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Chemical Physics Letters 452 (2008) 259–263

Hydrophobicity in modified water models

Giuseppe Graziano *

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

Received 3 December 2007; in final form 2 January 2008Available online 8 January 2008

Abstract

Molecular level explanation of the poor solubility in water of nonpolar species is still a debated topic. Recently, Lynden-Bell andHead-Gordon [R.M. Lynden-Bell, T. Head-Gordon, Mol. Phys. 104 (2006) 3593] have compared in detail, at room temperature andatmospheric pressure, the hydrophobicity of suitably modified water models with that of the realistic SPC/E water model. It is shownthat, by using the effective size of liquid molecules and the experimental density of water, scaled particle theory calculations reproducesatisfactorily the results of Lynden-Bell and Head-Gordon. Hydrophobicity is mainly determined by a trade-off between the effect of thesmall size of water molecules and the effect of the low value of its volume packing density.� 2008 Elsevier B.V. All rights reserved.

1. Introduction

Poor solubility of nonpolar molecules in water, hydro-phobicity, is a well known phenomenon, but the molecularfeatures determining this poor solubility are still a contro-versial subject [1–3]. In order to single out these molecularfeatures, some researches (a) have constructed water mod-els in which one or some of its molecular properties haveselectively been modified; (b) have tested their hydropho-bicity by calculating the work of cavity creation, the cavitysize distribution function, and the cavity contact correla-tion function [4–6]. Recently, Lynden-Bell and Head-Gor-don, LBHG [7] have thoroughly compared such functionsbetween SPC/E water [8], and two families of modifiedwater models. The family of hybrid models has intermolec-ular potentials which are hybrids between SPC/E waterand a pure Lennard-Jones liquid [9]; the family of bentmodels has a reduced bond angle [10]. I will focus my atten-tion on the hybrid model H30, in which the H-bondstrength amounts to 69% of its value in SPC/E water(i.e., the energy of a H-bond amounts to �37.1 kJ mol�1

in SPC/E water and �25.5 kJ mol�1 in the H30 model at

0009-2614/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2008.01.005

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

25 �C; see Table 2 in [7]); and on the bent model B60, inwhich the bond angle is 60� [7]. The modified water modelswere constructed to possess the same number density ofSPC/E water at 25 �C and 1 atm (that practically corre-sponds to the experimental one), and the same moleculardipole moment, by using the following van der Waalsdiameters for the oxygen atom: 3.165 A for both SPC/Eand H30, 2.92 A for B60 [7].

By means of molecular dynamics simulations, LBHGdetermined the oxygen-oxygen radial distribution function,OO-rdf, of the various water models [7]; they are shown inFig. 1. Close inspection of Fig. 1 indicates that: (a) the firstpeak maximum is located at 2.75 A for SPC/E water,2.95 A for the H30 model, and 2.55 A for the B60 model;(b) the first peak is sharper for SPC/E water with respectto that of the modified water models; (c) the second peakmaximum is located at 4.5 A for SPC/E water, 6 A forthe H30 model, and 5 A for the B60 model. These findingsmean that: (a) the effective size of SPC/E molecules is�2.8 A, markedly smaller than the van der Waals diameter,3.165 A, assigned to the oxygen atom in the SPC/E watermodel, due to the bunching up effect of H-bonds (the latter,due to their strength, are able to bunch up water moleculeswell below their van der Waals size) [4,11,12]; (b) bydecreasing the H-bond strength, the H30 model resemblesa pure Lennard-Jones liquid (i.e., the second peak in the

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

Hybrid - H30

Bent - B60

SPC/E

OO

- rd

f

r (angstrom)

Fig. 1. Oxygen–oxygen radial distribution functions of the SPC/E watermodel, of the bent B60 model, and of the hybrid H30 model at 25 �C and1 atm, calculated by Lynden-Bell and Head-Gordon using moleculardynamics computer simulations [7].

260 G. Graziano / Chemical Physics Letters 452 (2008) 259–263

OO-rdf occurs at twice the distance of the first peak), andthe effective size of H30 molecules is �3.0 A, intermediatebetween that of SPC/E molecules, 2.8 A, and the van derWaals diameter of the oxygen atom, 3.165 A; (c) bydecreasing the bond angle to 60�, the B60 model does notpossess a tetrahedral network, but it shows the tendencyto form chains, because both hydrogen atoms of a B60molecule can make H-bonds to the same neighbouringmolecule; the effective size of B60 molecules is �2.6 A,markedly smaller than the van der Waals diameter assignedto the oxygen atom in the B60 model, 2.92 A, due to thebunching up effect of H-bonds (i.e., the energy of a H-bondamounts to �31.9 kJ mol�1 in the B60 model at room tem-perature, not far from the value, �37.1 kJ mol�1, in SPC/Ewater; see Table 2 in [7]). The detected bunching up indi-cates that, while in a Lennard-Jones liquid the van derWaals diameter corresponds to the effective diameter ofmolecules, in liquids where strong electrostatic interactions,such as H-bonds, are in action, the effective moleculardiameter is smaller than the van der Waals one.

LBHG found that: (a) it is more costly to create a cavityin the H30 model than in SPC/E water; (b) it is less costlyto create a cavity in the B60 model than in SPC/E water [7].In order to explain these results they showed that: (a) thecavity size distribution function of H30 is sharper than thatof SPC/E water, which, in turn, is sharper than that of B60;(b) the cavity contact correlation function of H30 has ahigher maximum than that of SPC/E water, which, in turn,has a higher maximum than that of B60. LBHG concludedthat: (a) the H30 model is more hydrophobic than SPC/Ewater because the weakening of H-bonds leads to anincreased packing and a reduced probability of findingmolecular-sized cavities; (b) the B60 model is less hydro-phobic than SPC/E water because the destruction of thetetrahedral network leads to a more open structure andan increased probability of finding molecular-sized cavities[7].

I think that, in order to gain perspective and fully under-stand the interesting results of LBHG, it is necessary tocompare them with those obtained in reference hard spheremodels. Therefore, by means of scaled particle theory, SPT,calculations, in the present Letter it is shown that, notwith-standing the significant structural differences existingbetween the SPC/E, H30 and B60 models, their hydropho-bicity is, to a large extent, determined by two simple geo-metric features: the effective size of liquid molecules andthe volume packing density of the liquid. This is in line withthe original analysis by Lee [13,14], further strengthened bymy-self [11,12,15,16].

2. Scaled particle theory formulas

There are two measures of cavity size [4]: (a) the radiusof the spherical region from which all parts of the solventmolecules are excluded, indicated by rc; (b) the radius ofthe spherical region from which the centres of the solventmolecules are excluded, indicated by Rc. In the case ofspherical particles, the following relation holds: Rc = rc +r1, where r1 is the radius of the solvent molecules. Accord-ing to the statistical mechanical theory of fluctuations[17], the probability of finding a spherical region of radiusRc devoid of solvent molecule centres (also called thecavity insertion probability), at an arbitrarily fixed posi-tion, is related to the reversible work required to produce,at constant temperature and pressure, such constrainedmolecular configuration starting from the equilibriumone:

p0ðRcÞ ¼ exp½�DGcðRcÞ=RT � ð1ÞNote that DGc is a positive quantity also to create a pointcavity at a fixed position (i.e., rc = 0 implies Rc = r1), be-cause, even though such a cavity has a van der Waals sur-face area of zero, it gives rise to a non-zero solventexcluded volume of (4/3)p � r1

3.Geometric arguments lead to the following exact rela-

tionship for DGc when Rc 6 r1 (i.e., at most one molecularcentre can be found in the cavity for 0 6 Rc 6 r1):

DGc ¼ �RT � ln½1� ð4=3Þp � q1 � R3c � ð2Þ

where q1 = NAv/v1 is the solvent number density and v1 itsmolar volume. When Rc P r1, SPT provided the followingformula for DGc [18,19]:

DGc ¼ �RT � lnð1� nÞ þ RT � ½uðrc=r1Þ þ ðu=2Þ� ðuþ 2Þ � ðrc=r1Þ2� þ n � P � v1ðrc=r1Þ3 ð3Þ

in this relation R is the gas constant; n is the volume pack-ing density of pure solvent, which is defined as the ratio ofthe physical volume of a mole of solvent molecules over themolar volume of the solvent (i.e., n ¼ p � r3

1 � NAv=6 � v1Þand u = 3n/(1 – n); rc = 2 � rc and r1 = 2 � r1 are the hardsphere diameter of the cavity and of the solvent molecules,respectively; and P is the pressure. Since it has been sug-gested on theoretical grounds that the pressure has to be1 atm in performing calculations for real liquids [18], the

max

(r c

) (a

ngst

rom

-1 )

0.4

0.6

0.8

1.0SPT - 3.0

SPT - 2.8

SPT - 2.6

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0Hybrid - H30

SPC/E

Bent - B60

p max

(r c )

(an

gstr

om-1 )

rc (angstrom )

Fig. 2a. Cavity size distribution functions pmax(rc) of the SPC/E watermodel, of the bent B60 model, and of the hybrid H30 model at 25 �C and1 atm, calculated by Lynden-Bell and Head-Gordon using moleculardynamics computer simulations [7].

G. Graziano / Chemical Physics Letters 452 (2008) 259–263 261

cubic term in Eq. (3) proves to be very small and can beneglected.

The negative derivative of the insertion probability withrespect to the cavity radius gives the probability density ofthe radius of the largest cavity that can be successfullyinserted into a liquid, usually called the cavity size distribu-tion [20,21]:

pmaxðrcÞ ¼ �ðop0=orcÞ ð4ÞThis relationship indicates that pmax(rc) can be calcu-

lated from the cavity insertion probability, p0, that isrelated to DGc by means of Eq. (1). Thus, SPT providestwo analytical relationships to calculate the cavity size dis-tribution [22]:

pmaxð�r1 6 rc 6 0Þ ¼ 4p � q1 � ðrc þ r1Þ2 ð5Þwhere negative rc values are physically meaningful, but rc

cannot be smaller than �r1. The second SPT relationshipis an approximate formula valid over the size range rc P 0:

pmaxðrc P 0Þ ¼ 2½ðu=r1Þ þ ðuðuþ 2Þ=r21Þrc� � expð�DGc=RT Þ

ð6Þwhere the rc

2 term has been neglected for its smallnesswhen P = 1 atm.

The cavity contact correlation function G(Rc), which isthe conditional solvent density just outside a spherical cav-ity of radius Rc, is given by [20,21]:

GðRcÞ ¼ ð1=4p � q1 � R2cÞ � ½oðDGc=RT Þ=oRc� ð7Þ

By differentiating Eqs. (2) and (3), one obtains [22]:

Gð0 6 Rc 6 r1Þ ¼ 1=½1� ð4=3Þp � q1 � R3c � ð8Þ

GðRc P r1Þ¼ ð1=2p � q1 � R2

cÞ � fðu=r1Þ þ ½2uðuþ 2Þ=r21� � rcg ð9Þ

the rc2 term has been neglected in Eq. (9) for its smallness

when P = 1 atm. Eq. (8) is an always increasing function ofRc and does not possess a maximum. A search for amaximum in the expression of Eq. (9) gives [12]:

Rc;max ¼ r1 � ½1þ ðu=uþ 2Þ� ð10ÞGðRc;maxÞ ¼ uðuþ 2Þ2=2p � q1 � r3

1ðuþ 1Þ ð11Þ

In all SPT calculations I used the experimental density ofwater at 25 �C and P = 1 atm [23]. For the effective hardsphere diameter of water molecules, I selected three values:r1 = 2.6 A, 2.8 A and 3.0 A, respectively, which are close tothe first maximum in the OO-rdf of B60, SPC/E and H30models, respectively (see Fig. 1).

p

rc (angstrom)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.0

0.2

Fig. 2b. Cavity size distribution functions pmax(rc) calculated by means ofSPT, at 25 �C and 1 atm, with r1 = 2.6 A, 2.8 A and 3.0 A, respectively,and the experimental density of water.

3. Results and discussion

The SPC/E, H30 and B60 models possess the same num-ber density at 25 �C and 1 atm, but the location of the firstpeak in the OO-rdf demonstrates that their molecules havea different effective diameter: 2.8 A for SPC/E molecules,2.6 A for B60 molecules, and 3.0 A for H30 molecules

(see Fig. 1). On this basis I have used these diameter values,together with the experimental density of water at 25 �Cand 1 atm, to perform a complete set of SPT calculations.I have called such three models: SPT-2.6 A with n = 0.307,SPT-2.8 A with n = 0.383, and SPT-3.0 A with n = 0.471.The calculated cavity size distribution functions of thethree SPT models are shown in Fig. 2b; they have to becompared with those of SPC/E, H30 and B60 models deter-mined by LBHG, and reported in Fig. 2a. Note that, inreproducing the cavity size distribution functions of LBHGin Fig. 2a, I used rc and not Rc to measure the cavity sizebecause this is the correct procedure to perform a compar-ison between liquids whose molecules have a different effec-tive diameter [11,22].

There is a surprising agreement between the two sets ofpmax(rc) functions by considering that the hard sphere SPTmodels do not possess any of the structural features of

Table 1Values of the van der Waals diameter assigned to the oxygen atom in the three water models, of the location of the first peak maximum in the OO-rdf, ofthe maximum in the cavity size distribution and cavity contact correlation function, and of the work to create a cavity of Rc = 3 A (it should correspond torc = 3.2 A), at 25 �C and 1 atm. All the numbers come from [7]

r(A)

1st-peak(A)

rc,max

(A)pmax(rc,max)(A�1)

Rc,max

(A)G (Rc,max) DGc(Rc = 3 A)

(kJ mol�1)

B60 2.92 2.55 0.26 0.82 1.7 1.6 14.0SPC/E 3.165 2.75 0.16 0.84 2.3 1.9 19.1H30 3.165 2.95 0.13 0.96 2.3 2.7 26.0

Table 2Values of the maximum in the cavity size distribution and cavity contactcorrelation function and of the work to create a cavity of rc = 3.2 A,calculated via SPT, at 25 �C and 1 atm, by fixing r1 = 2.6 A, 2.8 A and3.0 A, respectively, and using the experimental number density of water

r1

(A)n rc,max

(A)pmax(rc,max)(A�1)

Rc,max

(A)G(Rc,max) DGc(rc = 3.2 A)

(kJ mol�1)

2.6 0.307 0.25 0.83 1.8 1.7 13.32.8 0.383 0.15 0.91 2.1 2.1 18.13.0 0.471 0.10 1.00 2.4 2.8 26.2

G (

Rc )

Rc (angstrom)

0.0 0.5 1.0 1.5 2.0 2.5 3.01.0

1.5

2.0

2.5

3.0

SPT - 2.6

SPT - 2.8

SPT - 3.0

Fig. 3b. Cavity contact correlation functions G(Rc) calculated by meansof SPT, at 25 �C and 1 atm, with r1 = 2.6 A, 2.8 A and 3.0 A, respectively,and the experimental density of water.

262 G. Graziano / Chemical Physics Letters 452 (2008) 259–263

SPC/E, H30 and B60 models. The agreement is confirmedby the values of rc,max and pmax(rc,max) listed in Tables 1and 2. The fundamental variables of SPT formulas are geo-metric: the effective diameter of liquid molecules and thevolume packing density of the liquid. In the present case,on increasing the hard sphere diameter from 2.6 A to3.0 A, and by keeping fixed the number density of waterso that n increases from 0.307 to 0.471, the pmax(rc) func-tion becomes sharper and the probability of finding molec-ular-sized cavities (i.e., cavities with rc = 1 A or larger; seeFig. 2b) decreases markedly.

The calculated cavity contact correlation functions ofthe three SPT models are shown in Fig. 3b; they have tobe compared with those of SPC/E, H30 and B60 modelsdetermined by LBHG [7], and reported in Fig. 3a. Not-

0.0 0.5 1.0 1.5 2.0 2.5 3.01.0

1.5

2.0

2.5

3.0

Bent - B60

Hybrid - H30

SPC/E

G (

Rc )

Rc (angstrom)

Fig. 3a. Cavity contact correlation functions G(Rc) of the SPC/E watermodel, of the bent B60 model, and of the hybrid H30 model at 25 �C and1 atm, calculated by Lynden-Bell and Head-Gordon using moleculardynamics computer simulations [7].

withstanding the structural differences between the watermodels constructed by LBHG and the hard sphere SPTmodels, there is agreement between the two sets of G(Rc)functions. This is confirmed by the values of Rc,max andG(Rc,max) listed in Tables 1 and 2. On increasing the hardsphere diameter from 2.6 A to 3.0 A, and keeping fixedthe number density of water, so that n increases from0.307 to 0.471, SPT Eqs. (10) and (11) indicate that Rc,max

increases from 1.8 A to 2.4 A, and G(Rc,max) increases from1.7 to 2.8. These SPT values are close to those of the B60and H30 models. It is worth noting that the cavity contactcorrelation function determined by Pratt and colleagues[21,24], for both TIP4P and SPC/E water models, at roomtemperature and atmospheric pressure, differs from thatobtained by LBHG in SPC/E water because the maximumis higher and located at a greater distance. Specifically,Pratt and colleagues obtained Rc,max � 2.8�3.0 A withG(Rc,max) � 2.2�2.3, whereas LBHG obtained Rc,max =2.3 A with G(Rc,max) = 1.9.

As a final comparison, I computed DGc to create a cavityof rc = 2 � rc = 3.2 A in the three SPT models at 25 �C and1 atm. It results DGc(rc = 3.2 A) = 13.3 kJ mol�1 in SPT-2.6 A, 18.1 kJ mol�1 in SPT-2.8 A, and 26.2 kJ mol�1 inSPT-3.0 A. These numbers can be compared with those cal-culated by LBHG to create a cavity of Rc = 3.0 A in theirmodels; see Table 3 in [7]. Since Rc = rc + r1 and r1 = 1.4 Afor water molecules, the cavities of Rc = 3.0 A should have

G. Graziano / Chemical Physics Letters 452 (2008) 259–263 263

rc = 3.2 A; LBHG obtained DGc = 14.0 kJ mol�1 in theB60 model, 19.1 kJ mol�1 in the SPC/E model, and26.0 kJ mol�1 in the H30 model [7]. The agreementbetween the SPT-calculated DGc values and those calcu-lated via molecular dynamics simulations by LBHG ismore than qualitative. This finding is a further demonstra-tion that the effective molecular diameter and the volumepacking density of the liquid are the fundamental determi-nants of the partitioning of void volume and of the occur-rence of molecular-sized cavities [11–16,22]. In this respectit is worth mentioning that LBHG found no clear correla-tion between the magnitude of the DGc values and (a) thestrength of H-bonds, (b) the surface tension, and (c) thedielectric constant of the SPC/E, H30 and B60 models[7]. Moreover, the results of LBHG directly show thatthe bunching up effect of H-bonds, by reducing the effectivesize of water molecules and so the volume packing density,leads to a decrease in the DGc magnitude, enhancing thesolubility of nonpolar species, as originally noted by Lee[4,13].

In conclusion, the present SPT analysis unequivocallydemonstrates that the effective size of liquid moleculesand the volume packing density obtained from the experi-mental density of the liquid are the fundamental determi-nants of the partitioning of void volume and the DGc

magnitude (i.e., differences in the two-particle and three-particle correlation functions are not so important, asemphasized by LBHG). Therefore, fixed the liquid density,simple geometric factors play the pivotal role. Hydropho-bicity is due to the markedly larger magnitude of DGc inwater with respect to common organic liquids, that, in

turn, is mainly caused by the small size of water molecules,notwithstanding the low packing density of the tetrahedralH-bonding network of water.

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