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2008, 112, 1049-1052 Published on Web 01/09/2008
Solvation in Mixed Aqueous Solvents from a Thermodynamic Cycle Approach Prateek P. Shah and Christopher J. Roberts* Department of Chemical Engineering, UniVersity of Delaware, 150 Academy Street, Newark, Delaware 19716 ReceiVed: July 23, 2007; In Final Form: December 13, 2007
A novel approach is presented for interpreting and potentially predicting values of the isothermal, isobaric transfer free energy, entropy, and enthalpy (∆µtr2 , ∆str2 , and ∆htr2 ) for a solute between water and watercosolvent mixtures. The approach explicitly accounts for volumetric properties of the solvent and solute (the equation of state, EoS) and casts the overall transfer process as a thermodynamic cycle with two stages: (1) isothermal solvent exchange from pure water to the cosolvent composition of interest at fixed mass density; (2) isothermal expansion or compression at the final solvent composition to recover the pressure of the initial state. Using molecular simulations with methane as the solute, the analysis is illustrated over a wide range of cosolvent concentrations for sorbitol-, ethanol-, and methanol-water binary mixtures. The EoS contribution semiquantitatively or quantitatively captures ∆µtr2 , ∆str2 , and ∆htr2 in almost all cases tested, highlighting the importance of considering the effects of changes in solvent density on the overall transfer process. The results also indicate that apolar solvation at these length scales is dominated by the work of cavity formation across a range of cosolvent species and concentrations.
I. Introduction The excess chemical potential of dissolved solutes in mixed aqueous solvents influences or controls a range of fundamentally and practically important processes, including solubility and bioavailability of small-molecule pharmaceuticals,1 the fate of organic compounds in the environment,2 protein-ligand binding,3 and protein folding.4,5 The term mixed aqueous solvent typically denotes a liquid mixture of water and a miscible or highly soluble cosolvent or osmolyte. In the Scatchard notation, water and the cosolvent are denoted as components 1 and 3, respectively, with 2 denoting the dissolved solute of interest.6 Common cosolvents include alcohols, polyhydroxy compounds such as sugar alcohols and carbohydrates, natural osmolytes such as amino acids and urea, as well as organic and inorganic salts.5,7 Transfer free energies (∆µtr2 ) are defined at infinite dilution of 2 as the difference between the solute chemical potential in the water-cosolvent binary mixture at a given temperature (T), pressure (p), and composition (e.g., molality, m3), and the corresponding solute chemical potential in pure water at that T and p.5,8 For any of the applications noted above, ∆µtr2 values provide the relevant equilibrium constants relative to pure water, while the transfer enthalpy (∆htr2 ) and/or entropy (∆str2 ) values provide the associated temperature dependences. Solvation thermodynamics and transfer free energies for mixed aqueous solvents have historically been interpreted based on a number of theoretical approaches. These include additive free energy or group-contribution models,9-11 Kirkwood-Buff theory and related models,4,12-18 and scaled particle theory and related approaches.19-23 These often provide useful molecular-scale insights into the physics and important contributions to the * Corresponding author. E-mail:
[email protected]. Phone: 302-831-0838. Fax: 302-831-1048.
10.1021/jp075783q CCC: $40.75
overall solvation process. Common limitations of most models include their reliance on solvent-specific, parameterized model coefficients9-11,21-24 or their need for accurate molecular paircorrelation functions or excess surface quantities.13-18,25 The former require experimental (excess) free energy values or differences for each solute-water-cosolvent combination at the T, p, and composition of interest. The latter are experimentally difficult or intractable to determine for most practical solutes except through a posteriori analysis of ∆µtr2 data.17 An alternative approach is presented here to aid in interpreting and potentially predicting the thermodynamics of solute transfer. It casts the transfer process as a two-stage thermodynamic cycle (Scheme 1) that explicitly accounts for isobaric changes in solvent density as cosolvents are added to water. It is inspired by recent results from molecular simulations of methane solvation in methanol- and sorbitol-water mixtures,26 and by earlier work for apolar solvation in pure water that highlights the importance of the contributions from the solvent equation of state (EoS), i.e., the density and thermodynamic response functions of the neat solvent.27-30 It is distinct from earlier approaches27-31 in that it focuses on the transfer process from water to a mixture, as opposed to the solute excess chemical potential in pure water or the solvation enthalpy in mixed aqueous solvents relative to an ideal gas. The approach is a general application of classical thermodynamics, and does not require molecular-scale information such as available space and solute excluded volume,24,26,32 or radial distribution functions and estimates of surface excess properties or compositions.13,14,18 It permits a portion of the transfer free energy to be determined simply from the binary solvent equation of state and the partial molar volume of the solute. The thermodynamic cycle analysis also provides quantitative and qualitative assessments of the degree to which the solvent EoS, © 2008 American Chemical Society
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Letters
SCHEME 1: Thermodynamic Cycle Separating the Isobaric, Isothermal Process of Transferring a Solute (2) at Infinite Dilution from Water (1) to a Water-Cosolvent (1-3) Mixture into Contributions from Solvent Exchange (SX) and Equation of State (EoS) Steps
as opposed to solvent rearrangement and direct solute-solvent interactions, influences solvation thermodynamics in mixed aqueous solvents relative to pure water. Finally, the EoS contribution provides a fundamental link between the thermodynamic effects of hydrostatic pressure and changes in cosolvent composition. II. Methods The isothermal, isobaric transfer of 2 at infinite dilution (N2 f 0) from water to a binary water-cosolvent mixture is described by the thermodynamic cycle depicted in Scheme 1; Ni denotes the number of molecules of component i, and V is the system volume. The first step (solvent exchange, SX) is exchange of water molecules with cosolvent molecules at fixed (N2 f 0, T, V) such that the resulting solvent composition is that for the final state in the overall transfer process, but the mass density remains fixed at that of pure water at the initial T and p. This exchange step is performed at fixed mass density (Fm) rather than fixed number density (F). For example, one sorbitol molecule must be exchanged for multiple water molecules due to the much larger mass and size of sorbitol compared to water. If the exchange were instead performed at constant number density, it would typically result in a significant increase in mass density and ultimately lead to unphysical volume fractions for most cosolvents of practical interest. The SX step does not formally rely on maintaining a fixed V, only fixed Fm, but it is shown in Scheme 1 with fixed V for ease of illustration. The results presented here are obtained by interpolating simulated state points across a range of Fm values at fixed composition and temperature, including those that correspond to the SX step in Scheme 1 for each cosolvent species and final composition (see also the Supporting Information). In addition, step one in Scheme 1 is performed at constant mass density for the following reason. Our previous results on methanol-water and sorbitol-water26 indicated that µex 2 values across a range of solvent composition, temperature, and pressure can be described by a common curve that depends primarily on packing fraction. Our results here for ethanolwater confirm this behavior (see the Supporting Information) and indicate that in many cases the SX step would add a small or negligible contribution to the transfer process if one could perform it at a constant packing fraction. However, packing fraction can be difficult to define unambiguously for experimental systems because one typically does not have a unique and rigorous method to experimentally assign the effective hardparticle or molecular volume of each species in the mixture as a function of solvent composition, temperature, and (number) density. For solvent molecules composed primarily of carbon and/or heteroatoms typical of aqueous and organic solvents (e.g., groups III to VIII in periods two and three), atomic volume scales essentially linearly with atomic mass.33 Thus, constant
Figure 1. Free energy, enthalpy, and entropy changes for the process of isothermal, isobaric (p ) 1 atm) transfer of methane from water to different cosolvent-water mixtures. The panels represent different cosolvent identities and/or temperatures; from top to bottom: sorbitol (320 K), ethanol (320 K), methanol (320 K), methanol (260 K). Colors represent different concentrations (numbers in units of w/w% cosolvent): 4 (red), 8 (yellow), 16 (blue), 30 (orange), 60 (green). Units on all vertical axes are cal mol-1.
mass density offers an experimentally tractable approximation for constant packing fraction. It is used here to illustrate a viable approach to quantitatively account for the effects of changes in solvent density and packing fraction on solvation thermodynamics, without requiring knowledge of the packing fraction or void distribution in the solvent. The exchange of cosolvent species in the SX step, as defined in Scheme 1, results in a pressure increase or decrease to p* in the intermediate, hypothetical mixture state. The second step (equation of state, EoS) is a density expansion (compression) at fixed T and m3 to reduce (raise) the pressure such that the overall transfer process is isobaric. The overall ∆µtr2 is then the sum of the change in solute chemical potential for the first and EoS second steps, ∆µSX 2 and ∆µ2 , respectively. The analogous tr process holds for ∆h2 and ∆str2 . Solute excess chemical potential values (µex 2 ) for methane were determined via Widom insertion during canonical ensemble molecular dynamics simulations of the initial, intermediate, and final solvent states in Scheme 1 at selected temperatures over a range of binary cosolvent compositions (p ) 1 atm for initial and final states). The cosolvents were methanol, ethanol, and sorbitol. Details of the simulation protocols are given in ref 26 and in the Supporting Information. ∆µSX and ∆µEoS were 2 2 between the calculated from the respective differences in µex 2 states according to Scheme 1. The enthalpic and entropic contributions (∆hR2 , ∆sR2 ; R ) tr, SX, and EoS) were calculated by numerical evaluation of the appropriate temperature derivatives of µex 2 corresponding to initial, final, and intermediate state points in Scheme 1. III. Results and Discussion Figure 1 shows the values of ∆µtr2 , ∆htr2 , and ∆str2 as a function of cosolvent concentration for each cosolvent type, including high and low temperatures for the case of methanol-
Letters
Figure 2. Equation of state contribution vs the overall transfer value for (A) free energy and (B) enthalpy and entropy. Each point corresponds to one of the state points in Figure 1. Dashed lines and the degree to which the EoS contributions capture the overall process are discussed in the text. The negative values of T∆str2 and T∆sEoS are 2 plotted to place them on comparable scales with the corresponding ∆ htr2 and ∆hEoS values. 2
water. The results qualitatively and semiquantitatively reproduce the experimental behavior for methane solubility in alcoholwater mixtures.34-36 While no comparable experimental data for sorbitol-water mixtures were found, the behavior is similar to that with other sugars.8,26,34 Figure 2 shows the EoS contributions to ∆µtr2 (Figure 2A), ∆htr2 (Figure 2B), and ∆str2 (Figure 2B) for each set of conditions in Figure 1. They are each plotted with respect to the corresponding value for the overall process in order to illustrate the degree to which the EoS contribution alone captures the overall process. The value of the SX contribution follows by difference. The dashed lines in Figure 2 correspond to yEoS ) ytr (y ) ∆µ2, ∆h2, or ∆s2) and indicate points for which the EoS contribution quantitatively determines the overall transfer process. The large majority of points in Figure 2A fall above this curve in the first quadrant and below it in the third quadrant. This illustrates that for most points the EoS and SX contributions are opposite in sign, and the EoS contribution overestimates the magnitude but determines the sign of ∆µtr2 , ∆htr2 , and ∆str2 . The only conditions for which the EoS contribution is qualitatively incorrect are for ∆µtr2 under the low-concentration, low-temperature conditions of
J. Phys. Chem. B, Vol. 112, No. 4, 2008 1051 methanol-water solutions (fourth quadrant in Figure 2A). The SX contribution is positive (∆µSX 2 > 0) in this case and slightly < 0) to yield dominates over the EoS contribution (∆µEoS 2 small positive ∆µtr2 values (, kBT; kB ) Boltzmann’s constant). Previous work26 suggests this behavior is a result of an unusual dependence of packing fraction on solvent composition that may be particular to low-concentration alcohol-water solutions. This behavior is not observed here for ethanol-water solutions, but experimental data36 and the results in ref 26 suggest it may be possible at lower temperatures. Figure 2A shows that ∆µtr2 is semiquantitatively captured by for sorbitol-water and high-temperature alcohol-water ∆µEoS 2 solutions. Figure 2B shows that ∆htr2 and ∆str2 are semiquantiand ∆sEoS for tatively or quantitatively captured by ∆hEoS 2 2 essentially all conditions in Figure 1. Together, the results in Figures 1 and 2 indicate the EoS contribution is a significant if not dominant factor in the thermodynamics of the transfer process for small apolar solutes for the range of cosolvent types and concentrations tested here. Essentially the same results are found for a solute with a purely repulsive soft-sphere potential and the same Lennard-Jones (LJ) diameter as that for methane. The soft-sphere µex 2 values (see the Supporting Information) account for 95% or more of the corresponding values for the full LJ potential over all conditions tested here. This clearly indicates the work of cavity formation is the dominant contribution to the solvation thermodynamics and transfer process, in keeping with observations for apolar molecular solvation in pure water,28,37-39 and in contrast to common interpretations of water-cosolvent solvation that rely on differences in solutewater and solute-cosolvent interactions for solutes such as hydrophobic side chains of proteins.9,40,41 The excess enthalpy (entropy) for insertion of methane or its soft-sphere equivalent is relatively large and positive (negative) under almost all conditions reported here (data not shown), as expected when cavity formation dominates the solvation process. The classical thermodynamic cycle analysis as depicted in Scheme 1 does not provide explicit relationships between molecular-scale details of cavity formation and the EoS contributions. Rather, its effectiveness is based on the idea that for molecular solutes the solvent mass density is a reasonable surrogate for packing fraction, and that the work of cavity formation is predominantly determined by the solvent packing fraction rather than the nature of solute-solvent interactions. The resulting picture is reminiscent of the EoS view of molecular solvation in pure water27-30 in that it is obtained without a need to invoke or have knowledge of changes in hydrogen bonding or structure of the solvent. It is potentially powerful in that it does not require knowledge or prediction of the relative strengths of interactions between solute, water, and cosolvent species.4,5 This is not to argue that details of the interactions among water, solute, and cosolvent are unimportant. They implicitly determine the density change and possibly also influence the partial molar volumetric properties of the solute (see below). They also inherently contribute to the SX step. The present results highlight the importance of incorporating the effects of solvent density changes into interpretations of experimental data and into molecular thermodynamic models of the transfer process, particularly when the work of cavity formation plays a significant role in determining the sign and/ or magnitude of solvation thermodynamics. Finally, the EoS contribution can be calculated without knowledge of free energies for any of the states in Scheme 1. EoS EoS The values of ∆µEoS are exactly those for an 2 , ∆h2 , ∆s2 isothermal, constant-composition, pressure change from p* to
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Letters
the pressure (p) for the initial and final states of the overall transfer process.
∆µEoS 2 (T,m3,p) ) ∆sEoS )2 ∆hEoS 2
)
∫p*p |∂p∂ |T,m ∆µEoS 2
+
( )
3
∂µ2 ∂T
p′,m3
T∆sEoS 2
)
∫p*p Vh 2 dp′
dp′ ) -
∫p*p Vh 2Rj2 dp′
∫p*Vh 2(1 - TRj2) dp′ p
(1a) (1b) (1c)
In eq 1, p* is the pressure in the binary solvent at (T, m3) such that its density is that of pure water at (T, p); with N2 f 0, the value of p* is determined solely by the isothermal compressibility (κT) of the binary solvent. V h 2 and R j 2 are, respectively, the partial molar volume and partial molar thermal expansion coefficient of the solute in the water-cosolvent mixture. We j 2 for were unable to locate experimental values for V h 2 and R methane in mixed aqueous solvents such as alcohol- or carbohydrate-water solutions. If one assumes V h 2 is of similar sign and magnitude to that in pure water (V h 2 ≈ 36 mL/mol), then the EoS contribution is found to overestimate the magnitude but capture the correct sign for ∆µtr2 of methane for the cosolvents34,42 ethanol, 1-propanol, and sucrose, similar to the results in Figure 2A. These estimates suggest the qualitative picture from analysis of our molecular simulation results is also reasonably predictive of experimental behavior. IV. Summary and Conclusions A thermodynamic cycle approach is presented to separate isothermal, isobaric transfer free energies for arbitrary solutes into contributions due to density changes at fixed composition (EoS), and to solvent exchange at fixed density. The approach is illustrated by analyzing transfer thermodynamics of methane between water and a range of water-cosolvent mixtures obtained from molecular dynamics simulations. Transfer free energies, enthalpies, and entropies are qualitatively and semiquantitatively captured for sorbitol-water and high-temperature alcohol-water solutions by the EoS contribution. The EoS contribution can be determined simply from the isothermal compressibility and density of the binary solvent of interest, along with the partial molar volumetric properties for the solute or reaction of interest. The results highlight the importance of considering how differences in solvent density affect the thermodynamics of the transfer process through the work of cavity formation. Acknowledgment. The authors gratefully acknowledge support from the Donors of the American Chemical Society Petroleum Research Fund, the University of Delaware Research Foundation, and Boehringer-Ingelheim Pharmaceuticals, Inc. Supporting Information Available: Descriptions, figures, and tables about MD simulations and calculation of excess
thermodynamic properties of methane, methane excess chemical potential versus solvent packing fraction, and comparison of methane solvation with solvation of equivalent soft-sphere solute. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Yalkowsky, S. H. Solubility and Solubilization in Aqueous Media; Oxford University Press: New York, 2000. (2) Sangster, J. Octanol-Water Partition Coefficients: Fundamentals and Physical Chemistry; John Wiley & Sons: Hoboken, NJ, 1997. (3) Timasheff, S. N. Biochemistry 2002, 41, 13473. (4) Schellman, J. A. Biopolymers 1978, 17, 1305. (5) Timasheff, S. N. Annu. ReV. Biophys. Biomol. Struct. 1993, 22, 67. (6) Scatchard, G. J. Am. Chem. Soc. 1946, 68, 2315. (7) Engberts, J. B. F. N. Mixed aqueous solvent effects on kinetics and mechanisms of organic reactions. In Water: A ComprehensiVe Treatise; Franks, F., Ed.; Plenum Press: New York, 1979; Vol. 6, p 139. (8) Xie, G. F.; Timasheff, S. N. Protein Sci. 1997, 6, 211. (9) Auton, M.; Bolen, D. W. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 15065. (10) Tanford, C. J. Am. Chem. Soc. 1964, 86, 2050. (11) Tanford, C. AdV. Protein Chem. 1970, 24, 1. (12) Schellman, J. A. Biophys. J. 2003, 85, 108. (13) Baynes, B. M.; Trout, B. L. J. Phys. Chem. B 2003, 107, 14058. (14) Chitra, R.; Smith, P. E. J. Phys. Chem. B 2002, 106, 1491. (15) Shulgin, I. L.; Ruckenstein, E. J. Phys. Chem. B 2006, 110, 12707. (16) Shulgin, I.; Ruckenstein, E. J. Chem. Phys. 2005, 123. (17) Ben-Naim, A. Statistical Thermodynamics for Chemists and Biochemists; Plenum Press: New York, 1992; Chapter 6. (18) Chitra, R.; Smith, P. E. J. Phys. Chem. B 2001, 105, 11513. (19) Minton, A. P. J. Pharm. Sci. 2005, 94, 1668. (20) Zimmerman, S.; Minton, A. Annu. ReV. Biophys. Biomol. Struct. 1993, 22, 27. (21) Boublik, T. Mol. Phys. 1986, 59, 775. (22) Minton, A. P. Biopolymers 1981, 20, 2093. (23) Berg, O. G. Biopolymers 1990, 30, 1027. (24) Tang, K.; Bloomfield, V. Biophys. J. 2000, 79, 2222. (25) Ben-Naim, A. Water and Aqueous Solutions; Plenum Press: New York, 1974. (26) Shah, P. P.; Roberts, C. J. J. Phys. Chem. B 2007, 111, 4467. (27) Garde, S.; Garcia, A. E.; Pratt, L. R.; Hummer, G. Biophys. Chem. 1999, 78, 21. (28) Hummer, G.; Garde, S.; Garcia, A.; Pohorille, A.; Pratt, L. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 8951. (29) Ashbaugh, H.; Truskett, T.; Debenedetti, P. J. Chem. Phys. 2002, 116, 2907. (30) Hummer, G.; Garde, S.; Garcia, A. E.; Pratt, L. R. Chem. Phys. 2000, 258, 349. (31) Peter, C.; van der Vegt, N. F. A. J. Phys. Chem. B 2007, 111, 7836. (32) Pohorille, A.; Pratt, L. J. Am. Chem. Soc. 1990, 112, 5066. (33) Kotz, J. C.; Purcell, K. F. Chemistry & Chemical ReactiVity; Saunders College Publishing: Philadelphia, PA, 1987. (34) Ben-Naim, A.; Yaacobi, M. J. Phys. Chem. 1974, 78, 170. (35) Yaacobi, M.; Ben-Naim, A. J. Phys. Chem. 1974, 78, 175. (36) Brandts, J. F.; Hunt, L. J. Am. Chem. Soc. 1967, 89, 4826. (37) Pratt, L. R. Annu. ReV. Phys. Chem. 2002, 53, 409. (38) Bishop, W. H. Biophys. Chem. 1987, 27, 197. (39) Lee, B. Biopolymers 1985, 24, 813. (40) Auton, M.; Bolen, D. W. Biochemistry 2004, 43, 1329. (41) van der Vegt, N. F. A.; van Gunsteren, W. F. J. Phys. Chem. B 2004, 108, 1056. (42) Yaacobi, M.; Ben-Naim, A. J. Solution Chem. 1973, 2, 425.
