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Chemical Physics Letters 446 (2007) 313–316

Cavity thermodynamics in the Gaussian model of particledensity fluctuations

Giuseppe Graziano *

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

Received 19 July 2007; in final form 21 August 2007Available online 25 August 2007

Abstract

The thermodynamics of cavity creation emerging from the Gaussian model of particle density fluctuations are investigated. The anal-ysis shows that the reversible work of cavity creation is dominated by entropy due to the exclusion of liquid particles from the cavityvolume. The enthalpy change is due to a structural reorganization of liquid particles that is distinct from the excluded volume effectand proves to be entirely compensated for by a corresponding entropy term. These results are in line with a previous general analysis[B. Lee, J. Chem. Phys. 83 (1985) 2421].� 2007 Elsevier B.V. All rights reserved.

1. Introduction

From the theoretical point of view the process of cavitycreation is an unavoidable and fundamental step for thesolubilization of a solute molecule in a liquid. Cavity crea-tion can be investigated solely by means of theoreticalapproaches or computer simulations in suitable models ofthe different liquids [1]. The reversible work associated withcavity creation is a large and positive quantity in all liquids[2–4]. There is a general consensus on the notion that thereversible work of cavity creation distinguishes water fromall the other common liquids and is the ultimate cause ofhydrophobicity, the low solubility of nonpolar compoundsin water [1–4].

Notwithstanding the great attention devoted in the lastyears to cavity creation, confusion still exists on cavitythermodynamics and its molecular origin [5]. Lee demon-strated that [6]: (a) the work of cavity creation is entropicin nature, due to the reduction in the size of the statisticalensemble of the liquid for the selection of solely the liquidconfigurations possessing the desired cavity (i.e., anexcluded volume effect); (b) the enthalpy change associated

0009-2614/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2007.08.063

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

with cavity creation is due to a structural reorganization ofliquid molecules, distinct from the excluded volume effect,and it is perfectly compensated for by a correspondingentropy contribution. In contrast, other scientists have sug-gested that the work of cavity creation should be governedby enthalpic terms, as recently written by van der Vegt andvan Gunsteren [7]: ‘cavities are formed by breaking sol-vent–solvent contacts’. For instance, in their pedagogicallyinsightful text [8], Dill and Bromberg have shown that, inorder to create a cavity in a liquid modelled as a lattice,molecules have to be removed by breaking intermolecularattractions. This picture, however, is not strictly correctbecause in actual liquids, that are far different from a lat-tice, no molecules have to be removed for cavity creation.In actual liquids the empty volume amounts on the averageto about 50% of the total volume [9,10], and cavities orig-inate from spontaneous equilibrium density fluctuations, asdirectly verified in computer simulations [3,11,12].

According to a general theorem of statistical mechanics[13,14], the reversible work of cavity creation is directlyrelated to the logarithm of the probability of finding noliquid particles in the volume corresponding to the desiredcavity, P(0;v). Pratt and colleagues have devised an infor-mation theory approach to obtain P(0;v), by calculatingthe probabilities of finding the centers of n particles

314 G. Graziano / Chemical Physics Letters 446 (2007) 313–316

P(n;v) inside a randomly positioned volume v of size corre-sponding to the liquid excluded volume of the desired cav-ity [15,16]. They showed that the use of a flat default modelin the information theory approach to determine the prob-abilities P(n;v) leads to discrete Gaussian distributions forthe P(n;v) themselves [17]. I would like to show that thethermodynamics of cavity creation originating from theGaussian model of particle density fluctuations is in linewith the general analysis performed by Lee.

2. Theoretical premise

The probability density of observing a specific configu-ration X of the liquid in the NPT ensemble, where X is avector describing the position of each of the N moleculesof the liquid, is given by [18]

qðXÞ ¼ exp½�HðXÞ=kT �=Z

exp½�HðXÞ=kT �dX ð1Þ

where H(X) = E(X) + P Æ V(X) is the enthalpy of the givenconfiguration, V(X) is the volume of the given configura-tion, P is the pressure of the liquid, k is the Boltzmann con-stant, and the denominator is the configurational partitionfunction. The probability of finding the centers of all the N

particles of the liquid excluded from the cavity volume v isobtained by summing over all the configurations possessingthe centers of all the N particles located in the volumeÆVæ � v [19]. In performing such a summation, it is impor-tant to recognize that the location of the cavity of excludedvolume v has to be fixed, but arbitrarily in the liquid vol-ume because the liquid density is uniform at equilibrium.In addition, note that, even though the total volume isnot strictly constant in the NPT ensemble, for a macro-scopic system, it is safe to consider that the total volumeV will assume, at equilibrium, values sharply close to theensemble average value ÆVæ. Thus one obtains:

P ð0; vÞ ¼ZhV i�v

qðX ÞdX ð2Þ

where the extrema of integration have the physical meaningclarified above. This rigorous definition of P(0;v) indicatesthat its evaluation corresponds to single out only a smallfraction (i.e., those possessing the desired cavity) of the to-tal liquid configurations [6]. This selection produces areduction in the size of the NPT ensemble that is entropicin nature, regardless of the interactions existing among li-quid particles (i.e., it is a geometric effect because one ob-tains a sub-ensemble of the original one) [6,20].Therefore, an approach aimed at evaluating directlyP(0;v), such as the assumption of Gaussian particle densityfluctuations, really accounts for the entropy decreasecaused by the excluded volume effect emphasized in Eq. (2).

3. Gaussian model

Following the results by Pratt and colleagues [15–17],the probability of finding the centers of exactly n particles

inside a randomly positioned volume v, when the liquidnumber density is q ” NAv/vm (i.e., NAv is the Avogadro’snumber and vm is the molar volume of the liquid), and pres-sure and temperature are held constant, is considered tofollow a Gaussian distribution

Pðn; vÞ ¼ ð2pr2nÞ�1=2 � expð�dn2=2r2

nÞ ð3Þ

where dn = n � Ænæ, Ænæ is the average number of particlecenters in the volume v, Ænæ = q Æ v, and r2

n ¼ hdn2i ¼hn2i � hni2 is the variance of the distribution, i.e., the meansquare fluctuation in the number of particle centers insidethe volume (the square of the standard deviation of theGaussian distribution). Since I am interested in the proba-bility of finding a cavity in the liquid, I have to know theprobability P(0;v) of finding zero particle centers (i.e.,n = 0) in the volume v:

Pð0; vÞ ¼ ð2pr2nÞ�1=2 � expð�hni2=2r2

nÞ

¼ ð2pr2nÞ�1=2 � expð�q2v2=2r2

nÞ ð4Þ

The probability of finding a cavity is related to equilib-rium particle density fluctuations, emphasizing that cavitycreation depends solely on the properties of the liquid itself[3–6]. An exact relationship of statistical mechanics statesthat the probability of observing a fluctuated configurationin a liquid is equal to the exponential of the reversible workrequired to produce the fluctuated configuration throughthe application of a constraint divided by kT [13,14]. Byapplying this theorem to equilibrium particle density fluc-tuations in an NPT statistical ensemble, the probabilityP(0; v) is given by

Pð0; vÞ ¼ expð�DGc=kT Þ ð5Þwhere DGc is the Gibbs energy change of cavity creation.By inserting Eq. (4) into Eq. (5), one obtains:

DGc ¼ ðkT=2Þ � lnð2pr2nÞ þ ðkT q2v2=2r2

nÞ ð6ÞEq. (6) indicates that DGc / 1=r2

n, and, since the isother-mal compressibility bT / r2

n [21], it means DGc � 1/bT. Inother words, the reversible work of cavity creation, mea-suring the entropy decrease associated with the excludedvolume effect, as indicated by Eq. (2), proves to be inverselyproportional to the isothermal compressibility of theliquid. This relationship, first derived by Pratt and col-leagues [15–17], does appear to be reliable because waterhas the smallest value of bT among all common liquids(at 25 �C, bT (in atm�1 105) = 4.58 for water, 9.80 for ben-zene, 11.55 for c-hexane, 16.27 for n-hexane, and 10.81 forcarbon tetrachloride [4]), and, in fact, it has the largest DGc

value for a given cavity size among all common liquids, asdemonstrated by computer simulation results [3,11,12].Since bT is a macroscopic thermodynamic quantity, it isnecessary to deepen the analysis in order to try to gain amicroscopic understanding of the difference between waterand the other liquids. According to its statistical mechani-cal definition [21], bT is a measure of the ensemble fluctua-tions in liquid particle density, bT ¼ vm � r2

n=hni2 � kT . The

G. Graziano / Chemical Physics Letters 446 (2007) 313–316 315

fact that water has the smallest bT value among all com-mon liquids is mainly because the molar volume of wateris the smallest among those of all common liquids: at25 �C, vm (in cm3 mol�1) = 18.07 for water, 89.41 for ben-zene, 108.75 for c-hexane, 131.62 for n-hexane, and 97.09for carbon tetrachloride [4]. The molecular origin of thisis because the effective size of water molecules is the small-est among those of all common liquids [2,4,9]; at 25 �C, theeffective hard sphere diameter is 2.80 A for water, 5.26 Afor benzene, 5.63 A for c-hexane, 5.92 A for n-hexane,and 5.37 A for carbon tetrachloride.

Direct application of fundamental relations of equilib-rium thermodynamics to Eq. (6) leads to the followingexpressions for the enthalpy and entropy changes associ-ated with cavity creation:

DH c ¼ �T 2½oðDGc=T Þ=oT �P¼ �ðkT 2=2Þ � fo½lnð2pr2

nÞ þ ðq2v2=r2nÞ�=oT gP

¼ �ðkT 2=2r2nÞ � f½1� ðq2v2=r2

nÞ� � ðor2n=oT ÞP

þ 2qv2 � ðoq=oT ÞPg ð7Þ

and

DSc ¼ �ðoDGc=oT ÞP¼ �ðk=2Þ � lnð2pr2

nÞ � ðkq2v2=2r2nÞ � ðkT=2r2

nÞ� f½1� ðq2v2=r2

nÞ� � ðor2n=oT ÞP þ 2qv2 � ðoq=oT ÞPg

ð8Þ

It is worth noting that, in performing the derivatives, Iconsidered r2

n a simple function of temperature (i.e., nota composite one), and the quantity v temperature indepen-dent. Eqs. (7) and (8) emphasize that: (a) the cavityenthalpy change is perfectly balanced by a correspondingterm in the cavity entropy change, and so there is no netenthalpic contribution to the reversible work of cavity cre-ation; (b) the cavity entropy change consists of a secondcontribution that is exactly given by �DGc/T, and accountsfor the excluded volume effect associated with cavity crea-tion in a liquid (i.e., a reduction in the size of the statisticalensemble of the liquid for the selection of the configura-tions possessing the desired cavity). In other words, Eqs.(6)–(8) confirm that the work of cavity creation is domi-nated by entropy, in line with the general analysis by Lee[6].

4. Discussion

These results emerge directly by assuming valid theGaussian model for the particle density fluctuations in aliquid [15–17]. In this respect, it is worth noting that Chan-dler demonstrated that the mean spherical approximationto the Ornstein–Zernike integral equations can be derivedon the assumption of Gaussian particle density fluctuations[22]. Moreover, the assumption that particle density fluctu-ations follow a Gaussian distribution is independent of thepotential energy existing among particles in the liquid. The

results of suitable computer simulations have shown that,for molecular-sized cavities, particle density fluctuationsobey Gaussian statistics in both hard sphere fluids [23],Lennard-Jones liquids [24], n-hexane and dimethyl sulfox-ide [25], and water [15–17]. Thus, the assumption of Gauss-ian statistics for particle density fluctuations should beconsidered a general and suitable ansatz.

Since in water r2n depends very little on temperature

(i.e., bT / r2n and the isothermal compressibility of water

stays almost constant over the 0–100 �C temperature range[26]), the derivative ðor2

n=oT ÞP should be a negligible quan-tity. On this basis, Eq. (7) can reliably be approximated by

DH c ffi �ðkT 2qv2=r2nÞ � ðoq=oT ÞP ¼ kT 2q2v2a=r2

n ð9Þwhere a = � (1/q) Æ (oq/oT)P is the thermal expansioncoefficient of the liquid. According to Eq. (9), DHc � a, inline with the relationship first derived by Pierotti in theframework of scaled particle theory [27]. In the other liq-uids the quantity r2

n depends on temperature, but the factor½1� ðq2v2=r2

nÞ� in Eq. (7) is not expected to be large, andEq. (9) should be a good approximation for all liquids.On this basis, and remembering that a is a measure ofthe ensemble average of the correlation between volumefluctuations and enthalpy fluctuations of the liquid [21], itshould be safe to state that DHc accounts for the structuralreorganization of liquid molecules associated with cavitycreation [2]. According to Eqs. (7) and (8), however, thelatter reorganization, which is distinct from the excludedvolume effect, is characterized by a complete enthalpy–en-tropy compensation [6,28].

Since a is very small for water at room temperature incomparison to organic liquids (at 25 �C, a (inK�1 103) = 0.257 for water, 1.240 for benzene, 1.214 forc-hexane, 1.407 for n-hexane, and 1.226 for carbon tetra-chloride [4]), DHc proves to be small in water and DGc isdominated by the excluded volume entropy contribution[28]. In organic liquids a is large at room temperature,and, for sufficiently great cavities, DHc proves to be largerthan DGc and does appear to dominate the cavity creationprocess [28]. However, the reversible work of cavity crea-tion is always associated with the excluded volume entropycontribution in all liquids, in view of the completeenthalpy–entropy compensation characterizing the struc-tural reorganization of liquid molecules due to cavity crea-tion [6,28].

The small value of a for water is a reflection of the plas-ticity of its 3D H-bonding network [2,29]: in water, at roomtemperature, a volume increase does not lead to a corre-lated increase in enthalpy, because water molecules are ableto avoid the loss of H-bonds due to their tetrahedral 2–2donor–acceptor functionalities. In other words, cavity cre-ation in water, at room temperature and atmospheric pres-sure, does not cause a net breaking of H-bonds and isdominated by the excluded volume effect exaggerated bythe small size of water molecules [2,28].

A final point: on the basis of Eqs. (8) and (9), also DSc

depends on the thermal expansion coefficient of the liquid.

316 G. Graziano / Chemical Physics Letters 446 (2007) 313–316

This result further connects the Gaussian model of particledensity fluctuations to the scaled particle theory, andshould be the basic origin of the ability of both approachesto reproduce the entropy convergence phenomenon in thehydration thermodynamics of noble gases, hydrocarbonsand n-alcohols [17,30,31].

In conclusion, I have shown that the thermodynamicfeatures of cavity creation obtained from the analysisof the Gaussian model of particle density fluctuationsare in line with the general results derived originally byLee with a direct and insightful statistical mechanicalprocedure.

References

[1] L.R. Pratt, Annu. Rev. Phys. Chem. 53 (2002) 409.[2] B. Lee, Biopolymers 24 (1985) 813.[3] A. Pohorille, L.R. Pratt, J. Am. Chem. Soc. 112 (1990) 5066.[4] G. Graziano, J. Phys. Chem. B 106 (2002) 7713.[5] V.V. Alexandrovsky, M.V. Basilevsky, I.V. Leontyev, M.A. Mazo,

V.B. Sulimov, J. Phys. Chem. B 108 (2004) 15830.[6] B. Lee, J. Chem. Phys. 83 (1985) 2421.[7] N.F.A. van der Vegt, W.F. van Gunsteren, J. Phys. Chem. B 108

(2004) 1056.[8] K.A. Dill, S. Bromberg, Molecular Driving Forces, Garland Science,

New York, 2003, p. 254.[9] E. Wilhelm, R. Battino, J. Chem. Phys. 55 (1971) 4012.

[10] K.E.S. Tang, V.A. Bloomfield, Biophys. J. 79 (2000) 2222.[11] B. Madan, B. Lee, Biophys. Chem. 51 (1994) 279.[12] I. Tomas-Oliveira, S.J. Wodak, J. Chem. Phys. 111 (1999) 8576.[13] R.C. Tolman, The Principles of Statistical Mechanics, Oxford

university Press, London, 1938, p. 636.[14] H. Reiss, Adv. Chem. Phys. 9 (1966) 1.[15] G. Hummer, S. Garde, A.E. Garcia, A. Pohorille, L.R. Pratt, Proc.

Natl. Acad. Sci. USA 93 (1996) 8951.[16] G. Hummer, S. Garde, A.E. Garcia, M.E. Paulaitis, L.R. Pratt, J.

Phys. Chem. B 102 (1998) 10469.[17] S. Garde, G. Hummer, A.E. Garcia, M.E. Paulaitis, L.R. Pratt, Phys.

Rev. Lett. 77 (1996) 4966.[18] G. Graziano, Chem. Phys. Lett. 429 (2006) 114.[19] A. Ben-Naim, Water and Aqueous Solutions. An Introduction to a

Molecular Theory, Plenum, New York, 1974, p. 114.[20] G. Graziano, J. Chem. Phys. 121 (2004) 1878.[21] T.L. Hill, An Introduction to Statistical Thermodynamics, Addison-

Wesley, Reading, 1960.[22] D. Chandler, Phys. Rev. E 48 (1993) 2898.[23] G.E. Crooks, D. Chandler, Phys. Rev. E 56 (1997) 4217.[24] D.M. Huang, D. Chandler, Phys. Rev. E 61 (2000) 1501.[25] C. Peter, N.F.A. van der Vegt, J. Phys. Chem. B 111 (2007) 7836.[26] G.S. Kell, J. Chem. Eng. Data 20 (1975) 97.[27] R.A. Pierotti, Chem. Rev. 76 (1976) 717.[28] G. Graziano, J. Phys. Chem. B 110 (2006) 11421.[29] R.D. Mountain, D. Thirumalai, Proc. Natl. Acad. Sci. USA 95 (1998)

8436.[30] G. Graziano, B. Lee, Biophys. Chem. 105 (2003) 241.[31] G. Graziano, J. Phys. Chem. B 109 (2005) 12160.