Scaled Particle Theory Study of the Length Scale Dependence of

McQuarry, D. A. Statistical Mechanics; Harper and Row: New York, 1976. .... Dill, K. A.; Bromberg, S. Molecular Driving Forces; Garland Science: New Y...
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J. Phys. Chem. B 2006, 110, 11421-11426

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Scaled Particle Theory Study of the Length Scale Dependence of Cavity Thermodynamics in Different Liquids Giuseppe Graziano* Dipartimento di Scienze Biologiche ed Ambientali, UniVersita` del Sannio, Via Port’Arsa 11-82100 BeneVento, Italy ReceiVed: December 7, 2005; In Final Form: March 9, 2006

It has been noted that the work of cavity creation in water exhibits a crossover behavior, in that its cavity size dependence changes from volume dependence for small cavities to area dependence for larger cavities [Lum, K.; Chandler, D.; Weeks, J. D. J. Phys. Chem. B 1999, 103, 4570]. It is shown here that this behavior can be reproduced using the scaled particle theory in a straightforward manner for six different liquids (water, methanol, ethanol, benzene, cyclohexane, and carbon tetrachloride). It has also been suggested that the crossover is due to a change in the physical mechanism of the process, from one entropy-dominated to another enthalpydominated. However, the crossover behavior can be produced using the scaled particle theory without invoking any change in any physical mechanism. Also, the crossover occurs at a length scale of the size of the liquid molecules, as has been pointed out by others. This is the length regime where the work of cavity creation bears little relation to the bulk liquid surface tension. In addition, it is pointed out that cavity creation can always be considered as a purely entropy-driven process, which is usually accompanied by another process with compensating enthalpy and entropy changes.

Introduction Weeks,1

Lum, Chandler, and LCW, pointed out that, for water at room temperature, the ratio of the work of cavity creation, ∆Gc, to the accessible surface area2 of the cavity, ASA, when plotted against the cavity radius shows an interesting trend. (There are two measures of the cavity size: (a) the radius of the spherical region from which all parts of the liquid molecules are excluded, indicated by rc; (b) the radius of the spherical region from which the centers of the liquid molecules are excluded, indicated by Rc. For spherical cavities, Rc ) rc + r, where r is the radius of the liquid molecules and ASA ) 4πRc2). The function ∆Gc/ASA versus Rc proves to be directly proportional to the cavity radius for small cavities and independent of cavity radius for large cavities (i.e., a large cavity had a radius of at least 10 Å according to LCW). The occurrence of the crossover between small and large cavities in the ∆Gc/ ASA function for water was considered to be the indication of a change in the thermodynamic nature of the cavity creation process, that, in turn, should reflect a change in the microscopic mechanism of cavity creation.3 The Gibbs energy change of cavity creation ∆Gc is a large positive quantity, strongly increasing with cavity radius in all liquids.4 This point is widely recognized, whereas there is no agreement in the scientific community on the enthalpic or entropic nature of the positive ∆Gc function.5 At room temperature in water, for molecular-sized cavities, ∆Sc is a large and negative quantity that determines the sign and magnitude of ∆Gc, so that cavity creation is entropy-dominated. By considering that the creation of a large cavity should correspond to the formation of a water-air interface and that the bulk surface tension of water decreases monotonically with temperature, Chandler3 suggested that, for large cavities, both ∆Hc and ∆Sc * Phone: +39/0824/305133. Fax: +39/0824/23013. E-mail: graziano@ unisannio.it.

should be positive quantities so that cavity creation would be enthalpy-dominated. The molecular-level explanation would be the following. When the cavity is very large, locally, the surrounding water molecules feel it as a flat surface and are not able to maintain a complete H-bonding network.3 Since a significant fraction of H-bonds would be broken, both ∆Hc and ∆Sc should be large and positive. In addition, to reduce the loss of H-bonds, water molecules would move away from the cavity surface producing a dewetting phenomenon, that would be the driving force for the collapse of hydrophobic chains in water and the folding of globular proteins.6 The enthalpy dominance in the work of cavity creation for large cavities in water has not been determined directly by Chandler and co-workers. It was only shown that the value of ∆Gc/ASA, in the limit of a cavity of infinite radius, corresponds to the surface tension of the water model used to perform the computer simulations.7,8 In addition, both the crossover and the dewetting were found by Huang and Chandler9 also in LennardJones liquids, suggesting that the breaking of H-bonds should not be the fundamental cause of the phenomenon. Recently, Rajamani, Truskett, and Garde,10 RTG, investigated the crossover and dewetting phenomena by means of molecular dynamics simulations in SPC/E water11 under different hydrostatic pressures and different temperatures in order to obtain estimates for ∆Hc and ∆Sc. At room temperature and 1 atm, RTG found that ∆Sc is large and negative over the whole cavity size range investigated, making the dominant contribution to ∆Gc even for cavity sizes beyond the crossover length, Rc ≈ 5 Å, determined by themselves. At room temperature and -1000 atm, RTG found enthalpy dominance on increasing the cavity size. Thus, the RTG results10 show that, in water at room temperature and 1 atm, for cavities as large as Rc ≈ 7 Å, there is no change in the thermodynamics of cavity creation. It seems evident that the situation is not settled down, and there is room for further investigations. In this perspective, I

10.1021/jp0571269 CCC: $33.50 © 2006 American Chemical Society Published on Web 05/20/2006

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Graziano

provide an analysis of cavity thermodynamics grounded on a basic statistical mechanical approach12,13 and scaled particle theory,14 SPT. The analysis shows that (a) the work of cavity creation, at constant temperature and pressure, is entirely entropic in all liquids, regardless of cavity size12,15 and (b) the SPT description is qualitatively correct. On this basis, SPT formulas are used to construct plots of ∆Gc/ASA, ∆Hc/ASA, and T‚∆Sc/ASA, at 25 °C, for the following liquids: water, methanol, ethanol, benzene, cyclohexane, and carbon tetrachloride. The SPT results indicate that (a) the crossover in ∆Gc/ ASA occurs in all the considered liquids without invoking any change in any physical mechanism; (b) the crossover is not related to a change in the thermodynamic nature of cavity creation from entropy-dominated to enthalpy-dominated; (c) the crossover occurs at a length scale of the size of the liquid molecules, as already found by LCW1 and RTG10 in the case of water, and by Chandler and co-workers7,9 in the case of Lennard-Jones liquids. Basic Theoretical Description The occurrence of a molecular-sized cavity in a liquid is due to molecular-scale density fluctuations at equilibrium.12,13 Thus, creation of a cavity in a liquid, at constant temperature, pressure, and number of molecules, corresponds to selecting the configurations possessing the desired cavity from the total liquid configurations. To perform such a selection, it is useful to introduce a function ζ(X) that is 1 if there is the desired cavity in the configuration X, and is 0 otherwise;16 the exact position of the cavity is not important due to the spatial homogeneity of the liquid. The selection gives rise to two statistical ensembles: (a) the total configurations of the pure liquid whose probability density function is

Fp(X) ) exp[-H(X)/RT]/

∫ exp[-H(X)/RT] dX ) exp[-H(X)/RT]/∆p (1)

where H(X) is the enthalpy of the configuration X of the liquid and ∆p is the isobaric-isothermal partition function of the pure liquid; (b) the liquid configurations possessing the desired cavity whose probability density function is

Fc(X) ) ζ(X) × exp[-H(X)/RT]/

∫ ζ(X) ×

exp[-H(X)/RT] dX ) ζ(X) × exp[-H(X)/RT]/∆c (2) where ∆c is the isobaric-isothermal partition function of such system. According to statistical mechanics,17 the reversible work of cavity creation is given by

∆Gc ) -RT × ln(∆c/∆p) ) -RT × ln〈ζ(X)〉p

(3)

The ensemble average of ζ(X) is equal to the probability that a spherical region of radius Rc is devoid of solvent molecular centers in the full statistical ensemble of the pure liquid.12,13,15,16 Application of equilibrium statistical mechanics leads to the following expressions for ∆Hc and ∆Sc:

∆Hc )

∫ H(X)[Fc(X) - Fp(X)] dX ) 〈H(X)〉c - 〈H(X)〉p (4)

∆Sc ) -R ×

∫ [Fc(X) × ln Fc(X) - Fp(X) × ln Fp(X)] dX

) R × ln〈ζ(X)〉p + [(〈H(X)〉c - 〈H(X)〉p)/T] ) ∆Sx + (∆Hc/T) (5)

The enthalpy change is due to the difference in the ensemble average enthalpy between the liquid configurations possessing the desired cavity and the total liquid configurations. For the entropy change associated with cavity creation, two contributions can be identified by means of a reliable and careful analysis, and indeed emerge from the statistical mechanical approach: (a) a reduction in the number of configurations because the liquid configurations possessing the desired cavity are only a very small fraction of the total liquid configurations; this reduction of configurations provides a negative contribution to the entropy in all liquids and is called excluded volume contribution, ∆Sx; (b) a qualitative difference between the two sets of liquid configurations because those possessing the cavity would have a different distribution of energy levels; this contribution is due to a purely structural difference in the spatial disposition of liquid molecules and is distinct from the excluded volume contribution; it is given by ∆Hc/T because this is the right measure of the entropy change due to the distribution over different energy levels between two ensembles, one of which is a subensemble of the other.12,15 There are no other entropy sources in the cavity creation process. Equations 4 and 5 show that the cavity enthalpy change is entirely compensated for by the entropy contribution arising from the purely structural reorganization of liquid molecules upon cavity creation. This is because cavity creation is a special process: The liquid configurations possessing the desired cavity are a subensemble of the total liquid configurations. As a consequence, ∆Gc ) -T × ∆Sx is entropic in nature and is due to the excluded volume effect associated with the reduction in the size of the configuration space accessible to liquid molecules.12,18 This result is valid for all cavities, regardless of their size, in all liquids as long as the cavity can be produced by molecular-scale density fluctuations at equilibrium. It is possible that in some liquids the ∆Hc term is larger than the ∆Gc one, masking the basic entropic nature of cavity creation. This fact has led to confusion because ∆Gc seems to be entropy-dominated in water but enthalpy-dominated in organic solvents.5 The confusion is only apparent and is due to the nonappreciation of the present basic approach. Scaled Particle Theory Formulas For cavities characterized by 0 e Rc e r, SPT provides exact relationships for ∆Gc, ∆Hc, and ∆Sc that have already been reported and analyzed.19 For cavities characterized by Rc g r, by neglecting the term of pressure-volume work which is absolutely negligible if the pressure is fixed at 1 atm, as is usually done and suggested on theoretical grounds,20 SPT provides the following expression for ∆Gc:4a,14

∆Gc ) RT × {-ln(1 - ξ) + [3ξ/(1 - ξ)] × (σc/σ) + (u/2)(u + 2)(σc/σ)2} (6) where u ) 3ξ/(1 - ξ); ξ is the volume packing density of pure liquid, which is defined as the ratio of the physical volume of a mole of liquid molecules over the molar volume of the liquid, V (i.e., ξ ) π × σ3 × Nav/6 × V); σ ) 2r is the hard-sphere diameter of the liquid molecules, and σc ) 2rc is the cavity diameter. Since SPT is a hard-sphere theory,14 the SPT formula for ∆Gc has to be purely entropic by definition; it is a measure of the reduction in the spatial configurations accessible to hard spheres upon cavity creation at the given packing density. In applying SPT to a real liquid, one considers a hard-sphere fluid having the same number density of the real liquid on the assumption that attractive interactions among molecules deter-

Length Scale Dependence of Cavity Thermodynamics

J. Phys. Chem. B, Vol. 110, No. 23, 2006 11423

mine the volume occupied by the liquid and so its density, while the size of the molecules determines its packing features.4a,14 Since ∆Gc depends on the molar volume of the liquid in the SPT framework, ∆Hc and ∆Sc can be calculated as appropriate temperature derivatives of ∆Gc at constant pressure4a

∆Hc ) -T2 × [∂(∆Gc/T)/∂T]P ) [ξR × RT2/(1 - ξ)3] × [(1 - ξ)2 + 3(1 - ξ)(σc/σ) + 3(1 + 2ξ)(σc/σ)2] (7) ∆Sc ) -[∂∆Gc/∂T]P ) -R × {-ln(1 - ξ) + [3ξ/(1 - ξ)] × (σc/σ) + (u/2)(u + 2)(σc/σ)2} + [ξR × RT/(1 - ξ)3] × [(1 - ξ)2 + 3(1 - ξ)(σc/σ) + 3(1 + 2ξ)(σc/σ)2] (8) where R is the thermal expansion coefficient of the liquid. Since R represents the NPT ensemble average correlation between enthalpy fluctuations and volume fluctuations of the liquid,21 it is reliable to consider the terms containing R as due to the purely structural reorganization of liquid molecules upon cavity creation. Since ∆Gc(SPT) is entirely entropic by definition, eqs 7 and 8 indicate that the purely structural reorganization of liquid molecules in response to cavity creation is a compensating process, in line with the basic theoretical description. Macroscopic arguments22 suggest that ∆Gc should be proportional to the bulk surface tension γ of the liquid

∆Gc ) 4πRc2 × γ

TABLE 1: Values of Some Physicochemical Properties at 25 °C for the Six Liquids Used in the Present Worka H2O V (cm3 mol-1) σ (Å) ξ F‚103 (molecules Å-3) R‚103 (K-1) βT‚105 (atm-1) ∆vapH (kJ mol-1) ced (J cm-3) γ (J Å-2 mol-1) βT‚γ (Å)  ∆Gc (kJ mol-1) ∆Hc (kJ mol-1) ∆Sc (J K-1 mol-1)

18.07 2.80 0.383 33.33 0.257 4.63 43.9 2294 433.5 0.329 78.5 489 88 -1350

CH3OH C2H5OH C6H6 CC6H12 CCl4 40.73 3.83 0.435 14.79 1.189 12.75 37.8 868 132.9 0.278 32.6 368 344 -81

58.68 4.44 0.470 10.26 1.089 11.53 42.5 682 132.3 0.250 24.3 343 319 -81

89.40 5.26 0.513 6.74 1.240 9.80 33.8 348 169.9 0.273 2.3 324 382 193

108.75 5.63 0.517 5.54 1.214 11.55 33.0 281 148.4 0.281 2.0 292 339 159

97.09 5.37 0.503 6.20 1.226 10.81 32.8 312 159.2 0.282 2.2 293 332 130

a V is the molar volume; σ is the effective hard-sphere molecular diameter; ξ is the volume packing density; F is the number density; R is the thermal expansion coefficient; βT is the isothermal compressibility; ∆vapH is the vaporization enthalpy change; ced ) (∆vapH - V)/RT is the cohesive energy density; γ is the bulk surface tension; the product βT‚γ is the Egelstaff-Widom length;  is the dielectric constant. All values come from refs 23, 25, 26, 27b, and 33. In the last three rows are listed the SPT values of ∆Gc, ∆Hc and ∆Sc for a cavity with rc ) 10 Å at 25 °C and 1 atm in the six liquids.

(9)

where 4πRc2 is the cavity ASA. To increase the contact surface between liquid and air, it is necessary to transfer some molecules from the bulk to the surface by breaking intermolecular interactions.22 If this picture is right, the surface tension should be dominated by an energetic term, with associated compensating enthalpy and entropy contributions due to liquid’s structural relaxation. Therefore, an increase in the liquid-air interface does not cause an entropy penalty due to the reduction in the configuration space accessible to liquid molecules. Since the excluded volume effect is not in action, eq 9 does not seem to provide a qualitatively correct picture of cavity creation physics. Scaled Particle Theory Results Cavity thermodynamics in several liquids have been investigated by means of SPT, at 25 °C and 1 atm, over a large cavity size range (i.e., 0 e Rc e ∼22 Å). The parameters used for SPT calculations are listed in Table 1; the experimental values of both molar volume and thermal expansion coefficient are used for all liquids. The effective hard-sphere molecular diameters selected are as follows: 2.80 Å for water,23 in line with the location of the first maximum in the oxygen-oxygen radial distribution function of liquid water;24 3.83 Å for methanol and 4.44 Å for ethanol, as obtained from the group contributions of Ben-Amotz and Willis;25 5.26 Å for benzene, 5.63 Å for cyclohexane, and 5.37 Å for carbon tetrachloride, as determined by Wilhelm and Battino26 from solubility measurements. The plots of ∆Gc/ASA, ∆Hc/ASA, and T‚∆Sc/ASA for the six liquids at 25 °C are shown in Figures 1-6. It is evident that (i) the crossover in ∆Gc/ASA upon increasing the cavity radius is a common feature for all liquids; (ii) the crossover occurs at a length scale of the size of the liquid molecules where

Figure 1. Water: plot of the functions ∆Gc (full curve), ∆Hc (dashed curve), and T‚∆Sc (dotted curve) normalized with respect to the cavity accessible surface area 4πRc2 versus the cavity radius Rc, calculated by means of SPT at 25 °C and 1 atm. The horizontal line represents the experimental liquid-air surface tension of water at 25 °C. SPT calculations were performed using the experimental density of water at 25 °C and σ ) 2.80 Å (ξ ) 0.383).

the bulk liquid surface tension should be basically irrelevant; (iii) the crossover is produced in a straightforward manner, without invoking any change in physical mechanism. Actually, even though all calculations have been performed with the same SPT formulas, the values of ∆Hc/ASA and T‚∆Sc/ASA change qualitatively passing from one liquid to another. Specifically, (a) in water, ∆Sc is always large and negative, while ∆Hc is always small and positive, so that T‚|∆Sc| . ∆Hc; (b) in methanol and ethanol, ∆Sc is always negative and small, while ∆Hc is always large and positive, so that ∆Hc > T‚|∆Sc| for large Rc; (c) in benzene, cyclohexane, and carbon tetrachloride, ∆Sc is small and negative up to Rc ≈ 5 Å, and then becomes small and positive, while ∆Hc is always large and positive so that ∆Hc . T‚|∆Sc| for large Rc. These results are qualitatively robust to changes in the effective hard-sphere diameter of liquid molecules, as shown in the Supporting Information. Within the SPT framework, cavity creation does appear entropy-dominated or enthalpy-dominated depending on27 (a) the values of σ and ξ that determine the ∆Gc magnitude and

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Graziano

Figure 2. Methanol: plot of the functions ∆Gc (full curve), ∆Hc (dashed curve), and T‚∆Sc (dotted curve) normalized with respect to the cavity accessible surface area 4πRc2 versus the cavity radius Rc, calculated by means of SPT at 25 °C and 1 atm. The horizontal line represents the experimental liquid-air surface tension of methanol at 25 °C. SPT calculations were performed using the experimental density of methanol at 25 °C and σ ) 3.83 Å (ξ ) 0.435).

Figure 4. Benzene: plot of the functions ∆Gc (full curve), ∆Hc (dashed curve), and T‚∆Sc (dotted curve) normalized with respect to the cavity accessible surface area 4πRc2 versus the cavity radius Rc, calculated by means of SPT at 25 °C and 1 atm. The horizontal line represents the experimental liquid-air surface tension of benzene at 25 °C. SPT calculations were performed using the experimental density of benzene at 25 °C and σ ) 5.26 Å (ξ ) 0.513).

Figure 3. Ethanol: plot of the functions ∆Gc (full curve), ∆Hc (dashed curve), and T‚∆Sc (dotted curve) normalized with respect to the cavity accessible surface area 4πRc2 versus the cavity radius Rc, calculated by means of SPT at 25 °C and 1 atm. The horizontal line represents the experimental liquid-air surface tension of ethanol at 25 °C. SPT calculations were performed using the experimental density of ethanol at 25 °C and σ ) 4.44 Å (ξ ) 0.470).

Figure 5. Cyclohexane: plot of the functions ∆Gc (full curve), ∆Hc (dashed curve), and T‚∆Sc (dotted curve) normalized with respect to the cavity accessible surface area 4πRc2 versus the cavity radius Rc, calculated by means of SPT at 25 °C and 1 atm. The horizontal line represents the experimental liquid-air surface tension of c-hexane at 25 °C. SPT calculations were performed using the experimental density of c-hexane at 25 °C and σ ) 5.63 Å (ξ ) 0.517).

(b) the value of R that determines the ∆Hc magnitude. If both σ and R are small, as in the case of water at room temperature (see Table 1), ∆Gc is very large and ∆Hc is very small, thus cavity creation does appear entropy-dominated. If both σ and R are large, as in the case of nonpolar organic liquids at room temperature (see Table 1), ∆Gc is large but ∆Hc is larger, and cavity creation does appear enthalpy-dominated. However, these appearances do not have a real meaning, because eqs 3-5 and 6-8 show that ∆Hc is compensated for by a corresponding entropy contribution, so that ∆Gc is entropic in nature. Water does appear different because (a) its molecules are very small in comparison to those of the other liquids and (b) the correlation between enthalpy fluctuations and volume fluctuations is very small with respect to the other liquids.27 The first thing is an intrinsic feature of water molecules (a water molecule for its size practically corresponds to an oxygen atom), amplified by the bunching-up effect of H-bonds28,29 (i.e., the van der Waals diameter of a water molecule is 3.2 Å, and decreases to 2.8 Å when it is H-bonded to other waters). The second thing is an intrinsic feature of the 3D H-bonding network of liquid water

that is able to absorb a volume increase without a significant change in the number and strength of H-bonds, as a consequence of its plasticity30 that, in turn, is a reflection of the 2-2 tetrahedral H-bonding functionalities of water molecules (R ) 0.257‚10-3 K-1 at 25 °C). For nonpolar liquids, such as benzene, cyclohexane, and carbon tetrachloride, there is a large correlation between enthalpy fluctuations and volume fluctuations, because the strength of van der Waals interactions is particularly sensitive to a volume increase (R ≈ 1.2‚10-3 K-1 at 25 °C); thus, ∆Hc is so large and positive as to overwhelm ∆Gc upon increasing the cavity size (see Figures 4-6). For polar liquids, such as alcohols, even though the molecules are connected by H-bonds, there is no 3D H-bonding network able to neutralize the effect of a volume increase on the internal energy, and R is large (R ≈ 1.1‚10-3 K-1 at 25 °C); thus, ∆Hc is large and positive but does not overwhelm ∆Gc upon increasing the cavity size (see Figures 2 and 3). According to SPT calculations, there is no clear relationship between the ∆Gc magnitude for large Rc values and the bulk

Length Scale Dependence of Cavity Thermodynamics

Figure 6. Carbon tetrachloride: plot of the functions ∆Gc (full curve), ∆Hc (dashed curve), and T‚∆Sc (dotted curve) normalized with respect to the cavity accessible surface area 4πRc2 versus the cavity radius Rc, calculated by means of SPT at 25 °C and 1 atm. The horizontal line represents the experimental liquid-air surface tension of CCl4 at 25 °C. SPT calculations were performed using the experimental density of CCl4 at 25 °C and σ ) 5.37 Å (ξ ) 0.503).

surface tension γ of all the six liquids considered in the present study (see Figures 1-6). This should reflect a quantitative deficiency of the SPT approach to provide an accurate prediction of the bulk surface tension of real liquids. It could also be an indication that the creation of a cavity in a liquid is different from the creation of an interface between liquid and air. Discussion SPT calculations confirm the original result of LCW theory1 that ∆Gc/ASA as a function of Rc is proportional to the cavity radius for small cavities and independent of cavity radius for large cavities. In addition, SPT calculations show that this crossover is a general feature of cavity thermodynamics in all liquids without invoking a change in the physical mechanism of cavity creation. Chandler3 suggested that the small-to-large length scale crossover in the trend of ∆Gc/ASA versus Rc should be a manifestation of the transition from entropy-dominated to enthalpy-dominated cavity thermodynamics in water: The formation of large cavities should require an interfacial thermodynamics framework for its description. These statements would be right if the crossover occurred at a macroscopic length scale. Instead, the crossover occurs at a length scale of the size of the liquid molecules according to both LCW theory,1,7,9 RTG simulations,10 and SPT calculations. This is the length regime where the Gibbs energy of cavity creation bears little relation to the bulk liquid surface tension. The water-cavity interface should resemble the water-air interface only when the cavity contact correlation function had the same value of the water density in the vapor phase in equilibrium with the liquid phase at that temperature (see also Appendix A). This is the idea originally put forward by Stillinger.20a The rightness of this idea is confirmed by the results obtained by Huang and Chandler:8 They found a real dewetting of cavity surface in water only for cavities of about 100 Å radius. Therefore, the small-to-large length scale crossover in the trend of ∆Gc/ASA versus Rc, occurring at a length scale of the size of the liquid molecules, does not seem to require an interfacial thermodynamics framework for its description. RTG10 found that, at room temperature and 1 atm, the crossover length is at Rc ≈ 5 Å, a subnanoscopic length scale using their words, and suggested that “the relatively small

J. Phys. Chem. B, Vol. 110, No. 23, 2006 11425 magnitude of the crossover lengthscale implies that, in general, thermodynamic and structural aspects of large-solute hydration are relevant even for subnanoscopic lengthscale”. However, the experimental hydration thermodynamics of neopentane (i.e., the latter would require a spherical cavity with Rc ≈ 4.5 Å for its accommodation in water) does not show any features of the thermodynamics associated with the formation of a water-air interface:31 The large and positive hydration Gibbs energy change is entropy-dominated at room temperature, and the hydration enthalpy change is large and negative. RTG10 devised also a theoretical procedure to determine the crossover length in water and found an intriguing connection between the crossover length and the Egelstaff-Widom length.32 In this respect, they10 wrote, “The fact that the crossover length is related to the Egelstaff-Widom length scale is physically intuitive, because the latter quantity contains information about both molecular (βT) and macroscopic (γ) solvation physics”. It is not so transparent, however, what type of information on the microscopic physical mechanism of the crossover can be provided by the Egelstaff-Widom length, because both the isothermal compressibility βT and the bulk surface tension γ are macroscopic thermodynamic quantities. For more on the Egelstaff-Widom length, see Appendix B. In conclusion, SPT calculations indicate that the crossover in the trend of ∆Gc/ASA versus Rc, first pointed out by LCW,1 is a general feature of all liquids and occurs, without invoking a change in the cavity creation mechanism, at a length scale of the size of the liquid molecules. The physics of cavity creation should necessarily change when the cavity size becomes macroscopic, but even a cavity of 10-20 Å radius cannot be considered macroscopic by any means. Acknowledgment. I would like to thank Dr. B. Lee (Center for Cancer Research, NCI, NIH, Bethesda, MD) for carefully reading a draft of the manuscript and making useful suggestions. Supporting Information Available: Additional plots of ∆Gc, ∆Hc, and T‚∆Sc for water, methanol, and ethanol (polar solvents). This material is available free of charge via the Internet at http://pubs.acs.org. Appendix A. Cavity-Dewetting in the SPT Framework The cavity contact correlation function G(Rc) is the conditional liquid density just outside a spherical cavity of radius Rc and is given by14

G(Rc) ) -[1/4π × F × Rc2] × (d ln P0/d Rc)

(A1)

where P0 is the probability of finding zero molecular centers in a spherical region of radius Rc, and according to statistical mechanics,14 ∆Gc ) -RT × ln P0. By applying eq A1 to the SPT expressions for ∆Gc, the SPT cavity contact correlation function G(Rc) is23a

G(0 e Rc e r) ) 1/[1 - (4/3)π × F × Rc3]

(A2)

G(Rc g r) ) [1/2π × F × Rc2] × {(u/σ) + [u(u + 2)/σ2]σc} (A3) where F is the number density of the liquid. It is worth noting that G(Rc) is related to the pair correlation function but it is not the same thing. In fact, Stillinger20a wrote, “The shape of the G(Rc) curve is intimately related to the occurrence of contact pairs of solvent molecules at the solute’s exclusion surface.”

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Graziano the bottom of Table 1, indicate unequivocally that the product of the isothermal compressibility times the surface tension of a liquid is a constant quantity also in the SPT framework. It is worth noting that Egelstaff and Widom32 reported eq B3 in their original article. References and Notes

Figure A1. Cavity contact correlation function for water (full curve) and carbon tetrachloride (dotted curve) calculated by means of SPT relationships, eqs A2 and A3, at 25 °C and 1 atm with the parameter values listed in Table 1.

The SPT G(Rc) functions calculated for water and carbon tetrachloride, using the parameter values reported in Table 1, are shown in Figure A1; they emphasize the occurrence of dewetting in both liquids upon increasing the cavity size (i.e., a decrease in the density of liquid molecules at contact with cavity surface). SPT predicts that in water G(Rc) < 1 for cavities larger than 8 Å radius, whereas in carbon tetrachloride, G(Rc) is still slightly larger than 1 for cavities of about 22 Å radius. It has already been pointed out that the SPT G(Rc) function calculated for water using σ ) 2.8 Å does not correspond in a quantitative manner to that determined by means of computer simulations.23a,34 The important point, however, is that SPT relationships work qualitatively well in reproducing both the crossover and dewetting phenomena associated with cavity creation without invoking any change in any physical mechanism. Appendix B. Egelstaff-Widom Length in the SPT Framework It should be interesting to try to verify the existence of the Egelstaff-Widom32 length also in the SPT framework. The SPT expressions for the isothermal compressibility βT and the surface tension γ of a liquid (using the van der Waals surface area of the cavity to define γ) are14

βT(SPT) ) πσ3‚[(1 - ξ)4/ξ(1 + 2ξ)2]/6kT

(B1)

γ(SPT) ) (kT/πσ2) ‚{[3ξ/(1 - ξ)] + [9ξ2/2(1 - ξ)2]} (B2) The product βT(SPT)‚γ(SPT) gives

βT(SPT)‚g(SPT) ) σ‚[(2 + ξ)(1 - ξ)2/4(1 + 2ξ)2] (B3) At first sight, it is not possible to establish if this product is a constant quantity for different liquids. Performing the simple calculations for the six liquids selected in this work, using data listed in Table 1, one obtains the following at 25 °C: βT(SPT)‚ γ(SPT) ) 0.204 Å for water, 0.213 Å for methanol, 0.205 Å for ethanol, 0.191 Å for benzene, 0.200 Å for c-hexane, and 0.206 Å for carbon tetrachloride. These numbers, even though smaller than the experimental ones listed in the fifth row from

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