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Chemical Physics Letters 442 (2007) 307–310

Cavity thermodynamics and surface tension of water

Giuseppe Graziano *

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

Received 19 April 2007; in final form 24 May 2007Available online 2 June 2007

Abstract

It is widely recognized that the air–water surface tension plays a role for the solubility of nonpolar gases in water. In particular, thework of cavity creation is considered to be governed by the large air–water surface tension. This relationship is investigated by calculatingthe cavity thermodynamics at room temperature on the basis of a direct proportionality between the work of cavity creation and the air–water surface tension. The obtained cavity entropy change is always positive regardless of the cavity size, in marked contrast with wellestablished results from computer simulations and theoretical approaches.� 2007 Elsevier B.V. All rights reserved.

1. Introduction

By a long time it is recognized that the solubility of gasesinto liquids is related to the liquid–vapour surface tensionof the same liquids [1–3]. This empirical finding has beeninterpreted as a manifestation of the dominance of surfacework in the Gibbs energy cost to create a spherical cavity ina liquid, DGc [4]. Scaled particle theory, SPT, providedstrong support to this interpretation [5,6]. Cavity creationis an unavoidable step of solvation because liquids area condensed state of the matter and a suitable void is nec-essary to insert a solute molecule. Recently, Lum et al.developed a theoretical treatment to calculate the work ofcavity creation over a very large cavity size range [7]. Theypointed out that the DGc/ASA function, where ASA ¼4pR2

c is the accessible surface area of the spherical cavity[8], and Rc is the cavity radius (i.e., the radius of the spher-ical region from which the centers of liquid molecules areexcluded), is directly proportional to Rc for small cavitiesand independent of Rc for large cavities. Such crossoverin the DGc/ASA function occurs at 5 A 6 Rc 6 10 A, andindicates that, for sufficiently large cavities, DGc is domi-

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doi:10.1016/j.cplett.2007.05.106

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

nated by the surface work [7]. In addition, Huang et al.[9] showed that the trend of the DGc/ASA function versusRc, calculated by means of umbrella sampling of MonteCarlo simulations in both SPC [10], and SPC/E [11] watermodels at 298 K and 1 atm, can be well reproduced bymeans of the relationship:

DGcðcÞ=ASA ¼ c � ½1� ð2d=RcÞ� ð1Þwhere c is the air–water surface tension, and d is the so-called Tolman length that accounts for curvature effectson the surface tension [12]. It resulted that: (a) the d estimateproved to be smaller than 1 A in both water models; (b) thec estimate, obtained by fitting the DGc/ASA values over the6 A 6 Rc 6 10.25 A range, agreed with that directly calcu-lated by means of suitable computer simulations [9]. There-fore, Chandler and co-workers concluded that the processof cavity creation for Rc P 6 A can be treated as the forma-tion of an air–water interface. On this basis, Chandler hasstrongly claimed the fundamental role played by air–watersurface tension in solvation phenomena [13,14]. Subse-quently, Rajamani et al. [15], RTG, confirmed that Eq.(1) is able to reproduce the DGc/ASA function determinedby means of a perturbation theory treatment of moleculardynamics simulations in SPC/E water at room temperatureand 1 atm for Rc P 3 A. Moreover, Ashbaugh and Pratt[16], A&P, used a revised/modified SPT approach, that

0 5 10 15 20

0

200

400

600

( J m

ol-1

angs

trom

-2)

Rc (angstrom)

Fig. 1. Plot of the functions DGc (full curve), DHc (dashed curve), andTDSc (dotted curve) normalized with respect to the cavity accessiblesurface area 4pR2

c versus the cavity radius Rc, calculated by means of Eqs.(1)–(3) at 25 �C and 1 atm. The horizontal line represents the experimentalair–water surface tension at 25 �C.

308 G. Graziano / Chemical Physics Letters 442 (2007) 307–310

incorporates Eq. (1) with the experimental air–water sur-face tension and d < 1 A, to calculate DGc/ASA values inwater for Rc P 3.5 A at room temperature.

It appears that Eq. (1) works well not only for large cav-ities, as originally claimed by Chandler, but also for small,molecular-sized cavities (note that the cavity necessary tohost methane in water should have Rc @ 3.3 A, and thatnecessary to host benzene in water should haveRc @ 4.1 A). On this basis it should be important to verifywhat type of cavity thermodynamics in water at room tem-perature emerges from the assumption that Eq. (1) is valid.This is the aim of the present analysis with the hope to shedlight on the relationship between the process of cavity cre-ation and the air–water surface tension.

2. Calculation procedure

I have used Eq. (1) to calculate DGc(c)/ASA at 25 �C inwater with the experimental air–water surface tension,c = 433 J A�2 mol�1 [17], and fixing the Tolman lengthd = 0.65 A (this d value is intermediate between that origi-nally obtained by Stillinger in classic/original SPT [18],�0.5 A, and those obtained by Chandler and co-workers[9], 0.76 A for SPC water, and 0.90 A for SPC/E water).It is evident that Eq. (1) is meaningless for Rc 6 2d = 1.3 A.For smaller cavities, I have calculated DGc/ASA using thefollowing exact relationship [5,6]:

DGc=ASA ¼ �ðRT =4pR2cÞ � ln½1� ð4=3Þp � qw � R3

c � ð2Þwhere qw = NAv/v(H2O) is the number density of water,and v(H2O) its molar volume. I have fixed the effective hardsphere diameter of water molecules r(H2O) = 2.9 A, be-cause the latter value, when inserted in the SPT formulafor c, reproduces the experimental air–water surface ten-sion at 25 �C [6]; r(H2O) is assumed to be temperature-independent. Eq. (2) is exact up to Rc = r(H2O)/2 =1.45 A; for larger cavities, I have assumed that Eq. (1)holds (i.e., the validity of Eq. (1) is extended down to verysmall cavities).

For the calculation of DSc(c)/ASA for Rc P 1.45 A, byperforming the temperature derivative of Eq. (1) oneobtains:

DScðcÞ=ASA ¼ �ðdc=dT Þ � ½1� ð2d=RcÞ� ð3Þwhere (dc/dT) = �0.933 J A�2 K�1 mol�1, on the basis ofexperimental data [17]. Clearly, to arrive at Eq. (3), I haveassumed that d is a temperature-independent quantity, inline with the results obtained by Stillinger [18] withinclassic/original SPT. A&P, with their revised/modifiedSPT approach [16], found that d is strongly tempera-ture dependent, decreasing on increasing temperature andchanging sign near 80 �C. In this respect, I would like topoint out that a strong temperature dependence of d doesappear to be unreliable in view of the geometric interpreta-tion attached to the Tolman length. For 0 6 Rc 6 1.45 A,DSc/ASA values are directly obtained by the temperaturederivative of Eq. (2), and so are proportional to the ther-

mal expansion coefficient of water. Finally, DHc(c)/ASA = [DGc(c) + TDSc(c)]/ASA.

In addition, I have used classic/original SPT to calculateDGc(SPT)/ASA, DHc(SPT)/ASA and TDSc(SPT)/ASA at25 �C and 1 atm [19–21]. In performing SPT calculations,I have used the experimental values of density and thermalexpansion coefficient of water at 25 �C and 1 atm [22], andr(H2O) = 2.9 A, assumed to be temperature-independent.Note that for 0 6 Rc 6 1.45 A, I have always used Eq. (2)and its temperature derivative which are exact.

3. Results and discussion

The DGc(c)/ASA, DHc(c)/ASA and TDSc(c)/ASA func-tions, calculated by means of Eqs. (1)–(3), are plotted overthe 0 6 Rc 6 20 A cavity size range in Fig. 1. It is evidentthe occurrence of a crossover in the DGc(c)/ASA values:linear dependence on Rc for small cavity radii and nodependence for large cavity radii, in line with the originalresults by Lum et al. [7]. Clearly, while there is a satisfac-tory continuity between the DGc/ASA values calculatedby means of Eq. (1) and those calculated by means ofEq. (2), such a continuity does not occur for DHc(c)/ASAand TDSc(c)/ASA. The fundamental finding is that theDSc(c) values are positive for Rc P 1.45 A, originatingfrom the negative temperature dependence of the experi-mental air–water surface tension. Over the whole cavitysize range where I have assumed that Eq. (1) is valid, theDSc(c) function is positive and favours the process of cavitycreation. Cavity thermodynamics are dominated by DHc(c)values that are positive and larger than those of DGc(c)(i.e., for instance, at Rc = 6 A, DGc(c)/ASA = 339 J A�2

mol�1, DHc(c)/ASA = 557 J A�2 mol�1, and TDSc(c)/ASA = 218 J A�2 mol�1).

In order to establish the reliability of such numbers, it isimportant to perform a closer scrutiny of the computa-tional results obtained by other scientists, because the ther-

0 5 10 15 20

-200

0

200

400

( J m

ol-1

angs

trom

-2)

Rc (angstrom)

Fig. 2. Plot of the functions DGc (full curve), DHc (dashed curve), andTDSc (dotted curve) normalized with respect to the cavity accessiblesurface area 4pR2

c versus the cavity radius Rc, calculated by means ofclassic/original SPT at 25 �C and 1 atm. The horizontal line represents theexperimental air–water surface tension at 25 �C. SPT calculations wereperformed using the experimental density of water at 25 �C andr(H2O) = 2.9 A.

0 5 10 15 20

-200

0

200

400

600

( J

mol

-1an

gstr

om-2

)

Rc (angstrom)

Fig. 3. Comparison between the functions DGc (full curve), DHc (dashedcurve), and TDSc (dotted curve) normalized with respect to the cavityaccessible surface area 4pR2

c of Fig. 1 (thick lines), and those of Fig. 2 (thinlines). The horizontal line represents the experimental air–water surfacetension at 25 �C.

G. Graziano / Chemical Physics Letters 442 (2007) 307–310 309

modynamic functions for cavity creation are not directlyaccessible to experimental determination. RTG [15], bynumerical differentiation of the DGc values determinedfrom molecular dynamics simulations in the SPC/E watermodel at different temperatures, obtained that DSc is a neg-ative quantity increasing in magnitude with cavity size upto Rc @ 7 A, at 300 K and 1 atm (see their Fig. 4, and notethat larger cavities were not studied). Therefore, the RTGresults indicate that DSc dominates the DGc values at roomtemperature in water. Huang and Chandler [23], H&C, per-forming calculations with the Lum–Chandler–Weeks the-ory [7], that uses the water’s density, oxygen–oxygenradial distribution function and surface tension as inputdata, found that DSc is a large and negative quantity upto Rc @ 10 A at room temperature and 1 atm in water (seetheir Fig. 2). A&P [16] determined DGc in water at 300 Kover a large cavity size range by means of direct computersimulations up to Rc @ 4 A, and of extrapolation to largercavity radii according to a revised/modified SPT approachthat incorporates Eq. (1) with the experimental air–watersurface tension. The DSc/ASA function calculated byA&P starts negative, passes through zero at Rc @ 6 A andbecomes large positive on further increasing the cavityradius (see their Fig. 11, that is qualitatively close to myFig. 1). The positive DSc/ASA values reported by A&Phave not been obtained directly from computer simula-tions, but are a consequence of the revised/modified SPTapproach selected to perform the extrapolation up to verylarge cavities.

By taking together the results of RTG, H&C and A&P,one can conclude that, in water at room temperature and1 atm, DSc/ASA is expected to be a negative quantity upto Rc @ 7–10 A, in marked contrast with the trend shownin Fig. 1 by the DSc(c)/ASA function. This means thatthe assumptions DGc � c and DSc � �(dc/dT) lead to reli-able DGc values, but to unreliable values for DSc. The abil-ity of Eq. (1) to reproduce DGc/ASA values obtained fromcomputer simulations is a necessary but not sufficient con-dition to state that Eq. (1) is a right description of cavitythermodynamics in water.

In order to try to gain a correct perspective, I have usedclassic/original SPT to calculate, at 25 �C and 1 atm, theDGc(SPT)/ASA, DHc(SPT)/ASA and TDSc(SPT)/ASAfunctions shown in Fig. 2 over the 0 6 Rc 6 20 A cavitysize range. In performing the SPT calculations, (a) I haveused the experimental values of density and thermal expan-sion coefficient of water at 25 �C and 1 atm [22]; (b) I havefixed r(H2O) = 2.9 A, a value that, inserted in the SPTexpression for c, reproduces the experimental air–watersurface tension at 25 �C [6], and considered it to be temper-ature-independent. While the Rc-dependence of DGc(SPT)/ASA is closely similar to that of DGc(c)/ASA, the Rc-dependence of TDSc(SPT)/ASA and DHc(SPT)/ASA doesnot correspond to that of TDSc(c)/ASA and DHc(c)/ASA,as clarified by their comparison in Fig. 3. In particular,DSc(SPT) is a large and negative quantity over the wholeinvestigated cavity size range, and it dominates the cavity

thermodynamics because DHc(SPT) is a small positivequantity (i.e., for instance, at Rc = 6 A, DGc(SPT)/ASA= 295 J A�2 mol�1, DHc(SPT)/ASA = 57 J A�2 mol�1, andTDSc(SPT)/ASA = �238 J A�2 mol�1). There is a strongqualitative agreement between the DSc(SPT) functionand the results by RTG [15] (they computed DSc up toRc @ 7 A at 300 K and 1 atm), by H&C [23] for cavitiesup to Rc @ 10 A at room temperature and 1 atm, and alsoby A&P [16] for cavities up to Rc 6 6 A. It is worth notingthat a different choice of the temperature-independentr(H2O) value leads to quantitative differences in theDGc(SPT)/ASA, DHc(SPT)/ASA and TDSc(SPT)/ASAfunctions, but their qualitative features are absolutely notaffected [20] (i.e., DSc(SPT) is always negative and increasesin magnitude with cavity size at room temperature).

310 G. Graziano / Chemical Physics Letters 442 (2007) 307–310

For nonpolar solutes at room temperature, the hydra-tion entropy change is large and negative, and the hydra-tion enthalpy change is negative [24]. The creation of acavity in water can be assimilated to the hydration of a cor-responding hard sphere. Classic/original SPT calculationspredict large and negative entropy changes, and smalland positive enthalpy changes for cavity creation in waterat room temperature. The qualitative agreement with thehydration entropy of nonpolar solutes is due to the domi-nance of the excluded volume effect [20,24,25], whereas thequalitative disagreement with the hydration enthalpy ofnonpolar solutes is simply due to the omission in classic/original SPT of the dispersion attractive interactions exist-ing between a nonpolar solute and surrounding water mol-ecules [24].

From the theoretical point of view, Lee has elegantlydemonstrated that DGc is a purely entropic function inall liquids [26], accounting for the excluded volume effectassociated with cavity creation. I have provided furthersupport to the validity of Lee’s analysis and conclusion,emphasizing the complete enthalpy–entropy compensa-tion characterizing the process of purely structural reor-ganization of liquid molecules associated with cavitycreation [20,27,28]. These features are because cavity cre-ation corresponds to the selection of a sub-ensemble ofmolecular configurations of the liquid (i.e., those contain-ing the desired cavity) from the whole ensemble ofmolecular configurations of the same liquid. Classic/ori-ginal SPT was developed for hard sphere fluids, and soits DGc formula accounts in a qualitatively correct man-ner for the excluded volume effect [5,6,21] (i.e., the pack-ing problem in a dense fluid). The extension of classic/original SPT to real liquids, by calculating DHc andDSc as appropriate temperature derivatives of DGc, withthe use of experimental data for the liquid density andthermal expansion coefficient, has led to reliable resultsalso in the case of water [19,20,24], as confirmed bythe present analysis.

The assumption that Eq. (1) is valid for molecular-sizedcavities leads to positive DSc values at 25 �C that are incontrast with the results of direct computer simulationsand experimental data for the hydration of small nonpolarsolutes. The air–water surface tension is strictly related tothe strength of water–water H-bonds and their temperature

dependence, and it should not be able to account for theexcluded volume effect associated with cavity creation, thatis the basic origin of the negative DSc values in water at25 �C. This means that Eq. (1) cannot be considered tobe right to describe the thermodynamics of cavity creationin water at room temperature for molecular-sized cavities.Finally, it is worth noting that the air–water surface ten-sion, being a macroscopic thermodynamic property, can-not provide any molecular insight into the physical originof cavity thermodynamics in water.

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