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Chemical Physics Letters 440 (2007) 221–223

A purely geometric derivation of the scaled particle theory formulafor the work of cavity creation in a liquid

Giuseppe Graziano *

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

Received 7 March 2007; in final form 12 April 2007Available online 19 April 2007

Abstract

A very simple statistical geometric approach is devised that allows the derivation of a formula for the work to create a spherical cavityin a liquid consisting of spherical molecules in the small cavity size limit. This formula exactly corresponds to the general expression forthe work of cavity creation provided by scaled particle theory. The origin of the success of the present statistical geometric approachneeds further insight.� 2007 Elsevier B.V. All rights reserved.

1. Introduction

Forty years ago, Reiss [1] noted that the major problemsin the theory of liquids (at least for simple liquids contain-ing spherical molecules) appear to be geometric (i.e., thepacking of molecules in the given volume). Accepting thispoint of view, I would like to show that a simple statisticalgeometric approach [2] can be devised to derive a formulafor the work of cavity creation in a liquid consisting ofhard spheres or spherical molecules possessing a definedhard sphere diameter. This formula corresponds to thatderived by Reiss and colleagues in the scaled particle the-ory [3,4], following an entirely different route.

The present statistical geometric approach provides asimple relationship for the probability of finding no parti-cles in a given spherical region of a liquid (i.e., a cavity).It distinguishes the empty volume of a liquid, which is ageneral and constant property of the liquid when the tem-perature and pressure are fixed, from the volume availableto create–insert a cavity of rc diameter, which is a verysmall fraction of the empty volume [5]. More importantly,it is grounded on the notion that the creation of a cavity ofrc diameter leads to the occurrence of a region that

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doi:10.1016/j.cplett.2007.04.048

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

becomes inaccessible to the centre of liquid moleculesbecause the latter can only at most touch the cavity surface[1,2]. This inaccessible volume is the fundamental geomet-ric factor to arrive at a formula for the work of cavitycreation.

2. Statistical geometric approach

According to a general theorem of statistical mechanics[6], the work of cavity creation is related to the probabilityof finding no solvent molecules in the desired sphericalregion of rc diameter (note that the location of the cavityhas to be fixed, but arbitrarily in the liquid volume becausethe liquid density is uniform at equilibrium) [7]:

W ðrcÞ ¼ �RT � ln p0ðrcÞ; ð1Þwhere p0(rc) can be expressed as the ratio between theensemble average volume available to insert such a cavityand the total volume of the liquid [7–10], so that

W ðrcÞ ¼ �RT � ln hV availðrcÞiV tot

� �: ð2Þ

It is fundamental to recognize that ÆVavail(rc)æ does not cor-respond to the empty volume of the liquid [5]: Vempty is alarge fraction of the total volume (i.e., 50–60% of the total),and is a general and constant property of a given liquid

222 G. Graziano / Chemical Physics Letters 440 (2007) 221–223

when the temperature and pressure are held constant;ÆVavail(rc)æ is a statistical mechanical term that measuresonly the void volume whose dimensions allow the occur-rence–insertion of a cavity of rc diameter; ÆVavail(rc)æ is nota constant quantity for a given liquid, but it depends onthe selected rc value. By inserting Vempty, Eq. (2) becomes

W ðrcÞ ¼ �RT � ln V empty

V tot

� �� RT � ln hV availðrcÞi

V empty

� �: ð3Þ

The first ratio in Eq. (3) can be transformed to

V empty

V tot

¼ V tot � V occupied

V tot

� �¼ 1� V occupied

V tot

� �¼ 1� n; ð4Þ

where n is the volume packing density of the liquid, mea-suring the fraction of occupied volume; for a liquid consti-tuted by hard spheres or spherical molecules, n ¼ pr3

1=6v1,where r1 is the diameter of the hard spheres or sphericalmolecules of the liquid; and v1 is the molar volume of theliquid, assumed to be the total volume [1–4].

The second ratio in Eq. (3) can be transformed by consid-ering that the ensemble average volume available for theoccurrence–insertion of a cavity having a rc diameter is sim-ply given by the total volume minus the unavailable volume

hV availðrcÞi ¼ V tot � hV unavailðrcÞi; ð5Þwhere ÆVunavail(rc)æ is an ensemble average quantity itself,and so the question is: how is it possible to quantifyÆVunavail(rc)æ in a liquid?

I have thought that, in the small cavity size limit, a sim-ple geometric estimate of ÆVunavail(rc)æ can be obtained bylooking at the existence of a single spherical cavity of rc

diameter in a configuration of the liquid. For the existenceof such a cavity, the spherical shell between the sphere of(rc + r1) diameter and that of rc diameter has to be devoidof liquid molecule centres (i.e., the liquid molecule centrescannot penetrate the spherical shell of thickness rc/2 to(rc + r1)/2, if the cavity must exist). In a first approxima-tion (i.e., an ensemble average quantity is determined bylooking at a single liquid configuration containing thedesired cavity), this spherical shell can be considered tobe an estimate of ÆVunavail(rc)æ:

hV unavailðrcÞi ¼pðrc þ r1Þ3

6

" #� pr3

c

6

� �

¼ pr2cr1

2

� �þ prcr2

1

2

� �þ pr3

1

6

� �: ð6Þ

The last term in Eq. (6) is the volume occupied by the liquidspherical molecules, and the other two terms represent theinaccessible volume (i.e., volume inaccessible to liquid mol-ecule centres for the presence of the cavity having a rc

diameter). As a consequence, the second ratio in Eq. (3)can be rearranged to

hV availðrcÞiV empty

¼ V tot � V occupied � V inaccessðrcÞV tot � V occupied

� �

¼ 1� V inaccessðrcÞV empty

� �: ð7Þ

By inserting Eqs. (4) and (7) into Eq. (3), the expression ofW(rc) becomes

W ðrcÞ ¼ �RT � lnð1� nÞ � RT � ln 1� V inaccessðrcÞV empty

� �� �:

ð8ÞBy using the MacLaurin power series expansion of thefunction ln(1 � j), valid for j ” (Vinaccess(rc)/Vempty)� 1,truncated at the quadratic term, one obtains

W ðrcÞ ffi �RT � lnð1� nÞ þ RT

� V inaccess

V empty

� �þ 1

2

V inaccess

V empty

� �2" #

: ð9Þ

By remembering that the first two terms of Eq. (6) repre-sent Vinaccess(rc), the ratio Vinaccess/Vempty is given by (i.e.,Vempty = v1(1 � n)):

V inaccess

V empty

¼ pr2cr1

2v1ð1� nÞ

� �þ prcr2

1

2v1ð1� nÞ

� �ð10Þ

and

1

2

V inaccess

V empty

� �2

¼ 1

2

pr2cr1

2v1ð1� nÞ

� �þ prcr2

1

2v1ð1� nÞ

� �� �2

: ð11Þ

By retaining only the term containing r2c (note that this

constraint is related to the validity of the conditionVinaccess(rc)� Vempty used to pass from Eq. (8) to Eq.(9), by expanding in power series the logarithmic term),Eq. (11) becomes

1

2

V inaccess

V empty

� �2

ffi p2r2cr

41

8v21ð1� nÞ2

: ð12Þ

According to this geometric approach, the expression ofW(rc) is

W ðrcÞ ¼ �RT � lnð1� nÞ þ RT

� prcr21

2v1ð1� nÞ

� �þ pr2

cr1

2v1ð1� nÞ

� �þ p2r2

cr41

8v21ð1� nÞ2

" #( ):

ð13ÞSince the volume packing density of the liquid n ¼ pr3

1=6v1,it is simple to show that the expression in curly bracketscan be rearranged to give

W ðrcÞ ¼ �RT � lnð1� nÞ þ RT � 3nð1� nÞ

� �� rc

r1

� ��

þ 3n2ð1� nÞ

� �� 3n

1� n

� �þ 2

� �� rc

r1

� �2): ð14Þ

This is exactly the expression of the work of cavity creationderived by means of scaled particle theory, SPT, by Reissand colleagues at constant volume and temperature [3,4].

In the condition of constant pressure and temperature,upon insertion of a cavity of rc diameter, the total volumeof the liquid increases by a quantity corresponding to thevolume of the inserted cavity. The latter volume multiplied

G. Graziano / Chemical Physics Letters 440 (2007) 221–223 223

by the pressure of the hard sphere fluid (or actual liquid)gives a further contribution that should be added to theexpression of Eqs. (13) and (14) to obtain the Gibbs energychange for cavity creation [1,3,4].

A final point merits attention. One of the two anony-mous reviewers has provided a slightly different geometricderivation of Eq. (14), that does appear to be shorter. Toa first approximation, valid at low fluid densities, the inser-tion probability p0(rc) can be related to the excluded vol-ume due to the existing particles by writing

p0ðrcÞ ¼V tot � N 1V exðrc; r1Þ

V tot

; ð15Þ

where Vex(rc, r1) is the excluded volume of two sphereswith diameters rc and r1, and N1 is the number of existingspheres in the volume Vtot. Since, Vex(rc, r1) = p(rc + r1)3/6, and using the definition of volume packing density, Eq.(15) becomes

p0ðrcÞ ¼ ð1� nÞ � 1� 3n1� n

� �� rc

r1

� �� 3n

1� n

� ��

� rc

r1

� �2

� n1� n

� �� rc

r1

� �3): ð16Þ

By inserting the latter into Eq. (1), and expanding the log-arithm in powers of (rc/r1) up to quadratic order, one ob-tains directly Eq. (14). Clearly, the approximation to writedown Eq. (6) corresponds to that to write down Eq. (15).

3. Discussion

I have tried to develop a purely geometric approach toarrive at a formula for the work of cavity creation. By con-sidering the volume unavailable to hard spheres for thepresence of a spherical cavity, and by entirely neglectingcorrelation effects among the positions occupied by thehard spheres or liquid molecules, I have obtained the sameformula derived by Reiss and colleagues in scaled particle

theory [1–4]. The present derivation works when the cavitydiameter rc is much smaller than the diameter of liquidmolecules r1 (i.e., Vinaccess(rc)� Vempty), because the possi-ble overlap between the inaccessible volume of differentcavities in the liquid, a contribution that should becomemore important on increasing the cavity size, has not beentaken into account. This supports the choice of neglectingcubic terms in rc in order to arrive at an expression forthe work of cavity creation.

Unfortunately, I am not able to offer more insight on therightness, even in approximation, of Eq. (6) that is the cor-nerstone of the present geometric approach. In any case, Ithink that, since the SPT formula is usually used when thecavity diameter rc is larger than the diameter of liquid mol-ecules r1 [4,11,12], it is interesting that the exact SPT for-mula for the work of cavity creation can be derived inthis simple geometric way for the small cavity size limit.

Acknowledgements

I would like to express my gratitude to the two anony-mous reviewers for the insightful and useful suggestions.

References

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University Press, London, 1938, p. 636.[7] B. Lee, J. Chem. Phys. 83 (1985) 2421.[8] A. Pohorille, L.R. Pratt, J. Am. Chem. Soc. 112 (1990) 5066.[9] B. Madan, B. Lee, Biophys. Chem. 51 (1994) 279.

[10] G. Hummer, S. Garde, A.E. Garcia, A. Pohorille, L.R. Pratt, Proc.Natl. Acad. Sci. USA 93 (1996) 8951.

[11] G. Graziano, J. Phys. Chem. B 109 (2005) 12160.[12] G. Graziano, J. Phys. Chem. B 110 (2006) 11421.