Association and Dissociation of Nonpolar Solutes in Super-and

The solvent-mediated part of the pair potential of mean force of methane in water was investigated by computer simulations over a wide range of thermo...
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10352

J. Phys. Chem. B 2000, 104, 10352-10358

Association and Dissociation of Nonpolar Solutes in Super- and Subcritical Water Nobuyuki Matubayasi* and Masaru Nakahara Institute for Chemical Research, Kyoto UniVersity, Uji, Kyoto 611-0011, Japan ReceiVed: June 12, 2000; In Final Form: July 26, 2000

The solvent-mediated part of the pair potential of mean force of methane in water was investigated by computer simulations over a wide range of thermodynamic conditions. It is shown that the solvent effect on the association and dissociation of a pair of methane molecules is weakly dependent on the temperature when the density of solvent water is fixed. The solvent-mediated attraction between the nonpolar solutes is strengthened when the solvent density increases. This density dependence is in contrast to that of the solvent-mediated interaction of oppositely charged ions, for which the association is normally strengthened by the density reduction at constant temperature. The effect of water on the formation of a chemical bond in a reaction of nonpolar nature is also examined schematically. It is shown that while supercritical water is a worse environment than ambient water with respect to the solvent effect on the reaction rate constant, it is a better medium for the reaction due to the high-temperature involved.

Introduction Water is recognized as an effective solvent for organic chemical reactions of synthetic, industrial, or environmental importance.1-4 When the system is kept at ambient conditions, however, the low solubilities of organic compounds restrict the utility of water as a medium for organic chemical reactions. Super- and subcritical water is a useful medium to overcome this restriction.5-11 When the temperature is elevated and the (water) density is reduced, nonpolar solutes are no longer “hydrophobic” and dissolve well into water. A characteristic feature of super- and subcritical water is the availability of a wide range of densities and temperatures, and a large variation in the density and temperature may lead to a drastic change in the role of water in an organic chemical reaction. Thus, to understand and control the organic chemical reaction in superand subcritical water, it is essential to establish a molecular picture of the solvent-mediated interaction of nonpolar solutes over a wide range of thermodynamic conditions. At ambient conditions, the phenomena involving nonpolar solutes in water are classified into two categories: hydrophobic hydration and hydrophobic interaction.12-17 Hydrophobic hydration refers to the thermodynamics and structure of the aqueous solution of a single nonpolar entity, and hydrophobic interaction is concerned with the mean force operating between two or more nonpolar solutes in water. A similar classification is obviously useful to analyze the behaviors of nonpolar solutes in hightemperature water. In previous papers, we studied the thermodynamics and structure of hydration of a nonpolar solute over a wide range of (water) densities and temperatures.18,19 On the thermodynamic side, it was shown that the probability of cavity formation is primarily determined by the density of water and is a weak function of the temperature. On the structural side, it was found even at supercritical conditions that a nonpolar solute strengthens the hydrogen bonding between a pair of water molecules in its hydration shell. * To whom correspondence should be addressed.

When the system is at an ambient liquid state, it is widely believed that the uniqueness of the water structure gives rise to hydrophobic interaction and is responsible for the structural stability of proteins, micelles, and membranes.12,13,15 Upon elevation of the temperature, on the other hand, it is recognized that water becomes less unique and acts more as a “nonpolar” solvent, as evidenced from the radial distribution functions and the dielectric constant.18-39 Thus, to reveal the relationship between the water structure and the solvent-mediated interaction between nonpolar solutes, it is insightful to examine the dependence of the pair potential of mean force of a nonpolar solute on the (water) density and temperature of the system. At ambient conditions, the potential of mean force between two spherical nonpolar solutes in water has been extensively studied as a prototype of hydrophobic interaction.12,13,16,17,40-64 In this work, we investigate the solvent-mediated interaction of a pair of nonpolar solutes in water over a wide range of thermodynamic conditions including both ambient and supercritical. Using methane as a model solute, we evaluate the pair potential of mean force by computer simulations and elucidate the role of water in the association and dissociation of the solute as a function of density and temperature. When the density of water is lower and/or the temperature is higher, it is normally observed that the association of oppositely charged ions is stronger.65-72 In this case, the ion-water interaction competes less effectively against the direct ion-ion interaction, and the process of ion pair formation is energetically more stabilized. A markedly different density and temperature dependence is expected, on the other hand, for the association of nonpolar solutes in water. By simulating the aqueous solution of benzene at an ambient state and at a supercritical state with a gaslike density, Gao observed that the association of benzene is weaker at the supercritical state.73 To comprehend the solventmediated interactions of nonpolar solutes in high-temperature water, however, it is desirable to systematically explore a wide range of density and temperature. In this and previous works,18,19,24,74,75 we fix the density of water when varying the temperature and fix the temperature when varying the density.

10.1021/jp002105u CCC: $19.00 © 2000 American Chemical Society Published on Web 10/19/2000

Nonpolar Solutes in Super- and Subcritical Water

J. Phys. Chem. B, Vol. 104, No. 44, 2000 10353

TABLE 1: The Average Pressure of Pure Solvent Water in the Units of Bara temperature (°C) density (g/cm3)

25

100

200

300

400

500

600

1.00 0.80 0.60 0.40 0.20

-b

993 ( 15

2949 ( 20

5109 ( 12 949 ( 5

7332 ( 11 2303 ( 7 663 ( 2 307 ( 2 225 ( 1

9547 ( 4 3672 ( 7 1422 ( 2 691 ( 2 389 ( 1

11714 ( 11 5033 ( 7 2200 ( 5 1081 ( 3 548 ( 1

a The simulation is not carried out at the sets of density and temperature for which the average pressure is left in blank. b At this state, the corresponding simulation in the isothermal-isobaric ensemble is performed at an input pressure of 1 bar.

By doing so, we may treat the density and temperature effects separately and identify the factor governing the solvent effect on the association and dissociation of nonpolar solutes. When a chemical reaction occurs in an aqueous medium, water acts in the reaction as an environment and/or as a reactant. As an environment, water affects the potential of mean force among the reactive species involved in the reaction. In this case, water can be treated as “chemically inert” and the hydration of the reactive species along the reaction coordinate controls the reaction rate and pathway. As a reactant, water needs to be treated as “chemically active” and the water molecule or its ionized species appears as a reactive species in the reaction scheme. When the temperature is high, in particular, water may be a reactive medium even without catalysts due to the enhanced reactivity of the water molecule itself. In this work, we study the association and dissociation of simple nonpolar solutes in water using a nonreactive potential model. Since our model study isolates the role of water as an environment and does not take into account the role as a reactant, it will be helpful for separating the two roles of high-temperature water in organic chemical reactions. Methods The potential of mean force w(r) between two methane molecules in water is the change in the free energy of the solution system when the methane molecules are brought from infinite separation to a distance r. w(r) is commonly treated as a sum of two terms. One is the direct interaction u(r) between the solutes and does not depend on the state of the system. The other is the solvent-mediated (indirect) part ∆w(r) and describes the solvent effect on the association and dissociation of the nonpolar solutes. The radial distribution function between the nonpolar solutes is given by exp(-βw(r)), where β is the inverse of the thermal energy (the product of the Boltzmann constant and the absolute temperature). Since the radial distribution function is simply equal to exp(-βu(r)) in the absence of solvent, the solvent effect on the pair correlation of the solutes is expressed in terms of exp(-β∆w(r)). The potential of mean force w(r) is obtained from computer simulations by calculating the changes in the free energy of the system upon insertion of a methane molecule in pure solvent water and in aqueous solution of methane. To calculate the free energy change of methane insertion in pure solvent water, we performed computer simulations of the pure solvent system and employed the particle insertion method.76-81 In this case, the free energy change is equal to the excess chemical potential ∆µ0 of methane at infinite dilution and is independent of the position of insertion. To calculate the free energy change of methane insertion in aqueous solution of methane, we simulated the solution system containing one methane molecule and implemented the particle insertion method by treating another methane molecule as a test particle. In this case, the free energy change depends on the distance r of the inserted test particle

from the methane molecule originally present in the solution. When the free energy change at the distance r is ∆µ(r), it is obvious that ∆µ(r) approaches the excess chemical potential ∆µ0 for large r. The potential of mean force w(r) between two methane molecules in water is then given by (∆µ(r) - ∆µ0). The pure solvent system was simulated in the canonical ensemble at densities of 1.00, 0.80, 0.60, 0.40, and 0.20 g/cm3. The temperature was set to 25, 100, 200, 300, 400, 500, and 600 °C at the highest density of 1.00 g/cm3, and to 300, 400, 500, and 600 °C at the density of 0.80 g/cm3. For each of the lower densities of 0.60, 0.40, and 0.20 g/cm3, the temperature was taken to be 400, 500, and 600 °C. The intermolecular interaction between water molecules was modeled with the SPC/E potential,82 and the spherical truncation was applied at 9.0 Å. At each set of density and temperature, a Monte Carlo simulation of pure solvent water was performed for one million passes by locating 648 water molecules in a cubic unit cell (one pass corresponds to the generation of 648 configurations). The standard Metropolis sampling scheme was implemented, and the periodic boundary condition in the minimum image convention was adopted.76 A methane molecule was treated as a test particle, and the particle insertion method was employed to calculate its excess chemical potential ∆µ0.76-81 The methanewater interaction was taken to be the Lennard-Jones potential with σMe-O ) 3.45 Å and Me-O ) 0.21 kcal/mol,55 and was also spherically truncated at 9.0 Å. Of course, when the system is large enough, ∆µ0 does not depend on the condition of solute insertion, for example, on whether the solute is inserted in the constant volume condition or in the constant pressure condition. For each simulation of pure solvent water in the canonical ensemble, a corresponding Monte Carlo simulation of aqueous solution of methane was performed in the isothermal-isobaric ensemble. In this case, the temperature of the solution system was taken to be the same as that of the pure solvent system, and the input pressure for the solution system was set to the average pressure of the pure solvent system listed in Table 1. The unit cell was cubic and contained 648 water molecules and one methane molecule fixed at the center of the unit cell. The interaction potentials and the boundary condition were identical to those for the simulations of pure solvent water. When the density of the pure solvent system is 1.00 g/cm3, the simulation of the corresponding solution system was performed for six million passes, and when the density is 0.80 g/cm3 or lower, the simulation of the corresponding solution system was performed for three million passes. The particle insertion method was employed to calculate the free energy change ∆µ(r) upon insertion of another methane molecule at the distance r from the methane molecule fixed at the center of the unit cell. In this case, the methane molecule inserted was treated as a test particle and the Boltzmann factor for its interaction energy with the molecules in the system was averaged to give ∆µ(r). The Lennard-Jones potential with

10354 J. Phys. Chem. B, Vol. 104, No. 44, 2000 σMe-Me ) 3.73 Å and Me-Me ) 0.29 kcal/mol was adopted for the methane-methane interaction,55 to which the spherical truncation was applied at 9.0 Å. Note that ∆µ(r) is independent of the choice of the ensemble when the system is large enough. In the following, we specify the state of the solution system simulated in the isothermal-isobaric ensemble by the (water) density and temperature of the corresponding pure solvent system, rather than by the input pressure and temperature. The potential of mean force w(r) or its solvent-mediated part ∆w(r) has been commonly calculated by the thermodynamic perturbation method and the thermodynamic integration method.76 In these methods, since ∆w(r) is equal to zero at infinite separation of the solutes, its calculated values are reliable when the distance r is large. Because of the propagation of errors, however, the error in ∆w(r) becomes larger as r is reduced, and it is inconvenient to focus on the behavior of ∆w(r) at short distances. Furthermore, since a molecular simulation always needs to be performed in a finite system, it is sometimes problematic to find an appropriate r beyond which ∆w(r) can be safely set to zero. In this work, we implemented the particle insertion method in calculating the potential of mean force. Although the evaluation of ∆w(r) at large distances is not statistically desirable in the particle insertion method, a reliable calculation of ∆w(r) is possible when the region of solvent exclusion by the inserted solute molecule overlaps significantly with that by the solute molecule originally present in the solution. In the present work, the SPC/E model was adopted as the intermolecular potential function between water molecules at all the thermodynamic states of interest. The critical point of this water model was determined by Guissani and Guillot and was found to be close to the experimental value.27 In real water, the dipole moment of a water molecule is a function of the thermodynamic state and is smaller at lower densities and/or higher temperatures. The SPC/E model is not very realistic, however, in that the state-dependent nature of the dipole moment is not incorporated. The state dependence of the dipole moment can be implemented by employing a polarizable model or an effective potential model with a modified dipole moment.24,33,36,38,39,83-91 Our use of the SPC/E model over a wide range of thermodynamic conditions is justified because the objective of this work is to elucidate the general trend of the density and temperature dependence of the potential of mean force between nonpolar solutes in water. A further justification is also provided in ref 24, which used experimentally determined proton chemical shifts to estimate that even at the low- to medium-density states of the supercritical region, the value of the dipole moment is closer to that in the SPC/E model than to that in the dilute gas phase.24,74,75 Results and Discussion Overall Trend of the Potential of Mean Force. The potential of mean force w(r) between two methane molecules in water is a sum of the direct interaction u(r) between the solutes and the solvent-mediated (indirect) part ∆w(r). Since u(r) is independent of the state of the system in this work, the solvent effect on the association and dissociation of the solutes is described in terms of ∆w(r). The physically accessible range of the methane-methane distance r is limited by the direct interaction u(r) due to the harshly repulsive nature of u(r) at short distances. In Figure 1(a), we show u(r) and w(r) in the physically accessible region of r at an ambient state of 1.0 g/cm3 and 25 °C, and at supercritical states of 1.0, 0.6, and 0.2 g/cm3 and 400 °C. It is seen at a fixed density of 1.0 g/cm3 that the

Matubayasi and Nakahara

Figure 1. (a) The direct interaction u(r) and the potential of mean force w(r) between two methane molecules in water at an ambient state of 1.0 g/cm3 and 25 °C, and at supercritical states of 1.0, 0.6, and 0.2 g/cm3 and 400 °C. The dotted line represents the direct interaction u(r), the dashed line represents the potential of mean force w(r) at the ambient state, and the solid lines represent w(r) at the supercritical states. The variation of w(r) against the distance r at the supercritical states involves a larger amplitude when the density is higher. (b) The normalized form β∆w(r) of the solvent-mediated part of the potential of mean force at the density of 1.0 g/cm3 and the temperatures of 25, 100, 400, and 600 °C, and at the densities of 0.6 and 0.2 g/cm3 and the temperatures of 400 and 600 °C.

minimum of w(r) corresponding to the contact pair (r ≈ 3.7 Å) deepens with the temperature elevation. In other words, the contact pair of the nonpolar solutes is stabilized at higher temperatures when seen in terms of the free energy change w(r). When the temperature is fixed, on the other hand, the density reduction flattens the variation of w(r) against the distance r and diminishes the detailed structure of w(r). The density and temperature dependence of w(r) observed in Figure 1(a) is entirely the effect of the solvent-mediated part ∆w(r) because the direct interaction u(r) between the solutes does not depend on the thermodynamic state. To compare the association and dissociation of the solutes at various temperatures in terms of the potential of mean force w(r), it is actually necessary to treat βw(r) explicitly, rather than w(r) itself, where β is the inverse of the thermal energy. Indeed, the radial distribution function between the solutes is given by exp(-βw(r)), and the potential of mean force appears in the form of βw(r) to express the equilibrium constant for the association and dissociation of the solutes. Correspondingly, the solvent effect on the association and dissociation needs to be described by the normalized form β∆w(r) of the solventmediated part of the potential of mean force. In Figure 1(b), we show β∆w(r) at the density of 1.0 g/cm3 and the temperatures of 25, 100, 400, and 600 °C, and at the densities of 0.6 and 0.2 g/cm3 and the temperatures of 400 and 600 °C. When the

Nonpolar Solutes in Super- and Subcritical Water

Figure 2. β∆w(r) for the contact pair (r ) 3.7 Å) as a function of the temperature T. When not shown, the error bar is smaller than the size of the corresponding symbol.

temperature is fixed, the solvent-mediated attraction of the nonpolar solutes is stronger at a higher density. This density dependence is in marked contrast to that of oppositely charged ions and of a pair of water molecules, for which the association is normally strengthened by the density reduction at constant temperature.18,24,65-72 When the density is fixed, the temperature variation in Figure 1(b) does not change the overall dependence of β∆w(r) on the methane-methane distance r. In the hightemperature regime, in particular, β∆w(r) is essentially a function of the density and is insensitive to the temperature variation. Contact Pair Formation. To model physical aggregates such as micelles in high-temperature water,92,93 it is essential to examine the density and temperature dependence of β∆w(r) at the contact distance of the nonpolar solutes. The distance of physical contact for a pair of methane molecules is r ≈ 3.7 Å. In this section, we focus on the solvent effect on the contact pair formation of the nonpolar solutes in water over a wide range of densities and temperatures. In Figure 2, we show β∆w(r) for r ) 3.7 Å as a function of the temperature T at each density simulated. When the temperature is varied between 25 and 100 °C at the highest density of 1.0 g/cm3, it is seen, in agreement with simulations by Skipper and Lu¨demann et al.,57,63,64 that β∆w(r) becomes more attractive (negative) with T. In other words, the association of the nonpolar solutes in liquid water is strengthened by the moderate temperature elevation at a fixed density. When the temperature exceeds 100 °C, the T dependence of β∆w(r) is significantly weaker.94 Indeed, while the temperature elevation from 25 to 100 °C changes β∆w(r) by ∼0.5, β∆w(r) at a temperature between 100 and 600 °C differs from that at 100 °C within ∼0.2, which in turn means that the solvent contribution to the equilibrium constant for the contact pair formation of the nonpolar solutes above 100 °C deviates within ∼20% from that at 100 °C. In this sense, the solvent effect on the contact pair formation responds weakly to the temperature variation above 100 °C. In the temperature range between ∼100 and ∼250 °C, Lu¨demann et al. also observed at 1.0 g/cm3 that β∆w(r) of the contact pair formation is essentially independent of the temperature.63,64 Figure 2 then shows that the weak temperature dependence at the liquidlike density is valid even up to supercritical conditions. In a previous paper, it was seen at 1.0 g/cm3 that water retains the tetrahedral structure below ∼200 °C and involves the simple liquid-like structure above ∼200 °C.18 Therefore, although the drastic transition in the water structure is evidenced in the elevation of the temperature from

J. Phys. Chem. B, Vol. 104, No. 44, 2000 10355

Figure 3. β∆w(r) for the contact pair (r ) 3.7 Å, the open symbols) and for the chemical bond (r ) 1.5 Å, the filled symbols connected by the dashed line) as functions of the density F at a temperature of 400 °C. When not shown, the error bar is smaller than the size of the corresponding symbol. The solid line represents the linear fit of β∆w(r) for the contact pair which passes β∆w(r) ) 0 at F ) 0.

a moderate to a supercritical, this temperature elevation leads only to a weak variation in the solvent effect on the formation of a contact pair of the nonpolar solutes. In the supercritical region, when the temperature is increased from 400 to 600 °C, the change in β∆w(r) of the contact pair formation is only less than ∼0.1 at each density shown in Figure 2 (note that the change in β∆w(r) by ∼0.1 corresponds to a ∼10% variation in the equilibrium constant for the contact pair formation). Therefore, over the commonly accessible range of temperature, the solvent contribution to the contact pair formation is primarily controlled by the density, and the temperature is effective only to tune the contribution from the solute-solute direct interaction.95 In Figure 3, we show β∆w(r) for the contact pair (r ) 3.7 Å) as a function of the density F at a supercritical temperature of 400 °C. It is easy to see that the magnitude of β∆w(r) increases monotonically with F, which extends Gao’s observation that the association of benzene is weaker at a supercritical state with a gaslike density than at an ambient state.73 Actually, β∆w(r) is linear in F with deviations of less than ∼0.1.96,97 In this case, since the temperature elevation from 100 to 400 °C at 1.0 g/cm3 changes β∆w(r) by ∼0.1 as shown in Figure 2, β∆w(r) at a lower density in the super- and subcritical region can be reproduced, within an error of ∼0.2, simply by scaling β∆w(r) at a moderate temperature of 100 °C with the density. It is therefore reasonable, at least as a first approximation, to model the solvent effect on the aggregation of nonpolar groups in super- and subcritical water from the density scaling of the well-established data at ∼100 °C in the liquid state. Formation of a Chemical Bond. The solvent effect on a bond-forming (or breaking) reaction is governed by β∆w(r) around the corresponding bond distance r. The distance of the carbon-carbon (single) bond is r ≈ 1.5 Å. Since β∆w(r) at r ≈ 1.5 Å mimics the solvent effect on the conversion of two methane molecules into a single ethane molecule, the difference between the excess chemical potential of ethane and twice the excess chemical potential of methane was employed by BenNaim as an indicator for hydrophobic interaction at ambient conditions.13,14,16 In this section, we focus on the solvent effect on the hypothetical “dimerization” reaction of methane in water over a wide range of densities and temperatures. As in the case of the contact pair (r ≈ 3.7 Å), Figure 1(b) evidences that the solvent effect on the formation of a chemical

10356 J. Phys. Chem. B, Vol. 104, No. 44, 2000 bond depends weakly on the temperature when the water density is fixed. The density dependence of β∆w(r) for the bond formation (r ) 1.5 Å) is shown in Figure 3 at a temperature of 400 °C. According to Figure 3, the density reduction is favorable for the dissociation of the nonpolar solutes, and β∆w(r) is more sensitive to density variation at a higher density. When the state represented by the water density F and the temperature T is moved from an ambient state (F ≈ 1.0 g/cm3 and T ≈ 25 °C) to a commonly accessible supercritical state (F j 0.7 g/cm3 and T ≈ 400 °C), Figures 1(b) and 3 show that β∆w(r) increases by an amount between ∼3 and ∼6 for the hypothetical “methane dimerization” reaction which involves the formation of a carbon-carbon bond (r ≈ 1.5 Å). In this case, the β∆w(r) contribution to the reaction barrier for the bond formation also increases by a similar amount, which in turn corresponds to the reduction in the reaction rate constant by a factor between ∼20 and ∼500. If the reaction barrier is 10 kcal/ mol in the absence of solvent, however, the reaction without solvent will be accelerated by a factor of more than 104 just upon the temperature elevation from 25 to 400 °C (the acceleration factor is larger when the barrier is higher). Therefore, while supercritical water is a worse environment than ambient water with respect to the solvent effect on the reaction rate constant, it is a better medium for the reaction due to the high temperature involved.98 Of course, the enhanced solubilities of nonpolar solutes are another feature of supercritical water as an effective solvent for organic chemical reactions. In a recent paper, Harano et al. performed a RISM-SCF study of a Diels-Alder reaction at an ambient state of 1.0 g/cm3 and 25 °C and at a supercritical state of 0.6 g/cm3 and 600 °C.99 It was then shown that the reaction is of nonpolar nature and that the solvent stabilization of the bond-forming process is less effective at the supercritical state than at the ambient state. In this case, the mechanism giving rise to a larger reaction rate constant at the supercritical state is similar to that described above for the hypothetical “methane dimerization” reaction. When the system is moved from an ambient state to a commonly accessible supercritical state, the accompanying change in the solvent-independent contribution (solute-solute direct interaction part) is responsible for the increase in the reaction rate constant of the bond-forming reaction and is partially canceled by the change in the solvent contribution. Water acts in a chemical reaction as an environment and/or as a reactant. As an environment, water affects the potential of mean force among the reactive species involved in the reaction. As a reactant, the water molecule or its ionized species appears as a reactive species in the reaction scheme. The present work isolates the role of water as an environment and does not take into account the role as a reactant. Since water is a reactive reagent, in particular at high temperatures, the role of water as a reactant needs to be treated explicitly when the variation of the thermodynamic state from an ambient to a supercritical accelerates a bond-forming reaction of nonpolar nature by a factor larger than that expected in the absence of solvent.

Matubayasi and Nakahara

Conclusions

when the independent variables controlling the solution system are taken to be the solvent density and temperature, not the pressure and temperature. The association of nonpolar solutes is strengthened when the density increases at constant temperature. This density dependence is in contrast to that of the association of oppositely charged ions and of a pair of water molecules.18,24,65-72 Thus, the solvent density dependence of the association and dissociation of solutes may be a useful indicator to classify whether the solvent-mediated interaction of the solutes is of nonpolar nature or of ionic (polar) nature. When the density decreases and/or the temperature increases, water becomes less unique as the hydrogen bond network is destroyed and the dielectric constant is reduced.12,20,24,74,75 Since the strong hydrogen bond network and the large dielectric constant are major unique characteristics of ambient water, water is viewed as more “oil-like” at lower densities and/or higher temperatures. In ref 18, however, it was seen that the increased oil-like nature of high-temperature water does not ensure the enhanced affinity (solubility) of a nonpolar solute for water. With respect to the affinity (solubility), the nonpolar solute is still “hydrophobic” at high temperatures when the density of water is in the liquidlike regime. In the present paper, we observed that the solvent-mediated attraction of nonpolar solutes in water is not appreciably weakened by the temperature elevation when the solvent density is fixed at a liquidlike value. In other words, the hydrophobic interaction is present in superand subcritical water with the liquidlike density in the sense that high-temperature and high-density water is not inferior to ambient water as a medium to promote the association of nonpolar solutes. Under the solute-solvent interaction potential treated in the present work, the effect of temperature variation at a fixed density on the solvent contribution to the equilibrium constant for the association and dissociation of the nonpolar solutes is related only to the corresponding change in the oxygen-oxygen radial distribution function of pure solvent water when the approximate integral equation in the energetic representation is employed.100 In subsequent work, we analyze the temperature dependence of the pair potential of mean force of a nonpolar solute in water within the framework of the integral equation in the energetic representation. To distinguish the nature of the solvent-mediated interactions of nonpolar solutes in high-temperature water and in ambient water, it will be useful to decompose the potential of mean force into the enthalpic and entropic components. According to ref 18, while the thermodynamics of hydration of a nonpolar solute is dominated by the entropic component at ambient conditions, it is enthalpic at high-temperature conditions. It is then expected that the solvent-mediated attraction of nonpolar solutes is enthalpy-driven in high-temperature water, whereas it is established at ambient conditions that the hydrophobic interaction is driven by the entropic component.12,13,54,55 Therefore, although the association and dissociation of nonpolar solutes is determined by the accompanying free energy change (potential of mean force), its connection to the structure of solvent water will be made clearer by studying the accompanying changes in the enthalpy and entropy.

The potential of mean force between two methane molecules in water was examined over a wide range of thermodynamic conditions. It was found that its solvent-mediated part normalized relative to the thermal energy (the product of the Boltzmann constant and the absolute temperature) is weakly dependent on the temperature when the density of water is fixed. In this case, a simple description will be possible for the solvent contribution to the association and dissociation of nonpolar solutes in water

Acknowledgment. This work is supported by the Research Grant-in-Aid from the Ministry of Education, Science, and Culture (No. 10304047) and by CREST (Core Research for Evolutional Science and Technology) of Japan Science and Technology Corporation (JST). N.M. is also grateful to the Research Grant-in-Aid from the Ministry of Education, Science, and Culture (No. 11740322) and to the Supercomputer Labora-

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10358 J. Phys. Chem. B, Vol. 104, No. 44, 2000 density derivative, it is sensible to state that the change in β∆w(r) upon the temperature elevation from 400 to 600 °C at a fixed density of 0.6 g/cm3 is smaller than that upon the density variation from 1.0 to 0.6 g/cm3 at a fixed temperature of 400 °C. (96) The linear dependence of β∆w(r) on the density F seen in Figure 3 is valid in the contact region expressed as 3.4 Å j r j 4.1 Å. The linearity is illustrated only at r ) 3.7 Å for brevity. (97) When the density is varied by ∼0.02 g/cm3 at ∼1.0 g/cm3, Lu¨demann et al. observed that β∆w(r) of the contact pair formation at ∼80 °C does not exhibit a statistically significant change.63,64 In contrast, Dang reported that the same density variation at ∼60 °C gives rise to a ∼30% change in β∆w(r).58 If the linear dependence of β∆w(r) on the density seen

Matubayasi and Nakahara in Figure 3 at supercritical conditions is valid at moderate conditions, a 2% variation in the density will lead simply to a 2% change in β∆w(r) and the result by Lu¨demann et al. is supported. This does not mean, of course, that the pressure effect is unimportant in the association and dissociation of the nonpolar solutes in water. Figure 3 simply indicates, in the present level of precision, that the density variation by ∼0.02 g/cm3 is not large enough to cause a statistically significant change in β∆w(r). (98) Our arguments are actually valid within the framework of the transition state theory. (99) Harano, Y.; Sato, H.; Hirata, F. J. Am. Chem. Soc. 2000, 122, 2289. (100) Matubayasi, N.; Nakahara, M. J. Chem. Phys. 2000, 113, 6070.