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Chemical Physics Letters 429 (2006) 420–424

Non-intrinsic contribution to the partial molar volume of cavitiesin water

Giuseppe Graziano *

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

Received 31 July 2006Available online 18 August 2006

Abstract

In general the partial molar volume of a cavity in a liquid or that of a solute in a solvent can be dissected in an intrinsic contributionand a non-intrinsic one. It is shown that, if the intrinsic contribution is given by the van der Waals volume of the cavity, the non-intrinsiccontribution is always a positive quantity in water, increasing with cavity size. This result has a general validity, being in line with thebasic notion that there is a lot of void volume in liquids, and water, in particular, has a small volume packing density. In addition, it ispointed out that the recent claim [F.M. Floris, J. Phys. Chem. B 108 (2004) 16244] that the non-intrinsic contribution changes sign, pass-ing from negative to positive values for sufficiently large cavities, is simply due to the choice of using the excluded volume of the cavityinstead of the van der Waals volume as the intrinsic contribution.� 2006 Elsevier B.V. All rights reserved.

1. Introduction

The partial molar volume, PMV, of a solute in a solventis an important quantity to shed light on the nature andeffectiveness of solute–solvent interactions [1]. Among sol-utes a hard sphere is the simplest one, especially when it isassimilated to a cavity existing in the solvent. The PMV ofa cavity in a solvent, indicated as Vcav, is not an experimen-tal quantity, but it can be determined by means of suitablecomputer simulations or theoretical expressions groundedon hard sphere theories or more sophisticated liquid statetheories [2–4].

In general, the PMV of a solute is divided in an intrinsiccontribution and a non-intrinsic one [5,6]; note that, byadopting the Ben-Naim standard [7], the contributionbT Æ RT to PMV, where bT is the isothermal compressibilityof the solvent, is excluded from the outset. The intrinsicpart of the PMV of a spherical cavity in a liquid corre-sponds to the geometric van der Waals volume of thecavity:

0009-2614/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2006.08.065

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

V i ¼ 4p � r3c=3 ð1Þ

where the subscript i stands for intrinsic, and rc is the ra-dius of the cavity, i.e., the radius of the spherical regionfrom which any part of liquid molecules is excluded [8].It is fundamental to recognize that there is another measureof the cavity radius indicated by Rc = rc + r1, where r1 isthe radius of the liquid molecules assimilated to spheres;Rc is the radius of the spherical region from which the cen-tres of liquid molecules are excluded [8] (i.e., the excludedvolume of the cavity). Obviously, the two cavity sizesshould not be confused.

The non-intrinsic contribution to the PMV of a spheri-cal cavity is simply given by

V ni ¼ V cav � V i ¼ V cav � ð4p � r3c=3Þ ð2Þ

where the subscript ni stands for non-intrinsic; this contri-bution comes from the spatial organization of liquid mole-cules around the cavity and reflects the features of bothintermolecular packing and interactions.

Recently, Floris [9] computed directly Vcav for cavitiesof very different sizes by means of Monte Carlo simula-tions, at room temperature, in TIP4P water [10] usingan NPT ensemble, and claimed that the non-intrinsic

0 100

500

1000

1500

2000

2500

Vca

v (cm

3 m

ol-1

)

rc (angstrom) 2 4 6 8

Fig. 1. Values of Vcav calculated by means of SPT equation (3) withr1 = 2.70 A, 2.80 A and 2.91 A (solid thick lines), and by means of MonteCarlo simulations by Floris [9] (solid thin line); values of Vi calculated bymeans of equation (1) using rc (dashed line), or Rc = rc + 1.4 A (dottedline). See text for further details.

G. Graziano / Chemical Physics Letters 429 (2006) 420–424 421

contribution to Vcav shows an interesting behaviour as afunction of cavity radius. Floris pointed out that: (a) theresults of Monte Carlo simulations indicate an inversionof the sign of Vni occurring for a cavity radius Rc between8 and 10 A from negative values to positive ones; (b) thisinversion of the sign does not occur if the scaled particletheory [11], SPT, is used to compute Vcav, since Vni provesto be a negative quantity over the whole investigated cavitysize range. These results would imply that: (a) somethingspecial should occur in water on increasing the cavityradius Rc beyond 8 A; (b) SPT, being a hard sphere theory,cannot be able to reproduce in a qualitatively correct man-ner the size dependence of Vcav in water.

I decided to test the correctness of these results becausethe Vcav formula derived from the SPT equation of state forbinary hard sphere mixtures in the infinite dilution limit [3]seems to work satisfactorily well in reproducing the PMVof both hydrocarbons and n-alcohols in water [12,13].The performed calculations show that the non-intrinsiccontribution Vni is always a positive quantity consideringboth the Vcav values obtained from the Monte Carlo simu-lations and those obtained from the SPT formula, if thecavity van der Waals volume is considered to be the intrin-sic contribution.

2. Results from scaled particle theory

Starting from the SPT equation of state for binary hardsphere mixtures, and calculating the derivative (oV/on2) inthe limit of infinite dilution, n2! 0, at constant tempera-ture, pressure and number of moles of the solvent, Lee[3] obtained the following analytical formula for Vcav:

V cavðSPTÞ ¼ ðx3 þ 3Ax2 þ 3 A2xþ A2BÞ � v1=C ð3Þwhere A = (1 � n)/(1 + 2n), B = (1 � n)/n, C = (1 + 3A +3A2 + A2B); n is the volume packing density of the puresolvent, which is defined as the ratio of the physical volumeof a mole of solvent molecules over the molar volume ofthe solvent, v1 (i.e., n ¼ p � r3

1 � N Av=6 � v1); r1 is the hardsphere diameter of the solvent molecules; and x = rc/r1,where rc = 2 Æ rc is the hard sphere diameter of the cavity.

In order to perform calculations, I used the experimentaldensity of water at 25 �C [14], and for the effective hardsphere diameter of water molecules, I selected three values:r1 = 2.70 A, 2.80 A and 2.91 A, respectively. The first valueis that suggested and used by Pratt and Pohorille [15]; thesecond one is close to the first maximum in the oxygen–oxygen radial distribution function of water [16], and seemsto work well in reproducing, by means of SPT, the cavitysize distribution in water models [17]; the third one hasbeen determined by Tomasi and co-workers [18] in orderto obtain a close agreement between the results of com-puter simulations for the work of cavity creation in TIP4Pwater and those obtained using SPT.

The SPT-calculated values of Vcav in water at 25 �C arereported in Fig. 1 over the cavity radius range0 6 rc 6 10 A (see the three solid thick lines in Fig. 1).

The three sets of numbers are close to each other eventhough the difference in the effective diameter of water mol-ecules is not small. Actually, the Vcav(SPT) values calcu-lated by fixing r1 = 2.70 A are the largest, and thosecalculated by fixing r1 = 2.91 A are the smallest. The fun-damental point is that all the three sets of Vcav(SPT) valuesare larger than the Vi numbers obtained by means of Eq.(1), over the whole cavity radius range 0 6 rc 6 10 A (com-pare the three solid thick lines with the dashed line inFig. 1). This means that the non-intrinsic contributionVni = Vcav(SPT) � Vi is a positive quantity over the wholecavity radius range 0 6 rc 6 10 A, in contrast with theclaim by Floris.

This result is absolutely not new. More than thirty yearsago, Edward and Farrell [19] showed that the PMV of non-polar solutes in water can be described by the equation:

PMV ¼ 4p � ðr2 þ dÞ3=3 ð4Þwhere r2 is the radius of the solute molecule and d is thethickness of a layer of empty volume surrounding the sol-ute molecule. Comparison with experimental data for alarge series of solutes indicated that d � 0.5 A in water atroom temperature [19,20]. On the same line, Lee [3] showedthat the Vcav(SPT) expression of Eq. (3) is consistent withEq. (4), and confirmed that d � 0.5 A in water at roomtemperature. Since Vni = Vcav(SPT) � Vi is a positive quan-tity, it is important to clarify the origin of the contrastingresults presented by Floris.

3. Results from the Kirkwood–Buff approach

The solid thin line in Fig. 1 corresponds to the Vcav val-ues obtained by means of Monte Carlo simulations inTIP4P water at room temperature by Floris [9], afterrescaling from Rc to rc,using r1 = 1.4 A; such Vcav(MonteCarlo) numbers are markedly larger than the Vcav(SPT)

422 G. Graziano / Chemical Physics Letters 429 (2006) 420–424

ones. The dotted line in Fig. 1 corresponds to the Vi num-bers obtained by means of Eq. (1) inserting in it the Rc val-ues instead of the rc ones (i.e., by using the Rc radiusinstead of the rc one, with r1 = 1.4 A in order to be consis-tent); these values will be indicated in the following as Vexcl.It is evident that: (a) Vexcl is always larger than Vcav(SPT);(b) Vexcl is larger than Vcav(Monte Carlo) up to rc � 6.5 A,while for larger rc values Vcav(Monte Carlo) proves to belarger than Vexcl. It should be clear that, by using Vexcl,the behaviour of the non-intrinsic contribution Vni

reported by Floris emerges in an exact manner: (a) Vni isnegative using the Vcav(SPT) estimates over the whole cav-ity size range; (b) Vni is negative up to rc � 6.5 A and thenbecomes positive using the Vcav(Monte Carlo) estimates.Therefore, according to these simple calculations, the inter-esting behaviour of Vni pointed out by Floris is the conse-quence of the choice adopted in order to compute theintrinsic volume of the cavity: Floris used the Rc radiusinstead of the rc one, even though it is the latter that mea-sures the spherical region that cannot be penetrated by anypart of the solvent molecules.

The choice of Floris to use the Rc radius instead of the rc

one in calculating Vi originated from the analysis per-formed by Matubayasi and Levy [21], M&L, in order todevelop a hydration shell approach for the calculation ofmethane PMV in water. According to the Kirkwood–Bufftheory of solutions [22], the PMV of a solute in water isgiven by

PMV ¼ �4p �Z 1

0

½gswðrÞ � 1� r2 dr ð5Þ

where gsw(r) is the solute–water radial distributionfunction. The Kirkwood–Buff formula is exact and has anon-local character by definition, because it requires theintegration up to infinity of the solute–solvent radial distri-bution function within the grand canonical ensemble. Incomputer simulations, in which a finite cutoff has to beintroduced, it is necessary to extend the integration up tolarge distances from the central solute molecule in orderto obtain reliable estimates of PMV. Fig. 1b of M&L [21]and Fig. 3b of Floris [9] unequivocally demonstrate thenon-local character of the Kirkwood–Buff formula forPMV.

In the case of a cavity, the cavity–water radial distribu-tion function gcw = 0 over the range 0 6 r 6 rc + rw, whererw is the radius of water molecules; for r = rc + rw, the so-called contact point, gcw reaches the maximum value and,

Table 1For three cavity sizes are listed the values of Vcav determined by means of MoWaals volume, VvdW, excluded volume, Vexcl, non-intrinsic contribution Vni(1) =first hydration shell water vhs, calculated by means of Eq. (7) and Vni(1) or V

Rc Vcav VvdW Vexcl

3.45 60 22 1047.05 826 455 884

10.0 2637 1604 2523

The Rc values are in A units, while all the volume values are in cm3 mol�1 un

on increasing r, it shows other smaller maxima. Therefore,Eq. (5) can be dissected in two parts:

V cav ¼ 4p �Z rcþrw

0

r2 dr � 4p �Z 1

rcþrw

½gcwðrÞ � 1�r2 dr ð6Þ

where the first term on the right-hand-side corresponds tothe excluded volume of the cavity V excl ¼ 4pðrcþrwÞ3=3 ¼ 4pR3

c=3; the second term on the right-hand-sideis the hydration shell contribution according to M&L. Thisdissection is feasible, even though its reliability has to bejudged depending on the physical scenario that providesfor Vcav in water. What should be clear from the outset isthat: (a) Vexcl contains a large contribution from the watermolecules in the first hydration shell (i.e., those contactingthe van der Waals surface of the cavity); (b) as a conse-quence, the actual meaning of the term identified as thehydration shell contribution in Eq. (6) proves to be notso transparent. It appears that the non-local character ofthe Kirkwood–Buff formula does not allow a non-ambigu-ous identification of the intrinsic volume of the cavity.

On this basis, I have analyzed the Vcav(Monte Carlo)values in order to see the emerging scenario for water mol-ecules in the first hydration shell. I selected cavities withRc = 3.45 A, 7.05 A, and 10.0 A (the rc values are 2.05 A,5.65 A, and 8.6 A). The corresponding values of Vcav,VvdW, Vexcl, Vni(1) = Vcav � VvdW, and Vni(2) = Vcav � Vexcl

are listed in the columns 2–6 of Table 1. The two sets ofVni estimates can be analyzed by means of the followingrelationship proposed by M&L – see Eq. (2.6) in [21]:

V ni ¼ NH2O � ðvhs � vbÞ ð7Þwhere NH2O is the number of water molecules in the firsthydration shell, 18, 68 and 133, respectively, for the se-lected cavities, according to Fig. 6a in [9]; vb is the molarvolume of bulk water, 18.1 cm3 mol�1 at 25 �C [14]; andvhs is the molar volume of the water molecules constitutingthe cavity first hydration shell. By using the two sets of Vni

estimates, two distinct sets of vhs values have been ob-tained, and are reported in the last two columns of Table 1.

According to vhs(1) estimates, the water molecules in thecavity first hydration shell have always a molar volume lar-ger than that of bulk water, and the difference increaseswith cavity size. For the largest cavity, Rc = 10 A,vhs(1) = 25.9 cm3 mol�1 versus vb = 18.1 cm3 mol�1 (thereis a 43% increase in molar volume). According to vhs(2)estimates, the water molecules in the cavity first hydrationshell have a molar volume smaller than that of bulk water

nte Carlo simulations by Floris [9], the corresponding estimates of van derVcav � VvdW, and Vni(2) = Vcav � Vexcl; estimates of the molar volume of

ni(2), are reported in the last two columns

Vni(1) Vni(2) vhs(1) vhs(2)

38 �44 20.2 15.7371 �58 23.6 17.3

1033 114 25.9 19.0

its.

G. Graziano / Chemical Physics Letters 429 (2006) 420–424 423

up to Rc � 8 A; for larger cavities vhs(2) becomes slightlylarger than vb. For the largest cavity, Rc = 10 A,vhs(2) = 19.0 cm3 mol�1 versus vb = 18.1 cm3 mol�1.

Lum et al. [23] have pointed out that, on increasing thecavity size, water molecules dewet the cavity surface inorder to avoid the waste of H-bonds. Floris [24] has sug-gested that the change in the sign of the non-intrinsic con-tribution to Vcav should be a direct consequence of cavitydewetting by water molecules. Cavity-dewetting shouldmanifest itself with a marked density decrease of watermolecules in the first hydration shell of the cavity. Such adensity decrease should correspond to a marked increaseof molar volume for water molecules in the first hydrationshell of the cavity with respect to that of bulk water. This isexactly the behaviour of vhs(1) estimates, obtained by fixingVi = VvdW and using Eq. (7). The scenario in which thehydration shell water is always less dense than bulk wateris physically reliable because the hydration shell corre-sponds to a disordered clathrate cage with the water–waterH-bonds more broken than those in the bulk, on the basisof both structural [25–27] and statistical thermodynamic[28] investigations, also in the case of small nonpolarsolutes. In addition, from the analysis of the experimentalvalues of PMV and adiabatic compressibility of a, x-ami-nocarboxylic acids in aqueous solutions [29], it emergedthat the density of water in the hydration shell of a meth-ylene group is 5–6% lower than that of bulk water.

In contrast, it does not appear physically reliable thatvhs(2) is smaller than vb up to Rc � 8 A because this behav-iour should originate from something like electrostriction.It is difficult to imagine that electrostriction is operativefor water molecules in the cavity first hydration shell (i.e.,there is not a strong attractive interaction between a cavityand surrounding water molecules, in contrast to the case ofions). In this respect it is worth noting that water in the3 -A thick hydration layer around globular proteins hasan average density approximately 10% larger than that ofbulk water [30]. Computer simulations have shown thatthis density increase is due to topographical features andelectrostatic properties of the protein surface [31]. The lat-ter mechanisms cannot be operative for water moleculesconstituting the hydration shell of spherical cavities or non-polar moieties, as confirmed by experimental structuraldata [25–27].

4. Discussion

The performed analysis should clarify that the non-intrinsic contribution to the PMV of a cavity in water isalways a positive quantity if the intrinsic contribution isgiven by the van der Waals volume of the cavity. A differ-ent choice, considering that the intrinsic contribution isgiven by the excluded volume of the cavity, leads to thestrange results reported by Floris [9], with an inversion ofthe sign for the non-intrinsic contribution on increasingthe cavity size. Such a different choice, grounded in theKirkwood–Buff theory of solutions, even though feasible,

is not useful to distinguish an intrinsic from a non-intrinsiccontribution to the PMV of a cavity or a nonpolar solute inwater. In fact, its use leads to artefacts, as further shownbelow.

By adopting the dissection scheme of Eq. (6), M&L [21]claimed that the intrinsic volume of methane shouldbe its excluded volume. If the effective radius of amethane molecule is 1.9 A and that of a water molecule is1.4 A, Vi(CH4) = Vexcl(CH4) = 4p Æ (1.9 + 1.4)3/3 = 90.7cm3 mol�1. Since the experimental partial molar volumeof methane PMV(CH4)exp = 36.2 cm3 mol�1 at 25 �C [21],the non-intrinsic contribution proves to be Vni(CH4) =36.2� 90.7 = �54.5 cm3 mol�1. From the M&L choice itdescends that the contribution of hydration shell watermolecules to methane PMV is a large and negative quan-tity. This negative value is because the water molecules incontact with the methane molecule contribute largely toVexcl(CH4), that, as a consequence, should not be consid-ered the intrinsic volume of the solute. Such a result is alsoin qualitative contrast with: (a) the experimental findingthat a disordered clathrate cage of water molecules sur-rounds the methane molecule [26] (b) the M&L simulationresult [21] that the solution of methane in TIP4P water ismore compressible than pure TIP4P water. In other words,the relative softness of water molecules in the hydrationshell of methane contrasts with the idea that they providea negative contribution to PMV. It is worth noting thatChalikian has already pointed out the incorrectness ofusing Vi = Vexcl in a theoretical analysis of volume changesaccompanying the conformational transitions of globularproteins [32].

A further test of the reliability of fixing Vi = Vexcl can beperformed by analyzing the molar volume of water. In theassumption that the effective radius of a water molecule is1.4 A,Vi(H2O) = Vexcl(H2O) = 4p Æ 2.83/3 = 55.4 cm3 mol�1.Since the experimental molar volume of water is 18.1cm3 mol�1 at 25 �C [14], the non-intrinsic contributionwould be Vni = 18.1 � 55.4 = �37.3 cm3 mol�1. This largeand negative value cannot have any special meaningbecause it is associated with the solvation of water in water.Actually, it is the consequence of the choice adopted to cal-culate Vi. By fixing Vi = Vexcl, one takes into account con-tributions coming from the water molecules in contact withthe central one (i.e., roughly speaking, about half of thevolume of each of the four molecules constituting the firsthydration shell contributes to Vi = Vexcl).

On the other hand, by using Eq. (1), the intrinsic molarvolume of water is Vi = VvdW = 6.9 cm3 mol�1 and thenon-intrinsic one is Vni = 18.1 � 6.9 = 11.2 cm3 mol�1.These values are compatible with Eq. (4) usingd = 0.53 A. In this manner, Vni is a positive quantity asexpected on the basis of physical intuition and liquid statetheories, because the volume packing density of liquids atroom temperature is in general around 0.5, and that ofwater is smaller, amounting to 0.38 with r1 = 2.80 A(clearly, 6.9/18.1 = 0.38) [3,15,17]. In other words, thereis a lot of void volume around a water molecule in water,

424 G. Graziano / Chemical Physics Letters 429 (2006) 420–424

and this is true also for nonpolar solutes or cavities inwater. Therefore, this analysis supports the validity ofEq. (1) to calculate Vi because the van der Waals surfaceof a solute molecule or of a cavity is the surface wherethe local packing density of water molecules is zero, andthere cannot be a contribution from water molecules to Vi.

It is important to underline that, even though SPT-cal-culated quantities are qualitatively right for water, theyare not quantitatively close to those derived from MonteCarlo simulations by Floris [9]. The quantitative failureof SPT in the case of water cannot come as a surprise.SPT is a hard sphere theory that uses only two rough struc-tural input data to characterize a real liquid: the experi-mental density and the effective diameter of liquidmolecules. At the same time, the results of computer simu-lations should not be considered exact because, especiallyin the case of very large cavities, they could be affectedby the limited size of the simulated system.

In conclusion, the present analysis indicates that, atroom temperature, the non-intrinsic contribution to thePMV of a cavity in water is a positive quantity increasingin magnitude with cavity size, if the intrinsic contributioncorresponds to the van der Waals volume of the cavityitself. The latter choice does appear to be the right onefrom the physical point of view, because the van der Waalsvolume does not take into account any contribution fromwater molecules in contact with the cavity surface.

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