Water: cavity size distribution and hydrogen bonds
Chemical Physics Letters 396 (2004) 226–231 www.elsevier.com/locate/cplett
Water: cavity size distribution and hydrogen bonds Giuseppe Graziano
*
Department of Biological and Environmental Sciences, University of Sannio, Via PortArsa 11, 82100 Benevento, Italy Received 7 June 2004; in final form 30 July 2004
Abstract ˚ ; (b) There are two sizes for water molecules: (a) the distance of closest approach between two hydrogen bonded molecules, 2.8 A ˚ , corresponding to the van der Waals diameter the distance of closest approach between two non-hydrogen bonded molecules, 3.2 A of the oxygen atom. This fact is due to the bunching up effect of hydrogen bonds. Correspondingly, when the hydrogen bonding potential is turned off in computer models of water, the effective size of water molecules increases. It is shown by means of scaled particle theory calculations that this basic point has profound effects on the cavity size distribution if the number density is kept constant. The recognition of the bunching up effect of hydrogen bonds is a key factor in order to address the role played by hydrogen bonds in the partitioning of void volume in liquid water. 2004 Elsevier B.V. All rights reserved.
1. Introduction Liquids are characterized by the presence of a significant fraction of void volume. By performing the ratio of the van der Waals volume occupied by a mole of liquid molecules to the molar volume of the liquid itself, one obtains the so-called volume packing density n. For common organic liquids at room temperature, n is around 0.5, indicating that about half of the molar volume is not occupied [1]. However, the total void volume is not the relevant quantity to shed light on solvation phenomena [2,3]. What is really important is the manner in which the void volume is partitioned. In other words, it would be necessary to know the cavity size distribution of the liquid. Unfortunately, the determination of the cavity size distribution cannot be accomplished by experimental measurements and has to be performed by means of computer simulations. In 1990, Pohorille and Pratt [4,5] were the first to report the cavity size distribution of several liquids,
*
Fax: +39-0824-23013. E-mail address: [email protected].
0009-2614/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.07.126
including water and n-hexane, determined from a detailed analysis of a large number of configurations generated by means of molecular dynamics simulations. It resulted that the most part of the void volume is parti˚ diameter, tioned in very small cavities, less than 1 A and that the number of large cavities is exceedingly small. In particular, the number of molecular-sized ˚ diameter) cavities proved to be markedly smaller (2–5 A in water than in n-hexane or other non-polar solvents. From a microscopic point of view, this should be the origin of the poor solubility of non-polar compounds in water [2,4,5]. Obviously, a molecular level explanation of the Pohorille and Pratt results has to be provided. We have shown that the cavity size distribution of water determined by Pohorille and Pratt can be satisfactorily reproduced by means of scaled particle theory [6], SPT, assigning to water molecules an effective diameter of ˚ [7]. This means that the void volume in water is 2.8 A partitioned in small cavities because the size of water molecules, the liquid length scale, is small [8]. In this respect, it is worth noting that the effective diameter of water molecules is markedly smaller than that of common organic solvents [9].
G. Graziano / Chemical Physics Letters 396 (2004) 226–231
Recently, Sanchez and co-workers [10,11] determined the cavity size distribution analysing liquid configurations generated by means of Monte Carlo simulations in the canonical ensemble, using a newly developed algorithm. They considered three different fluid types: (a) a hard sphere, HS, fluid with no attractions between particles; (b) a Lennard-Jones, LJ, fluid whose particles interact according to the Lennard–Jones potential; (c) the extended simple point charge [12], SPC/E, model of water that treats explicitly the presence of H-bonds by means of electrostatic interactions between partial charges. It resulted that, at densities corresponding to the liquid state, the cavity size distribution of HS fluid is very similar to that of LJ fluid (i.e., the two fluids have the same number density and their particles have the same size), consistent with the notion that the structure of simple liquids is mainly determined by repulsive interactions, the excluded volume effect [13,14]. On the other hand, by recognizing that SPC/E water is an LJ fluid with added electrostatic interactions to account for the presence of H-bonds, Sanchez and coworkers [10,11] tried to address the role played by Hbonds on the cavity size distribution by deleting the partial charges of the SPC/E model. In this manner, they generated an LJ fluid having the same number density of water at 25 C, but with the H-bonding potential turned off (it will be indicated as the corresponding LJ fluid hereafter). The cavity size distributions of the two liquids were both unimodal, but that of SPC/E water proved to be broader and peaked at a larger size than that of the corresponding LJ fluid [10,11]. The two distributions are shown in Fig. 1 that corresponds to Fig. 5 in [10] and Fig. 11 in [11]. The maximum occurs at 0.7 Æ r in SPC/E water and at 0.6 Æ r in the LJ fluid, where r is ˚ the van der Waals diameter of the oxygen atom, 3.16 A
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in the SPC/E model. The maximum is higher in the LJ fluid than in SPC/E water and the latter liquid has a greater number of molecular sized cavities. Sanchez and co-workers [10,11] rationalized their results by stating that strongly attractive interactions such as the H-bonds in water promote clustering phenomena that lead to the existence of larger cavities with respect to the corresponding LJ fluid. In other words, even though the two liquids have the same number density, the void volume should be distributed in larger packets in the liquid characterized by stronger interactions. In fact, they wrote [11]: Hydrogen bonding causes water to aggregate into clusters that produce a few large cavities rather than many smaller cavities and This result indicates that applying SPT, which completely ignores clustering effects, to describe the solvation properties of water is questionable. The only way to compensate for clustering in SPT is to use a smaller effective diameter for water that makes water appear to be low density. The finding that the cavity size distribution in SPC/E water is broader, smaller and shifted toward larger cavities with respect to that of the corresponding LJ fluid seems to contrast with the ability of SPT to reproduce the cavity size distribution in TIP4P water determined by Pohorille and Pratt [4,5] and Graziano [7]. The SPT success would imply that SPC/E water and a properly selected HS fluid should have similar cavity size distributions. We would like to show that this is indeed true. On this basis, an entirely different explanation of the results obtained by Sanchez and co-workers can be provided. The fundamental argument is that the role of the H-bonding potential in determining the cavity size distribution in water cannot be addressed by simply deleting the partial charges in the SPC/E water model and keeping fixed the number density.
2. Methods 2,5
1,0
SPT provides two analytical relationships to calculate the cavity size distribution pmax(rc), which is the probability density of the radius of the largest cavity that can be successfully inserted into a liquid [6,7]. Simple geometric arguments lead to the following exact relationship valid over the size range r16rc60:
0,5
pmax ðr1 6 rc 6 0Þ ¼ 4p q1 ðrc þ r1 Þ2 ;
probability density
LJ 2,0 1,5
0,0 0,0
water
0,5
1,0
cavity size in units of σ
1,5
Fig. 1. Cavity size distributions determined by Sanchez and coworkers in SPC/E water and in the corresponding LJ fluid at 25 C (see Fig. 5 in [10] and Fig. 11 in [11]). The corresponding LJ fluid was obtained by turning off the electrostatic interactions in SPC/E water and keeping fixed the number density at the experimental value of water at 25 C.
ð1Þ
where q1 = NAv/v1 is the number density of the solvent and v1 its molar volume; r1 is the radius of the solvent molecules and rc is the radius of the cavity (i.e., the radius of the spherical region from which any part of any solvent molecules is excluded). In this respect, it is worth noting that negative rc values are physically meaningful, but rc cannot be smaller than r1 [6]. The second SPT relationship is an approximate formula valid over the size range rc P 0 [6,7]:
G. Graziano / Chemical Physics Letters 396 (2004) 226–231
pmax ðrc P 0Þ ¼ 2½ðK 1 =r1 Þ þ ð2K 2 =r21 Þrc þ ð3K 3 =r31 Þr2c expðDGc =RT Þ; ð2Þ where K1 = u = 3n/(1n); K2 = u(u + 2)/2; K3 = n Æ P Æ v1/ RT; and DGc is the work of cavity creation as expressed by means of SPT [15]. In these relations R is the gas constant; n is the volume packing density of pure solvent, which is defined as the ratio of the physical volume of a mole of solvent molecules over the molar volume of the solvent ði:e:; n ¼ p r31 N Av =6 v1 Þ; rc and r1 are the hard sphere diameter of the cavity and of the solvent molecules, respectively; and P is the pressure. In all SPT calculations, the experimental density of liquid water at 25 C is used [16], and the pressure is fixed at 1 atm [15,17,18].
3. Results The starting point is the recognition that the size of molecules depends on the interactions in which the molecules are involved. For water there are two relevant sizes: (a) the distance of closest approach between two ˚ , on the basis of H-bonded water molecules, r @ 2.8 A the location of the first peak in the oxygen–oxygen radial distribution function, rdf, of water [19–21]; (b) the distance of closest approach between two non H-bonded water molecules, corresponding to the van der Waals ˚ [22]. Madan diameter of an oxygen atom, r @ 3.2 A and Lee [23] determined, by means of Monte Carlo simulations, that when the H-bonding potential between water molecules in the TIP4P model [22] is turned off and the density is kept constant, the distance of closest ˚ . Practically, they approach increases from 2.8 to 3.2 A found that the first peak of the oxygen–oxygen rdf ˚ upon turning off the H-bonding passes from 2.8 to 3.2 A potential [23]. This is because H-bonds are so strong to bunch up water molecules beyond their van der Waals size. Since in the TIP4P model the van der Waals diam˚ , when the eter of the oxygen atom is fixed at 3.15 A H-bonding potential is turned off, the distance of closest approach becomes close to this number. Exactly the same thing should occur in the SPC/E model, where ˚ , when the size of the oxygen atom is fixed at 3.16 A the partial charges are stripped and the density is kept constant. It is simple to grasp why an LJ liquid obtained by simply turning off the H-bonding potential in TIP4P or SPC/E model cannot be the sole reference liquid to assess the role played by H-bonds in determining the partitioning of void volume in water. If the number density is kept constant at the value of water at 25 C, but the effective size of the molecules increases from 2.8 to ˚ , the void volume existing in the liquid necessarily 3.2 A decreases. Such a decrease of void volume manifests itself in a marked increase of the volume packing density.
˚ versus Specifically, at 25 C, n = 0.383 for r = 2.8 A ˚ n = 0.572 for r = 3.2 A, using the experimental molar volume of water, v = 18.07 cm3 mol1 [16]. The increase in volume packing density, in turn, causes a sharpening of the cavity size distribution and an increase of the magnitude of the work of cavity creation. This reasoning has a general validity because it is grounded on simple geometric and physical principles [24]. It can be illustrated by means of SPT calculations. We have calculated pmax(rc) by means of Eqs. (1) and (2), using the two different values for the diameter of ˚ . The SPT pmax(rc) water molecules: r = 2.8 and 3.2 A functions are shown in Fig. 2 together with the distribution determined by Pohorille and Pratt [4,5] in TIP4P water at room temperature. The latter should be considered the experimental cavity size distribution in water, because it was determined by means of a brute force procedure [4,5]. The SPT cavity size distribution calculated for r = 2.8 ˚ is in satisfactory agreement with the experimental A ˚ does not one, whereas that calculated for r = 3.2 A agree with the experimental one. Since the cavity size distribution of water can be reproduced by means of ˚ , the effective size of water moleSPT, using r = 2.8 A cules, the latter has to be the principal factor in the partitioning of void volume in water, when the density is kept fixed, as already pointed out [7]. On increasing the diameter of water molecules, the calculated pmax(rc) distribution becomes sharper and the maximum shifts to smaller rc values, as anticipated. The two SPT distributions are similar to those determined by Sanchez and co-workers [10,11] for SPC/E
1.2
pmax (rc) (angstrom-1)
228
1.0
SPT-3.2
SPT-2.8
0.8 0.6 0.4 0.2 0.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
rc ( angstrom ) Fig. 2. Comparison of the cavity size distribution determined by Pohorille and Pratt [4,5] in TIP4P water at 300 K (empty squares), and ˚ (blue line), and 3.2 A ˚ those calculated by means of SPT using r = 2.8 A (black line). On increasing r, the SPT distribution becomes sharper and peaked at smaller sizes. In all calculations, the experimental density of water at 25 C is used, and the pressure is fixed at 1 atm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
G. Graziano / Chemical Physics Letters 396 (2004) 226–231
water and the corresponding LJ fluid, respectively. The SPT functions are shown in Fig. 3 versus the cavity size ˚ by means of the relation expressed in units of r = 3.2 A (rc + 1.6)/3.2. The comparison suggests that the corresponding LJ fluid consists of molecules having an effective diameter larger than that of molecules in SPC/E water. Notwithstanding the claim by Sanchez and coworkers, SPT calculations reproduce their results, providing an explanation grounded on basic physical ideas. However, the comparison cannot be quantitative. Even though the distribution determined by Sanchez and co-workers should be a volume distribution [10], giving the probability p(vc) dvc that a cavity has volume between vc and vc + dvc, we suspect that it is similar to, but not exactly, p(vc) dvc. On the other hand, the distribution determined by Pohorille and Pratt and the SPT ones provide the probability pmax(rc) drc that a cavity has radius between rc and rc + drc. Furthermore, comparison of the cavity size distribution in SPC/E water of Sanchez and co-workers with that in TIP4P water of Pohorille and Pratt indicates that the algorithm devised by Sanchez and co-workers overestimates the cavity diameters, as already noted by the authors [10]. The argument that the molecular diameter is the fundamental factor controlling the division of void volume in water, when the density is kept constant, can be further validated. Madan and Lee [23] showed that by ˚ and using adjusting the size of LJ particles at 2.8 A the same number density of water at 25 C, the DGc magnitude in such LJ fluid is comparable to that in TIP4P water. Similarly, Pratt and Pohorille [5] found that the cavity contact correlation function in an LJ fluid, with particles of the same size of water ones and having the same number density of water, is similar to that determined in TIP4P water.
pmax ( rc ) (angstrom-1)
1,2 1,0
SPT-3.2
0,8
SPT-2.8 0,6 0,4 0,2 0,0 0,0
0,2
0,4
0,6
0,8
1,0
cavity size in units of σ Fig. 3. SPT cavity size distributions calculated using the experimental density of water at 25 C and two different diameters for the particles ˚ , respectively, reported as a function of cavity size 2.8 and 3.2 A ˚ by means of the relation (rc + 1.6)/3.2. expressed in units of r = 3.2 A In this manner, the similarity with the distributions of Fig. 1 should be more evident.
229
The SPT estimates of the work to create a cavity of 4 ˚ diameter in water at 25 C are: 26.0 kJ mol1 for A ˚ and 61.1 kJ mol1 for r = 3.2 A ˚ . The reliabilr = 2.8 A ˚ is readily ity of the SPT value calculated using r = 2.8 A ˚ verified. To create a 4 A diameter cavity: (a) Guillot and Guissani found DGc = 26 kJ mol1 in SPC/E water at 300 K [25]; (b) van Gunsteren and co-workers calculated DGc = 27 kJ mol1 in SPC water at 300 K [26], upon accounting for the transformation from the cavity thermal diameter to the hard sphere one [27]; (c) Pohorille and Pratt found DGc = 27 kJ mol1 in TIP4P water at 300 K [4]. The corresponding cavity insertion probabil˚ , and ity, po = exp(DGc/RT) = 2.8 · 105 for r = 2.8 A 11 ˚ 2.0 · 10 for r = 3.2 A. The latter number is nine orders of magnitude larger than the SPT estimate, 3 · 1020, reported by Sanchez and co-workers [11] to in˚ diameter cavity into an HS fluid with particles sert a 4 A ˚ of 3.16 A diameter and having the same number density of water at 25 C. The origin of the huge discrepancy is because Sanchez and co-workers calculated DGc using the pressure of the HS fluid as determined by SPT [6], 27 600 atm. In this respect, several authors [15,17,18] have indicated that SPT calculations of DGc have to be performed at the experimental pressure of the real liquid, not at the pressure of the HS fluid. For this reason, we used P = 1 atm in SPT calculations.
4. Discussion The present analysis indicates that, if the number density is kept fixed, the cavity size distribution of water can be reproduced by means of SPT assigning to water ˚ . The latter nummolecules an effective diameter of 2.8 A ber corresponds to the location of the first peak in the oxygen–oxygen rdf of water and so to the distance between two H-bonded water molecules [19–21]. This means that H-bonds play an indirect role in determining the cavity size distribution by reducing the size of water ˚. molecules from 3.2 to 2.8 A In other words, the partitioning of void volume in water is determined by the size of the molecules, but the latter has to be considered with care. This is due to the bunching up effect of H-bonds that are so strong to allow the interacting molecules to come closer than their van der Waals size [23]. Even though all the standard computer models of water use a van der Waals ˚ [12,22], the diameter for the oxygen atom of 3.1–3.2 A obtained oxygen–oxygen rdf is always peaked at ˚ , in line with X-ray and neutron scattering meas2.8 A urements [19–21]. The bunching up effect of H-bonds is operative also in liquid methanol and ethanol. For ˚ [31], methanol, the van der Waals diameter is 4.10 A ˚ [1], or 3.83 A ˚ whereas the effective diameter is 3.69 A [32]. For ethanol, the van der Waals diameter is 4.66
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G. Graziano / Chemical Physics Letters 396 (2004) 226–231
˚ [31], whereas the effective diameter is 4.34 A ˚ [1], or A ˚ 4.44 A [32]. Therefore, in order to address the role of H-bonds in determining the cavity size distribution in water, it is necessary, upon turning off the H-bonding potential, to reduce the diameter of LJ particles in order to obtain the same effective size of water molecules and so the same amount of void volume in the simulated liquid. This need was pointed out by Madan and Lee [23], and is now well recognized by scientists working in the field [5,28–30]. In contrast, the corresponding LJ fluid of Sanchez and co-workers differs from the SPC/E water model not only for the absence of the H-bonding potential, but also for the size of the molecules. This is the origin of the finding that SPC/E water has larger cavities than the corresponding LJ fluid, a result readily rationalized when the molecular size is properly taken into account, as shown by means of SPT in Figs. 2 and 3. Since an HS fluid having the same number density of water and consisting of particles of the same size of water molecules is characterized by a cavity size distribution close to that of water, the space occupation in water is largely determined by the effective molecular size. It is important to note that the real situation is more subtle: the two sizes characterizing water molecules operate simultaneously in water. The average coordination number in water at room temperature is about 5 ˚ that are [19–21]. There are 4 water molecules at 2.8 A H-bonded to the central one; and there is also an inter˚ that is not H-bonded to the censtitial molecule at 3.2 A tral one, but occurs in its first coordination shell [33]. This implies that an LJ fluid consisting of molecules having a well defined diameter could not be able to exactly account for the partitioning of void volume in water. Similarly, SPT should be considered in a way unsatisfactory to describe the packing properties of water because it can make use of only one size at a time. Even though this is strictly true, the computer simulation results of both Pohorille and Pratt [4,5], and Madan and Lee [23] indicate that an LJ fluid with particles of ˚ diameter and the same number density of water 2.8 A at 25 C should possess a cavity size distribution close to that of SPC/E or TIP4P water models. This is further supported by the finding that SPT, an approximate theory of HS fluids, is able to reproduce in a satisfactory manner the experimental cavity size distribution determined by Pohorille and Pratt in TIP4P water [7], by ˚ for water molecules. The excluded volusing r = 2.8 A ume effect related to the effective size of water molecules, in part determined by the bunching up effect of H-bonds, plays the pivotal role in the partitioning of void volume in liquid water. Clearly, if the effective size of water molecules were equal to the van der Waals diameter of the oxygen atom, ˚ , water would be dense with n = 0.572. The solu3.2 A
bility of non-polar compounds in this hypothetical dense water would be even poorer than in water, because DGc would be larger, as emphasized by the cavity size distribution determined by Sanchez and co-workers for the corresponding LJ fluid, and the SPT one calcu˚ . The H-bonds, by rendering smaller lated for r = 3.2 A the water molecules and so less costly the process of cavity creation, should aid the solubility of non-polar compounds in water. Similar arguments were already discussed by Lee [9,23] and by us [34]. In conclusion, the present analysis points out that if the H-bonding potential is turned off, the effective size ˚ . An LJ of water molecules increases from 2.8 to 3.2 A fluid obtained by turning off the H-bonding potential and keeping fixed the number density has less void volume than water and cannot be the sole reference fluid to clarify the role of H-bonds in determining the cavity size distribution of water. Another LJ fluid should be considered: after turning off the H-bonding potential, the particle size has to be reduced to the effective diameter of water molecules so to have the same void volume of water. The role played by the diameter assigned to water molecules in determining the cavity size distribution is clarified by means of SPT calculations. The latter show unequivocally that, when the number density is fixed, the molecular diameter plays the fundamental role for the partitioning of void volume and the magnitude of the work of cavity creation in water. The H-bonds play an indirect role by determining the effective size of water molecules. The present results support the small size of water molecules as the origin of the hydrophobic effect [7–9,29,34], but do not support the dense packing effect as proposed by Sanchez and co-workers.
Acknowledgements I thank Dr. B. Lee (Center for Cancer Research, NCI, NIH, Bethesda, MD) for carefully reading many earlier drafts of the manuscript. This work is supported by the COFIN 2002 grant from the Italian Ministry of Instruction, University, and Research (M.I.U.R., Rome).
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Water: cavity size distribution and hydrogen bonds Giuseppe Graziano
*
Department of Biological and Environmental Sciences, University of Sannio, Via PortArsa 11, 82100 Benevento, Italy Received 7 June 2004; in final form 30 July 2004
Abstract ˚ ; (b) There are two sizes for water molecules: (a) the distance of closest approach between two hydrogen bonded molecules, 2.8 A ˚ , corresponding to the van der Waals diameter the distance of closest approach between two non-hydrogen bonded molecules, 3.2 A of the oxygen atom. This fact is due to the bunching up effect of hydrogen bonds. Correspondingly, when the hydrogen bonding potential is turned off in computer models of water, the effective size of water molecules increases. It is shown by means of scaled particle theory calculations that this basic point has profound effects on the cavity size distribution if the number density is kept constant. The recognition of the bunching up effect of hydrogen bonds is a key factor in order to address the role played by hydrogen bonds in the partitioning of void volume in liquid water. 2004 Elsevier B.V. All rights reserved.
1. Introduction Liquids are characterized by the presence of a significant fraction of void volume. By performing the ratio of the van der Waals volume occupied by a mole of liquid molecules to the molar volume of the liquid itself, one obtains the so-called volume packing density n. For common organic liquids at room temperature, n is around 0.5, indicating that about half of the molar volume is not occupied [1]. However, the total void volume is not the relevant quantity to shed light on solvation phenomena [2,3]. What is really important is the manner in which the void volume is partitioned. In other words, it would be necessary to know the cavity size distribution of the liquid. Unfortunately, the determination of the cavity size distribution cannot be accomplished by experimental measurements and has to be performed by means of computer simulations. In 1990, Pohorille and Pratt [4,5] were the first to report the cavity size distribution of several liquids,
*
Fax: +39-0824-23013. E-mail address: [email protected].
0009-2614/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.07.126
including water and n-hexane, determined from a detailed analysis of a large number of configurations generated by means of molecular dynamics simulations. It resulted that the most part of the void volume is parti˚ diameter, tioned in very small cavities, less than 1 A and that the number of large cavities is exceedingly small. In particular, the number of molecular-sized ˚ diameter) cavities proved to be markedly smaller (2–5 A in water than in n-hexane or other non-polar solvents. From a microscopic point of view, this should be the origin of the poor solubility of non-polar compounds in water [2,4,5]. Obviously, a molecular level explanation of the Pohorille and Pratt results has to be provided. We have shown that the cavity size distribution of water determined by Pohorille and Pratt can be satisfactorily reproduced by means of scaled particle theory [6], SPT, assigning to water molecules an effective diameter of ˚ [7]. This means that the void volume in water is 2.8 A partitioned in small cavities because the size of water molecules, the liquid length scale, is small [8]. In this respect, it is worth noting that the effective diameter of water molecules is markedly smaller than that of common organic solvents [9].
G. Graziano / Chemical Physics Letters 396 (2004) 226–231
Recently, Sanchez and co-workers [10,11] determined the cavity size distribution analysing liquid configurations generated by means of Monte Carlo simulations in the canonical ensemble, using a newly developed algorithm. They considered three different fluid types: (a) a hard sphere, HS, fluid with no attractions between particles; (b) a Lennard-Jones, LJ, fluid whose particles interact according to the Lennard–Jones potential; (c) the extended simple point charge [12], SPC/E, model of water that treats explicitly the presence of H-bonds by means of electrostatic interactions between partial charges. It resulted that, at densities corresponding to the liquid state, the cavity size distribution of HS fluid is very similar to that of LJ fluid (i.e., the two fluids have the same number density and their particles have the same size), consistent with the notion that the structure of simple liquids is mainly determined by repulsive interactions, the excluded volume effect [13,14]. On the other hand, by recognizing that SPC/E water is an LJ fluid with added electrostatic interactions to account for the presence of H-bonds, Sanchez and coworkers [10,11] tried to address the role played by Hbonds on the cavity size distribution by deleting the partial charges of the SPC/E model. In this manner, they generated an LJ fluid having the same number density of water at 25 C, but with the H-bonding potential turned off (it will be indicated as the corresponding LJ fluid hereafter). The cavity size distributions of the two liquids were both unimodal, but that of SPC/E water proved to be broader and peaked at a larger size than that of the corresponding LJ fluid [10,11]. The two distributions are shown in Fig. 1 that corresponds to Fig. 5 in [10] and Fig. 11 in [11]. The maximum occurs at 0.7 Æ r in SPC/E water and at 0.6 Æ r in the LJ fluid, where r is ˚ the van der Waals diameter of the oxygen atom, 3.16 A
227
in the SPC/E model. The maximum is higher in the LJ fluid than in SPC/E water and the latter liquid has a greater number of molecular sized cavities. Sanchez and co-workers [10,11] rationalized their results by stating that strongly attractive interactions such as the H-bonds in water promote clustering phenomena that lead to the existence of larger cavities with respect to the corresponding LJ fluid. In other words, even though the two liquids have the same number density, the void volume should be distributed in larger packets in the liquid characterized by stronger interactions. In fact, they wrote [11]: Hydrogen bonding causes water to aggregate into clusters that produce a few large cavities rather than many smaller cavities and This result indicates that applying SPT, which completely ignores clustering effects, to describe the solvation properties of water is questionable. The only way to compensate for clustering in SPT is to use a smaller effective diameter for water that makes water appear to be low density. The finding that the cavity size distribution in SPC/E water is broader, smaller and shifted toward larger cavities with respect to that of the corresponding LJ fluid seems to contrast with the ability of SPT to reproduce the cavity size distribution in TIP4P water determined by Pohorille and Pratt [4,5] and Graziano [7]. The SPT success would imply that SPC/E water and a properly selected HS fluid should have similar cavity size distributions. We would like to show that this is indeed true. On this basis, an entirely different explanation of the results obtained by Sanchez and co-workers can be provided. The fundamental argument is that the role of the H-bonding potential in determining the cavity size distribution in water cannot be addressed by simply deleting the partial charges in the SPC/E water model and keeping fixed the number density.
2. Methods 2,5
1,0
SPT provides two analytical relationships to calculate the cavity size distribution pmax(rc), which is the probability density of the radius of the largest cavity that can be successfully inserted into a liquid [6,7]. Simple geometric arguments lead to the following exact relationship valid over the size range r16rc60:
0,5
pmax ðr1 6 rc 6 0Þ ¼ 4p q1 ðrc þ r1 Þ2 ;
probability density
LJ 2,0 1,5
0,0 0,0
water
0,5
1,0
cavity size in units of σ
1,5
Fig. 1. Cavity size distributions determined by Sanchez and coworkers in SPC/E water and in the corresponding LJ fluid at 25 C (see Fig. 5 in [10] and Fig. 11 in [11]). The corresponding LJ fluid was obtained by turning off the electrostatic interactions in SPC/E water and keeping fixed the number density at the experimental value of water at 25 C.
ð1Þ
where q1 = NAv/v1 is the number density of the solvent and v1 its molar volume; r1 is the radius of the solvent molecules and rc is the radius of the cavity (i.e., the radius of the spherical region from which any part of any solvent molecules is excluded). In this respect, it is worth noting that negative rc values are physically meaningful, but rc cannot be smaller than r1 [6]. The second SPT relationship is an approximate formula valid over the size range rc P 0 [6,7]:
G. Graziano / Chemical Physics Letters 396 (2004) 226–231
pmax ðrc P 0Þ ¼ 2½ðK 1 =r1 Þ þ ð2K 2 =r21 Þrc þ ð3K 3 =r31 Þr2c expðDGc =RT Þ; ð2Þ where K1 = u = 3n/(1n); K2 = u(u + 2)/2; K3 = n Æ P Æ v1/ RT; and DGc is the work of cavity creation as expressed by means of SPT [15]. In these relations R is the gas constant; n is the volume packing density of pure solvent, which is defined as the ratio of the physical volume of a mole of solvent molecules over the molar volume of the solvent ði:e:; n ¼ p r31 N Av =6 v1 Þ; rc and r1 are the hard sphere diameter of the cavity and of the solvent molecules, respectively; and P is the pressure. In all SPT calculations, the experimental density of liquid water at 25 C is used [16], and the pressure is fixed at 1 atm [15,17,18].
3. Results The starting point is the recognition that the size of molecules depends on the interactions in which the molecules are involved. For water there are two relevant sizes: (a) the distance of closest approach between two ˚ , on the basis of H-bonded water molecules, r @ 2.8 A the location of the first peak in the oxygen–oxygen radial distribution function, rdf, of water [19–21]; (b) the distance of closest approach between two non H-bonded water molecules, corresponding to the van der Waals ˚ [22]. Madan diameter of an oxygen atom, r @ 3.2 A and Lee [23] determined, by means of Monte Carlo simulations, that when the H-bonding potential between water molecules in the TIP4P model [22] is turned off and the density is kept constant, the distance of closest ˚ . Practically, they approach increases from 2.8 to 3.2 A found that the first peak of the oxygen–oxygen rdf ˚ upon turning off the H-bonding passes from 2.8 to 3.2 A potential [23]. This is because H-bonds are so strong to bunch up water molecules beyond their van der Waals size. Since in the TIP4P model the van der Waals diam˚ , when the eter of the oxygen atom is fixed at 3.15 A H-bonding potential is turned off, the distance of closest approach becomes close to this number. Exactly the same thing should occur in the SPC/E model, where ˚ , when the size of the oxygen atom is fixed at 3.16 A the partial charges are stripped and the density is kept constant. It is simple to grasp why an LJ liquid obtained by simply turning off the H-bonding potential in TIP4P or SPC/E model cannot be the sole reference liquid to assess the role played by H-bonds in determining the partitioning of void volume in water. If the number density is kept constant at the value of water at 25 C, but the effective size of the molecules increases from 2.8 to ˚ , the void volume existing in the liquid necessarily 3.2 A decreases. Such a decrease of void volume manifests itself in a marked increase of the volume packing density.
˚ versus Specifically, at 25 C, n = 0.383 for r = 2.8 A ˚ n = 0.572 for r = 3.2 A, using the experimental molar volume of water, v = 18.07 cm3 mol1 [16]. The increase in volume packing density, in turn, causes a sharpening of the cavity size distribution and an increase of the magnitude of the work of cavity creation. This reasoning has a general validity because it is grounded on simple geometric and physical principles [24]. It can be illustrated by means of SPT calculations. We have calculated pmax(rc) by means of Eqs. (1) and (2), using the two different values for the diameter of ˚ . The SPT pmax(rc) water molecules: r = 2.8 and 3.2 A functions are shown in Fig. 2 together with the distribution determined by Pohorille and Pratt [4,5] in TIP4P water at room temperature. The latter should be considered the experimental cavity size distribution in water, because it was determined by means of a brute force procedure [4,5]. The SPT cavity size distribution calculated for r = 2.8 ˚ is in satisfactory agreement with the experimental A ˚ does not one, whereas that calculated for r = 3.2 A agree with the experimental one. Since the cavity size distribution of water can be reproduced by means of ˚ , the effective size of water moleSPT, using r = 2.8 A cules, the latter has to be the principal factor in the partitioning of void volume in water, when the density is kept fixed, as already pointed out [7]. On increasing the diameter of water molecules, the calculated pmax(rc) distribution becomes sharper and the maximum shifts to smaller rc values, as anticipated. The two SPT distributions are similar to those determined by Sanchez and co-workers [10,11] for SPC/E
1.2
pmax (rc) (angstrom-1)
228
1.0
SPT-3.2
SPT-2.8
0.8 0.6 0.4 0.2 0.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
rc ( angstrom ) Fig. 2. Comparison of the cavity size distribution determined by Pohorille and Pratt [4,5] in TIP4P water at 300 K (empty squares), and ˚ (blue line), and 3.2 A ˚ those calculated by means of SPT using r = 2.8 A (black line). On increasing r, the SPT distribution becomes sharper and peaked at smaller sizes. In all calculations, the experimental density of water at 25 C is used, and the pressure is fixed at 1 atm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
G. Graziano / Chemical Physics Letters 396 (2004) 226–231
water and the corresponding LJ fluid, respectively. The SPT functions are shown in Fig. 3 versus the cavity size ˚ by means of the relation expressed in units of r = 3.2 A (rc + 1.6)/3.2. The comparison suggests that the corresponding LJ fluid consists of molecules having an effective diameter larger than that of molecules in SPC/E water. Notwithstanding the claim by Sanchez and coworkers, SPT calculations reproduce their results, providing an explanation grounded on basic physical ideas. However, the comparison cannot be quantitative. Even though the distribution determined by Sanchez and co-workers should be a volume distribution [10], giving the probability p(vc) dvc that a cavity has volume between vc and vc + dvc, we suspect that it is similar to, but not exactly, p(vc) dvc. On the other hand, the distribution determined by Pohorille and Pratt and the SPT ones provide the probability pmax(rc) drc that a cavity has radius between rc and rc + drc. Furthermore, comparison of the cavity size distribution in SPC/E water of Sanchez and co-workers with that in TIP4P water of Pohorille and Pratt indicates that the algorithm devised by Sanchez and co-workers overestimates the cavity diameters, as already noted by the authors [10]. The argument that the molecular diameter is the fundamental factor controlling the division of void volume in water, when the density is kept constant, can be further validated. Madan and Lee [23] showed that by ˚ and using adjusting the size of LJ particles at 2.8 A the same number density of water at 25 C, the DGc magnitude in such LJ fluid is comparable to that in TIP4P water. Similarly, Pratt and Pohorille [5] found that the cavity contact correlation function in an LJ fluid, with particles of the same size of water ones and having the same number density of water, is similar to that determined in TIP4P water.
pmax ( rc ) (angstrom-1)
1,2 1,0
SPT-3.2
0,8
SPT-2.8 0,6 0,4 0,2 0,0 0,0
0,2
0,4
0,6
0,8
1,0
cavity size in units of σ Fig. 3. SPT cavity size distributions calculated using the experimental density of water at 25 C and two different diameters for the particles ˚ , respectively, reported as a function of cavity size 2.8 and 3.2 A ˚ by means of the relation (rc + 1.6)/3.2. expressed in units of r = 3.2 A In this manner, the similarity with the distributions of Fig. 1 should be more evident.
229
The SPT estimates of the work to create a cavity of 4 ˚ diameter in water at 25 C are: 26.0 kJ mol1 for A ˚ and 61.1 kJ mol1 for r = 3.2 A ˚ . The reliabilr = 2.8 A ˚ is readily ity of the SPT value calculated using r = 2.8 A ˚ verified. To create a 4 A diameter cavity: (a) Guillot and Guissani found DGc = 26 kJ mol1 in SPC/E water at 300 K [25]; (b) van Gunsteren and co-workers calculated DGc = 27 kJ mol1 in SPC water at 300 K [26], upon accounting for the transformation from the cavity thermal diameter to the hard sphere one [27]; (c) Pohorille and Pratt found DGc = 27 kJ mol1 in TIP4P water at 300 K [4]. The corresponding cavity insertion probabil˚ , and ity, po = exp(DGc/RT) = 2.8 · 105 for r = 2.8 A 11 ˚ 2.0 · 10 for r = 3.2 A. The latter number is nine orders of magnitude larger than the SPT estimate, 3 · 1020, reported by Sanchez and co-workers [11] to in˚ diameter cavity into an HS fluid with particles sert a 4 A ˚ of 3.16 A diameter and having the same number density of water at 25 C. The origin of the huge discrepancy is because Sanchez and co-workers calculated DGc using the pressure of the HS fluid as determined by SPT [6], 27 600 atm. In this respect, several authors [15,17,18] have indicated that SPT calculations of DGc have to be performed at the experimental pressure of the real liquid, not at the pressure of the HS fluid. For this reason, we used P = 1 atm in SPT calculations.
4. Discussion The present analysis indicates that, if the number density is kept fixed, the cavity size distribution of water can be reproduced by means of SPT assigning to water ˚ . The latter nummolecules an effective diameter of 2.8 A ber corresponds to the location of the first peak in the oxygen–oxygen rdf of water and so to the distance between two H-bonded water molecules [19–21]. This means that H-bonds play an indirect role in determining the cavity size distribution by reducing the size of water ˚. molecules from 3.2 to 2.8 A In other words, the partitioning of void volume in water is determined by the size of the molecules, but the latter has to be considered with care. This is due to the bunching up effect of H-bonds that are so strong to allow the interacting molecules to come closer than their van der Waals size [23]. Even though all the standard computer models of water use a van der Waals ˚ [12,22], the diameter for the oxygen atom of 3.1–3.2 A obtained oxygen–oxygen rdf is always peaked at ˚ , in line with X-ray and neutron scattering meas2.8 A urements [19–21]. The bunching up effect of H-bonds is operative also in liquid methanol and ethanol. For ˚ [31], methanol, the van der Waals diameter is 4.10 A ˚ [1], or 3.83 A ˚ whereas the effective diameter is 3.69 A [32]. For ethanol, the van der Waals diameter is 4.66
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G. Graziano / Chemical Physics Letters 396 (2004) 226–231
˚ [31], whereas the effective diameter is 4.34 A ˚ [1], or A ˚ 4.44 A [32]. Therefore, in order to address the role of H-bonds in determining the cavity size distribution in water, it is necessary, upon turning off the H-bonding potential, to reduce the diameter of LJ particles in order to obtain the same effective size of water molecules and so the same amount of void volume in the simulated liquid. This need was pointed out by Madan and Lee [23], and is now well recognized by scientists working in the field [5,28–30]. In contrast, the corresponding LJ fluid of Sanchez and co-workers differs from the SPC/E water model not only for the absence of the H-bonding potential, but also for the size of the molecules. This is the origin of the finding that SPC/E water has larger cavities than the corresponding LJ fluid, a result readily rationalized when the molecular size is properly taken into account, as shown by means of SPT in Figs. 2 and 3. Since an HS fluid having the same number density of water and consisting of particles of the same size of water molecules is characterized by a cavity size distribution close to that of water, the space occupation in water is largely determined by the effective molecular size. It is important to note that the real situation is more subtle: the two sizes characterizing water molecules operate simultaneously in water. The average coordination number in water at room temperature is about 5 ˚ that are [19–21]. There are 4 water molecules at 2.8 A H-bonded to the central one; and there is also an inter˚ that is not H-bonded to the censtitial molecule at 3.2 A tral one, but occurs in its first coordination shell [33]. This implies that an LJ fluid consisting of molecules having a well defined diameter could not be able to exactly account for the partitioning of void volume in water. Similarly, SPT should be considered in a way unsatisfactory to describe the packing properties of water because it can make use of only one size at a time. Even though this is strictly true, the computer simulation results of both Pohorille and Pratt [4,5], and Madan and Lee [23] indicate that an LJ fluid with particles of ˚ diameter and the same number density of water 2.8 A at 25 C should possess a cavity size distribution close to that of SPC/E or TIP4P water models. This is further supported by the finding that SPT, an approximate theory of HS fluids, is able to reproduce in a satisfactory manner the experimental cavity size distribution determined by Pohorille and Pratt in TIP4P water [7], by ˚ for water molecules. The excluded volusing r = 2.8 A ume effect related to the effective size of water molecules, in part determined by the bunching up effect of H-bonds, plays the pivotal role in the partitioning of void volume in liquid water. Clearly, if the effective size of water molecules were equal to the van der Waals diameter of the oxygen atom, ˚ , water would be dense with n = 0.572. The solu3.2 A
bility of non-polar compounds in this hypothetical dense water would be even poorer than in water, because DGc would be larger, as emphasized by the cavity size distribution determined by Sanchez and co-workers for the corresponding LJ fluid, and the SPT one calcu˚ . The H-bonds, by rendering smaller lated for r = 3.2 A the water molecules and so less costly the process of cavity creation, should aid the solubility of non-polar compounds in water. Similar arguments were already discussed by Lee [9,23] and by us [34]. In conclusion, the present analysis points out that if the H-bonding potential is turned off, the effective size ˚ . An LJ of water molecules increases from 2.8 to 3.2 A fluid obtained by turning off the H-bonding potential and keeping fixed the number density has less void volume than water and cannot be the sole reference fluid to clarify the role of H-bonds in determining the cavity size distribution of water. Another LJ fluid should be considered: after turning off the H-bonding potential, the particle size has to be reduced to the effective diameter of water molecules so to have the same void volume of water. The role played by the diameter assigned to water molecules in determining the cavity size distribution is clarified by means of SPT calculations. The latter show unequivocally that, when the number density is fixed, the molecular diameter plays the fundamental role for the partitioning of void volume and the magnitude of the work of cavity creation in water. The H-bonds play an indirect role by determining the effective size of water molecules. The present results support the small size of water molecules as the origin of the hydrophobic effect [7–9,29,34], but do not support the dense packing effect as proposed by Sanchez and co-workers.
Acknowledgements I thank Dr. B. Lee (Center for Cancer Research, NCI, NIH, Bethesda, MD) for carefully reading many earlier drafts of the manuscript. This work is supported by the COFIN 2002 grant from the Italian Ministry of Instruction, University, and Research (M.I.U.R., Rome).
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