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Chemical Physics Letters 432 (2006) 84–87

Cavity contact correlation function of water from scaled particle theory

Giuseppe Graziano *

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

Received 20 September 2006; in final form 4 October 2006Available online 13 October 2006

Abstract

The ratio of the number density of solvent molecules just outside a cavity of radius Rc to that in the bulk represents the cavity contactcorrelation function G(Rc). Experimental determination of G(Rc) for water by means of computer simulations has led to the claim thatthe shape of this function is a basic manifestation of the structural features of the H-bonded network of water [H.S. Ashbaugh, L.R.Pratt, Rev. Mod. Phys. 78 (2006) 159]. In this Letter, I would like to show that classic scaled particle theory is able to reproduce in asatisfactory manner the experimental G(Rc) values by taking into account two things: (a) the size of a water molecule depends on theinteractions in which it is involved, so that the effective diameter of H-bonded molecules is 2.8 A, whereas the van der Waals diameterof a water molecule is 3.2 A; (b) at any time, liquid water contains both H-bonded molecules and molecules interacting by means of vander Waals forces.� 2006 Elsevier B.V. All rights reserved.

1. Introduction

Scaled particle theory, SPT, is a hard sphere theory offluids developed by Reiss, Frisch and Lebowitz on the basisof simple geometric, physical and statistical principles [1,2].Over the years SPT has proven to be very powerful also todescribe phenomena such as the solubility of nonpolar spe-cies in water [3–5], the cavity size distribution in water [6,7],and the occurrence of entropy convergence in the solvationthermodynamics of several species in water [8,9], well out-side the realm of hard sphere fluids. A central function ofSPT is the so-called cavity contact correlation function[1,2], G(Rc), where Rc is the radius of the cavity (note thatthere are two measures of the cavity radius: (a) rc is theradius of the spherical region from which any part of liquidmolecules is excluded; (b) Rc is the radius of the sphericalregion from which the centres of liquid molecules areexcluded, Rc = rc + r1, where r1 is the radius of the liquidmolecules assimilated to spheres). The function G(Rc) isthe conditional pair distribution that characterizes theprobability of finding a molecular centre in the spherical

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doi:10.1016/j.cplett.2006.10.016

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

shell of thickness Rc to Rc + dRc, given that the sphere ofradius Rc is devoid of molecular centres (i.e., it is a cavity)[2]. More simply, G(Rc) is the ratio of the number densityof solvent molecules in contact with a cavity of radius Rc

to that in the bulk and is given by the rate of change ofthe work of cavity creation with respect to the cavityradius. The G(Rc) function can readily be determined bymeans of computer simulations because its value for agiven Rc corresponds to the value of the radial distributionfunction at the contact distance Rc between a central hardsphere of radius rc and the surrounding solvent molecules[10].

The first determination of the function G(Rc) for waterwas performed by Pratt and Pohorille [11], using moleculardynamics simulations of TIP4P water [12] at room temper-ature over the cavity size range 0 6 Rc 6 3.4 A. They foundthat the function has a maximum of about 2.3 at Rc @ 2.8 Aand pointed out that the trend of G(Rc) cannot be repro-duced by classic SPT using an effective diameter for watermolecules rw = 2.7 A, because in the SPT function themaximum occurs at Rc @ 2.0 A. More recently, Floris[13], by means of Monte Carlo simulations of TIP4P waterat 25 �C and 1 atm, has determined the G(Rc) function overthe cavity size range 0 6 Rc 6 10 A. The latter function

G. Graziano / Chemical Physics Letters 432 (2006) 84–87 85

presents a maximum of 2.2 at Rc = 2.8 A and thendecreases in a continuous manner, amounting to 0.9 atRc = 10 A. The values determined by Floris are shown asopen squares in Fig. 1; they are in line with those deter-mined by Pratt and Pohorille over the cavity size rangeinvestigated by the latter authors [11]. In addition, veryrecently, Ashbaugh and Pratt [14] have used the trend ofthe G(Rc) function of water as the basis to develop a revisedSPT approach, following in spirit an earlier proposal byStillinger [15]. Stillinger developed a revised SPT by incor-porating the experimental oxygen–oxygen radial distribu-tion function of water and the experimental bulk surfacetension of water with the aim to take into account the tet-rahedral coordination of the H-bonded network, by con-sidering the possible arrangements of triplets of oxygenatoms (i.e., the structural features of water differ fromthose of a hard sphere liquid). However, the G(Rc) functioncalculated by Stillinger at 4 �C showed a maximum of 2.4at Rc = 2.1 A [15], that is not in line with the ‘experimental’datum that the maximum of G(Rc) is located at Rc @ 2.8 A.According to Ashbaugh and Pratt [14], the shape of G(Rc)is a fundamental feature of water structure, determiningthe energetics of cavity creation, that cannot be reproducedin a quantitative manner by classic SPT that uses as inputdata only the solvent number density and the size of sol-vent molecules.

In the present Letter, I would like to show that classicSPT can be used to reproduce in a satisfactory mannerthe general trend of G(Rc) versus Rc in water by taking intoaccount two fundamental points: (a) the size of a watermolecule depends on the interactions in which it isinvolved; the effective diameter of H-bonded molecules isrw = 2.8 A, whereas the van der Waals diameter of a watermolecule is rw = 3.2 A [16,17]; (b) at any time, liquid water

0 2 4 6 8 10

1.0

1.5

2.0

2.5

3.0

3.5

4.0

c

b

a

G (

Rc

)

Rc (angstrom )

Fig. 1. Comparison between the cavity contact correlation function ofTIP4P water at 25 �C and 1 atm (empty squares), determined by Floris viaMonte Carlo simulations [13], and those calculated using classic SPT Eqs.(4) and (5) for rw = 2.8 A (curve a) and rw= 3.2 A (curve b), and usingEq. (8) (curve c). The experimental number density of water at 25 �C isused in all calculations.

contains both H-bonded molecules and molecules interact-ing by means of van der Waals forces.

2. Scaled particle theory relationships

Direct geometric arguments lead to the following exactrelationship for the work of cavity creation when Rc 6 r1

(i.e., at most one molecular centre can be found in the cav-ity for 0 6 Rc 6 r1):

DGc ¼ �RT � ln½1� ð4=3Þp � q1 � R3c � ð1Þ

where q1 = NAv/v1 is the number density of the solvent andv1 its molar volume. When Rc P r1, Reiss and colleagues[1,2] provided the following formula for DGc:

DGc ¼ �RT � lnð1� nÞ þ RT � ½uðrc=r1Þþ ðu=2Þ � ðuþ 2Þ � ðrc=r1Þ2� þ n � P � v1ðrc=r1Þ3 ð2Þ

in this relation R is the gas constant; n is the volume packingdensity of pure solvent, which is defined as the ratio of thephysical volume of a mole of solvent molecules over the mo-lar volume of the solvent, (i.e., n ¼ p � r3

1 � N Av=6 � v1), andu = 3n/(1 � n); rc = 2 Æ rc and r1 = 2 Æ r1 are the hard spherediameter of the cavity and of the solvent molecules, respec-tively; and P is the pressure. Since it has been suggested ontheoretical grounds [3,15,18] that the pressure has to be1 atm in performing calculations for real liquids, the cubicterm in Eq. (2) proves to be very small and can be neglected.

The cavity contact correlation function G(Rc), which isthe conditional solvent density just outside a spherical cav-ity of radius Rc, is given by [8]:

GðRcÞ ¼ ð1=4p � q1 � R2cÞ � ½@ð�Gc=RT Þ=@Rc� ð3Þ

Performing the derivatives one obtains:

Gð0 6 Rc 6 r1Þ ¼ 1=½1� ð4=3Þp � q1 � R3c � ð4Þ

GðRc P r1Þ ¼ ð1=2p � q1 � R2cÞ � fðu=r1Þ

þ ½2uðuþ 2Þ=r21�rcg ð5Þ

Note that the r2c term has been neglected in Eq. (5) for its

smallness when P = 1 atm. Eq. (4) is an always increasingfunction of Rc and does not possess a maximum. A searchfor a maximum in the expression of Eq. (5) has given:

Rc;max ¼ r1 � ½1þ ðu=uþ 2Þ� ð6ÞGðRc; maxÞ ¼ uðuþ 2Þ2=2p � q1 � r3

1ðuþ 1Þ ð7Þ

In order to perform calculations, I used the experimentaldensity of water at 25 �C [19], and for the effective hardsphere diameter of water molecules, I selected two values:rw = 2.8 A and 3.2 A, respectively. The first value is closeto the first maximum in the oxygen–oxygen radial distribu-tion function of water [20], and seems to work well inreproducing, by means of SPT, the cavity size distributionin water models [8,9]; the second one corresponds to thevan der Waals diameter of a water molecule [16,17]. Thedifference between the two diameters is due to the strengthof H-bonds which are able to bunch up water molecules

Table 1Values of DGc/ASA at 25 �C calculated for cavities of different size byconsidering rw = 2.8 A (n = 0.383), and rw = 3.2 A (n = 0.572)

rc

(A)DGc(rw = 2.8 A)/ASA(J A�2 mol�1)

DGc(rw = 3.2 A)/ASA(J A�2 mol�1)

DG/ASA(J A�2 mol�1)

10 299.6 728.0 363.925 334.6 838.1 410.150 348.4 881.9 428.4

100 356.5 906.0 438.9

The numbers obtained by means of Eq. (9) are listed in the last columnand should be compared to the experimental bulk surface tension of waterat 25 �C, 433.5 J A�2 mol�1 [31]. Since ASA = 4p (rc + rw)2, different ASAvalues are obtained by fixing rw = 1.4 or 1.6 A; the differences, however,are very small in view of the large size of the selected cavities.

86 G. Graziano / Chemical Physics Letters 432 (2006) 84–87

well beyond their van der Waals diameter, as convincinglyshown by Madan and Lee [16]. By turning off the partialcharges in the TIP4P water model, while keeping fixedthe number density, Madan and Lee showed that the loca-tion of the maximum in the oxygen–oxygen radial distribu-tion function increases from �2.8 A to �3.2 A [16]. Inaddition, far-infrared vibration-rotation tunnelling spec-troscopy measurements demonstrated that the averageoxygen–oxygen distance decreases on increasing the num-ber of H-bonded water molecules in a cluster [21].

3. Results and discussion

The G(Rc) functions calculated by means of classic SPTEqs. (4) and (5) for rw = 2.8 A (n = 0.383) and rw = 3.2 A(n = 0.572) at 25 �C are reported in Fig. 1, as curve a andcurve b, respectively. Both curves present a maximum:G(rw = 2.8 A) = 2.1 at Rc,max = 2.08A, and G(rw =3.2 A) = 4.2 at Rc,max = 2.67 A. However, no one of thesetwo curves reproduces the ‘experimental’ values of G(Rc)determined by Floris [13], confirming the failure of classicSPT pointed out by Pratt and colleagues [11,14].

It is possible to do better by recognizing that in waterthere is a fraction of interstitial molecules not connectedto the others by means of H-bonds. Sharp and Madanfound that in the first peak of the oxygen–oxygen radialdistribution function of water at room temperature thereare about four molecules H-bonded to the central oneand about one interstitial molecule not H-bonded to thecentral one [22]. In addition, from a two-state analysis ofboth Raman spectra [23] and configurational heat capacityof water [24], it resulted that the fraction of broken H-bonds is about 0.30 at 25 �C [25,26] (i.e., the fraction ofwater molecules not involved in H-bonds should be 0.15).These structural data coupled with the notion that theeffective size of a water molecule is 2.8 A when it formsH-bonds and 3.2 A when it does not form H-bonds[16,17], support the assumption that the cavity contact cor-relation function of water can be represented as

GðRcÞ ¼ 0:85 � Gðrw ¼ 2:8 AÞ þ 0:15 � Gðrw ¼ 3:2 AÞ ð8ÞThis function is reported in Fig. 1 labeled as curve c: it pre-sents a maximum of 2.39 at Rc,max = 2.2–2.4 A. It is evidentthat the G(Rc) of Eq. (8) reproduces in a satisfactory man-ner the ‘experimental’ values determined by Floris [13]. Theuse of a linear, weighted average in Eq. (8) does not have atheoretical basis, and has to be considered an heuristic ap-proach. It belongs to the family of two-state models, that,in several cases, have proven to be a useful approximationfor the continuum of H-bonding states of water [23–26].

A significant discrepancy still exists over the cavity sizerange 1.8 A 6 Rc 6 2.6 A, but the overall agreement canbe considered to be good in view of the simple procedureadopted. In all probability the remaining discrepancyshould be due to the tetrahedral coordination of water thatmanifests itself over the Rc range corresponding to the pos-sible distances among three H-bonded water molecules

(i.e., see the location of the second peak in the oxygen–oxy-gen radial distribution function of water and its rationali-zation in molecular terms [27]). Moreover, Finney hasargued that also trigonal coordination should play animportant role in determining the water structure [17,28].Clearly, a hard sphere theory such as SPT cannot describetetrahedral and/or trigonal coordination. Heying and Cortihave recently performed a revisitation of classic SPT inorder to obtain a more accurate determination of G(Rc)by taking into account, in an approximate way, thethree-particle correlation functions [29]. It is not the aimof the present analysis to test the performance of SPT revis-ited by Heyden and Corti.

The reliability of Eq. (8) with the fraction of not H-bonded water molecules equal to 0.15 at 25 �C can be fur-ther validated by calculating, using classic SPT Eq. (2), theratio of the work of cavity creation in water to the accessi-ble surface area, ASA, of the cavity as

DGc=ASA ¼ 0:85 � ½DGcðrw ¼ 2:8 AÞ=ASA�þ 0:15 � ½DGcðrw ¼ 3:2 AÞ=ASA� ð9Þ

where ASA ¼ 4pR2c . As emphasized by Chandler [30], for

very large cavities this ratio should be close to the experi-mental bulk surface tension of water, 433.5 J A�2 mol�1

at 25 �C [31]. The calculated numbers are reported in Table1 and indicate that DGc/ASA from Eq. (9) amounts to428 J A�2 mol�1 for rc = 50 A and 439 J A�2 mol�1 forrc = 100 A, in good agreement with the bulk surface ten-sion of water. The ability of classic SPT to reproduce thebulk surface tension of a complex liquid such as watershould be considered a surprising result because SPT is ahard sphere theory that uses as input data only the exper-imental number density of a liquid and the effective size ofliquid molecules. Therefore the criticism that classic SPTpredicts an improper temperature dependence of the bulksurface tension of water [14] is absolutely too strong andunjustified. In addition, with respect to the temperaturedependence, I would like to point out that the effective sizeof liquid molecules is expected to decrease on increasingtemperature, as well explained by Reiss forty years ago [2].

A final point merits attention. Ashbaugh and Pratt [14]used their revised SPT approach to calculate the G(Rc)

Table 2Values of Rc,max and G(Rc,max), calculated by means of Eqs. (6) and (7) forrw = 2.8 and 3.2 A, respectively, at 25 and 100 �C, using the experimentalnumber density of water

T (�C) rw (A) n Rc,max (A) G(Rc,max)

25 2.8 0.383 2.08 2.113.2 0.572 2.67 4.21

100 2.8 0.368 2.05 2.023.2 0.550 2.64 3.83

G. Graziano / Chemical Physics Letters 432 (2006) 84–87 87

function of water over a large temperature range: thecurves have a very similar shape but the location of themaximum and its height decrease with temperature.According to Fig. 7 of Ashbaugh and Pratt [14], the max-ima are: Rc,max = 2.8–3.0 A with G = 2.25 at 300 K, andRc,max = 2.6 A with G = 2.15 at 360 K. These valuesemphasize that the temperature dependence of G(Rc) inwater should be small. Ashbaugh and Pratt stated that[14]: ‘Classic SPT fails to describe this temperature depen-dence of G(Rc).’ I have used SPT Eqs. (6) and (7) to calcu-late Rc,max and G(Rc,max) at 25 �C and 100 �C forrw = 2.8 A and 3.2 A, keeping always the experimentalnumber density of water at the respective temperature.The numbers are listed in Table 2 and indicate that thetemperature dependence predicted by classic SPT, assum-ing that the effective size of water molecules is tempera-ture-independent, is very small, but in qualitativeagreement with the results of Ashbaugh and Pratt: the val-ues of both Rc,max and G(Rc,max) slightly decrease. Obvi-ously, in order to use something like Eq. (8) at 100 �C,one should know the fraction of water molecules notinvolved in H-bonds at that temperature.

In conclusion, I have tried to show that the ‘experimen-tal’ G(Rc) function of water at 25 �C can be reproduced in asatisfactory manner by means of classic SPT. In this respectit is important to recognize that I have not tried to best fitthe ‘experimental’ G(Rc) values using the effective diameterof water molecules as the fitting parameter. The two diam-eters assigned to water molecules are the right values corre-sponding to two distinct physical situations in which thewater molecules can exist, forming H-bonds or not formingH-bonds. The assumption that the fraction of water mole-cules not forming H-bonds at room temperature is 0.15appears to be reliable on the basis of available structuralinformation. The G(Rc) function calculated by means of

Eq. (8) shows how much of the ‘experimental’ G(Rc) curveof water can be accounted for by simple geometric packingfeatures. The remaining discrepancy should be a directmanifestation of the structural constraints imposed by H-bonds.

Acknowledgement

I thank Dr. B. Lee (Center for Cancer Research, NCI,NIH, Bethesda, MD) for reading an earlier draft of themanuscript.

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