accessible surface of thecavity in order to have the cavity region really void. Thus, one saysthat cavity creation produces a solvent-excluded volume effect(i.e., a reduction in the configurational space available to solventmolecules) that, in turn, causes a loss of translational entropy forsolvent molecules. In water the inaccessible shell region can reli-ably be approximated by WASAc (the latter is a measure of the sol-vent-excluded volume), and so DGc has to depend on WASAc (more

correctly, keeping the cavity van der Waals volume fixed, DGc de-pends on the cavity shape, because, on changing the latter, alsoWASAc changes).

This solvent-excluded volume effect produces a strengtheningof pairwise HI in all the aqueous salt solutions considered byZHB simply because the DGc magnitude increases on adding saltsto SPC/E water (see Table 2), and there is a large WASA decreaseupon plate association. Classic SPT calculations show that suchan effect produces DdGc(HI) values possessing the right order-of-magnitude with respect to the ZHB results. Clearly, the DdGc(HI)numbers can explain the HI strengthening, but not the HI weaken-ing. For the latter, one has to simply remember that Eq. (7) is exact,and it would be necessary to account for the loss of energetic inter-actions between the plate surface and solvent particles (both ionsand water molecules) occurring upon dimerization as a conse-quence of the WASA decrease. The DdGa(HI) estimates prove tobe particularly large positive for salts with |q| = 0.5e and 0.6e, be-cause the corresponding ions possess a so low-charge density(i.e., they have the largest reff values and the lowest charge) tohave electrostatic interactions with water molecules much weakerthan the water–water H-bonds, so that the water molecules tendto push such ions towards the plate surfaces. In addition, low-charge density ions, in view of their size, should be able to do mul-tiple dispersion interactions with the carbon atoms of the platesurface. Even though the latter situation may seem almost unphys-ical because real ions do not possess a fractional charge smallerthan 1e, there are real ions possessing a low-charge density. For in-stance, the guanidinium ion is large, has a single positive chargedelocalized over the whole planar structure, is weakly hydrated[34], and shows preferential interaction with hydrocarbon mole-cules [35,36]. The behaviour of urea is similar: its ability to in-crease the water solubility of hydrocarbons is due to a bindingmechanism [37,38].

In conclusion, the rationalization of ZHB results emerged fromthe present analysis is the following. The solvent-excluded volumeeffect (i.e., WASA decrease) is the driving force for the associationof the two large hydrophobic plates in both SPC/E water and all theaqueous salt solutions (i.e., the contact configuration of the twoplates has the minimum Gibbs energy in all cases). High chargedensity ions, having strong electrostatic attractions with watermolecules, render more costly the process of cavity creation andcause a strengthening of pairwise HI. Low-charge density ions,being preferentially bound to plate surfaces, have to be removedto arrive at the contact configuration, and cause a weakening ofpairwise HI.

Acknowledgement

I would like to thank Dr. Ronen Zangi for sending me thenumerical values of the water oxygen-ion rdfs computed fromMD simulations.

Appendix A. On the effective size of low charge ions

The finding that, for cations with q < 0.8e, reff is larger thanrLJ = 5.0 Å, should emerge from the fact that I have assigned towater molecules reff(H2O) = 2.74 Å. The latter value is correct forboth water–water H-bonding distances and strong ion–water elec-trostatic interaction distances. However, when the ion charge islow, the effective diameter of water molecules interacting withsuch ions should be more close to rLJ(H2O) � 3.2 Å [10,29], becausethe ion–water electrostatic interactions are not strong enough tocontrast the repulsion of LJ interactions. By assigning to water mol-ecules reff(H2O) = 3.2 Å, one obtains: (a) reff = 5.0 Å for the cation,and 4.36 Å for the anion when |q| = 0.5e (i.e., the cation would have

58 G. Graziano / Chemical Physics Letters 491 (2010) 54–58

reff = rLJ); (b) reff = 4.8 Å for the cation, and 4.2 Å for the anionwhen |q| = 0.6e. The point is that, in such cases, there is not aclear-cut way to determine what is the right reff(H2O) value, sincethe latter depends on the strength of the interactions in whichwater molecules are involved [28–30].

For the aqueous salt solution with |q| = 0.5e, by using the abovereff values for ions, and by fixing that the fraction of watermolecules possessing reff = 2.74 Å is 0.8, so that the fraction ofwater molecules contacting ions (those having reff = 3.2 Å) is 0.2,it results g = 0.411, and DGc = 130.3 kJ mol�1 for rc = 10 Å, and278.9 kJ mol�1 for rc = 15 Å, at 300 K and 1 atm. These DGc valuesare larger than those calculated in SPC/E water (see Table 2), andimply that, also with this different assignment of reff to ions andwaters, the aqueous salt solution with |q| = 0.5e produces a largenegative DdGc(HI) term (i.e., its salting-in ability should be dueto preferential binding). The assumption that 20% of waters hasreff = 3.2 Å represents a lower limit because it means that, on aver-age, the hydration shell of the 60 ions consists of less than fourwater molecules.

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