Free Submitted: 04 October 2016 Accepted: 16 November 2016 Published Online: 01 December 2016
J. Chem. Phys. 145, 201102 (2016); https://doi.org/10.1063/1.4971205
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• 1Department of Chemical and Biomolecular Engineering, Tulane University, New Orleans, Louisiana 70118, USA
• 2Department of Chemical and Biomolecular Engineering, University of Delaware, Newark, Delaware 19716, USA
• a)Author to whom correspondence should be addressed. Electronic mail:

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• Henry S. Ashbaugh
• J. Wesley Barnett
• Alexander Saltzman
• Mae E. Langrehr
• Hayden Houser

We probe the anomalous compressibilities of dilute mixtures of alcohols and alkane gases in water using molecular simulations. The response to increasing solute concentration depends sensitively on temperature, with the compressibility decreasing upon solute addition at low temperatures and increasing at elevated temperatures. The thermodynamic origin of stiffening is directly tied to the solute’s partial compressibility, which is negative at low temperatures and rises above water’s compressibility with increasing temperature. Hydration shell waters concurrently tilt towards clathrate-like structures at low temperatures that fade with heating. Kirkwood-Buff theory traces the solute’s partial compressibility to changes in the solute-water association volume upon heating and incongruous packing of waters at the boundary between the more structured hydration shell and bulk water.
The hydration of non- and minimally polar species is accompanied by a host of anomalies whose origins have been subject to intense inquiry as part of the larger effort to understand aqueous phase and biological assembly. Historic interpretations of hydrophobic hydration have attributed the characteristic negative hydration entropies and large positive heat capacities to microscopic “iceberg” stabilization within the solute’s hydration shell.1,21. H. S. Frank and M. W. Evans, J. Chem. Phys. 13, 507 (1945). https://doi.org/10.1063/1.17239852. D. N. Glew, J. Phys. Chem. 66, 605 (1962). https://doi.org/10.1021/j100810a008 While influential, this model has proven less convincing in light of more recent experimental, theoretical, and simulation studies.3–73. D. T. Bowron, A. Filipponi, M. A. Roberts, and J. L. Finney, Phys. Rev. Lett. 81, 4164 (1998). https://doi.org/10.1103/PhysRevLett.81.41644. D. Laage, G. Stirnemann, and J. T. Hynes, J. Phys. Chem. B 113, 2428 (2009). https://doi.org/10.1021/jp809521t5. P. N. Perera, K. R. Fega, C. Lawrence, E. J. Sundstrom, J. Tomlinson-Phillips, and D. Ben-Amotz, Proc. Natl. Acad. Sci. U. S. A. 106, 12230 (2009). https://doi.org/10.1073/pnas.09036751066. G. Hummer, S. Garde, A. E. Garcia, M. E. Paulaitis, and L. R. Pratt, J. Phys. Chem. B 102, 10469 (1998). https://doi.org/10.1021/jp982873+7. H. S. Ashbaugh and L. R. Pratt, Rev. Mod. Phys. 78, 159 (2006). https://doi.org/10.1103/RevModPhys.78.159 Nevertheless, hints of structured waters encircling non-polar moieties persist,8–108. J. G. Davis, K. P. Gierszal, P. Wang, and D. Ben-Amotz, Nature 491, 582 (2012). https://doi.org/10.1038/nature115709. N. Galamba, J. Phys. Chem. B 117, 2153 (2013). https://doi.org/10.1021/jp310649n10. K. Meister, S. Strazdaite, A. L. DeVries, S. Lotze, L. L. C. Olijve, I. K. Voets, and H. J. Bakker, Proc. Natl. Acad. Sci. U. S. A. 111, 17732 (2014). https://doi.org/10.1073/pnas.1414188111 suggesting a more nuanced understanding is necessary.
While temperature-dependent solubility data may have inspired the iceberg model, this is not its only evidence. One of the more counterintuitive supporting experiments has been measurement of the speed of sound (u) in alcohol/water mixtures. While u is greater in water than in alcohol, u initially increases with increasing alcohol content before dropping to its minimum for pure alcohol (e.g., Figure 1(a)). Based on the Newton-Laplace equation
 $u=(ρκS)−1/2,$ (1)
where $ρ$ is the mass density and $κS$ is its isentropic compressibility (= $−V−1(∂V/∂P|S)$), Franks and Ives1111. F. Franks and D. J. G. Ives, Q. Rev. Chem. Soc. 20, 1 (1966). https://doi.org/10.1039/QR9662000001 interpreted u’s increase as if “some compression resistant structure were being formed or fortified.” Empirical relations invoking icebergs have subsequently been proposed to explain the anomalous increase in u, which begins with the first drop of alcohol.12,1312. G. Onori, J. Chem. Phys. 87, 1251 (1986). https://doi.org/10.1063/1.45330713. A. Burakowski and J. Glinski, Chem. Rev. 112, 2059 (2012). https://doi.org/10.1021/cr2000948 To date, however, limited molecular-scale studies have scrutinized this phenomenon.
In this communication, we report molecular simulations of alcohols and short alkanes (methanol (MeOH), ethanol (EtOH), 1-propanol (1-PrOH), 2-propanol (2-PrOH), tert-butyl alcohol (TBA), methane, and ethane) modeled using the TraPPE-UA force field14,1514. B. Chen, J. J. Potoff, and J. I. Siepmann, J. Phys. Chem. B 105, 3093 (2001). https://doi.org/10.1021/jp003882x15. M. G. Martin and J. I. Siepmann, J. Phys. Chem. B 102, 2569 (1998). https://doi.org/10.1021/jp972543+ in TIP4P/2005 water1616. J. L. F. Abascal and C. Vega, J. Chem. Phys. 123, 234505 (2005). https://doi.org/10.1063/1.2121687 to examine aqueous solution stiffening. These models were chosen since they accurately reproduce the pure liquid properties. Isothermal-isobaric ensemble simulations were performed using GROMACS1717. B. Hess, C. Kutzner, D. van der Spoel, and E. Lindahl, J. Chem. Theory Comput. 4, 435 (2008). https://doi.org/10.1021/ct700301q over the temperature range 5 °C to 75 °C. Full computational details are provided in the supplementary material.
To begin we evaluated u in ethanol/water mixtures at 25 °C over the full concentration range. The speed of sound was evaluated using Eq. (1), with the isentropic compressibility determined as $κS$ = $κT−Tα2/ρC^P$ (the isothermal compressibility ($κT$), thermal expansivity ($α$), and heat capacity ($C^P$) were evaluated from fluctuation relationships). The speed of sound determined from simulation is in semi-quantitative agreement with experiment (Figure 1(a)),1818. J. Lara and J. E. Desnoyers, J. Solution Chem. 10, 465 (1981). https://doi.org/10.1007/BF00652081 capturing the non-monotonic concentration dependence. While the maximum predicted from simulation is at a lower concentration, the slope $∂u/∂xsol$ at infinite dilution appears accurate. Indeed, we find that the simulation derivative is in excellent agreement with experiment over a broad temperature range (Figure 1(b)). Both experiment19,2019. J. Tong, M. J. W. Povey, X. Zou, B. Ward, and C. P. Oates, J. Phys.: Conf. Ser. 279, 012023 (2011). https://doi.org/10.1088/1742-6596/279/1/01202320. R. Ruiz, F. J. Hoyuelos, A. M. Navarro, J. M. Leal, and B. Garcia, Phys. Chem. Chem. Phys. 17, 2025 (2015). https://doi.org/10.1039/C4CP03459G and simulation find that the stiffening effect diminishes with temperature, with $∂u/∂xsol$ extrapolated from simulation changing sign just above 75 °C. We additionally find excellent agreement comparing experimental results for $∂u/∂xsol$ for a number of alcohols1818. J. Lara and J. E. Desnoyers, J. Solution Chem. 10, 465 (1981). https://doi.org/10.1007/BF00652081 against simulation at 25 °C (Figure 1(c)), with the derivative magnitudes increasing with solute size. The comparisons give confidence in the fidelity of our simulations.
Since the alcohol/water mixture density is monotonic with concentration, we expect that the maximum in $u$ can be tied to an extremum in $κS$. In the liquid state $κS$ is dominated by $κT$ (i.e., $κT$ $≫$ $Tα2/ρC^P$), which is simpler to evaluate. Consequently, positive values for $∂u/∂xsol$ are expected to be mirrored by negative values of $∂κT/∂xsol$ as a result of their inverse relationship. Figure 2(a) reports $∂κT/∂xsol$ at infinite dilution as a function of temperature for the simulated alcohols and alkanes. At low temperatures the slope is negative, in agreement with our expectation. In difference to $∂u/∂xsol$, $∂κT/∂xsol$ changes sign in the neighborhood of 45 °C to 60 °C, lower than projected from extrapolating $∂u/∂xsol$ (Figure 1(b)). The projected crossover of $∂u/∂xsol$ at higher temperatures can be rationalized by the fact that $∂u/∂xsol=0$ is satisfied when $∂κS/∂xsol=−(κS/ρ)∂ρ/∂xsol$. Since mixture densities decrease with alcohol concentration, $∂ρ/∂xsol$ is negative. As a result $∂u/∂xsol$ changes sign where $∂κS/∂xsol$ (and similarly $∂κT/∂xsol$) is positive, above the crossover temperatures in Figure 2(a).
The magnitude of $∂κT/∂xsol$ appears to grow with the solute size in Figure 2(a), consonant with the derivatives reported in Figure 1(c). This can be understood by recognizing that the compressibility derivative at infinite dilution is (see the supplementary material for derivation)
 $∂κT∂xsol|T,P∞=ρ0νsol∞(−1νsol∞∂νsol∞∂P|T−κT0),$ (2)
where $ρ0$ and $κT0$ are the number density and isothermal compressibility of pure water, while $νsol∞$ is the solute’s partial molar volume at infinite dilution. The term –$(νsol∞)−1(∂νsol∞/∂P|T)$, referred to here as the solute’s partial compressibility ($κsol∞$), is analogous to $κT$ but applied to the solute’s partial molar volume. Assuming $κsol∞$ for small, chemically related solutes is similar, $∂κT/∂xsol$ is expected to be proportional to the solute volume. Dividing $∂κT/∂xsol$ by $ρ0νsol∞$ (i.e., $∂κT*/∂xsol$ $=(ρ0νsol∞)−1∂κT/∂xsol=κsol∞−κT0$) the compressibility derivatives for the alcohols and alkanes appear to collapse onto two lines (Figure 2(b)). The lines for the alcohols and alkanes cross zero at 58.5 ± 0.9 °C and 47.1 ± 1.2 °C, respectively. The crossover temperature corresponds to the point at which $κsol∞=κT0$, above which $κsol∞$ is greater than $κT0$. The greater crossover temperature for the alcohols compared to the alkanes may result from the –OH group stabilizing hydration shell waters, although this effect likely diminishes for larger alcohols. Perhaps more interestingly, $κsol∞$ is negative when $∂κT*/∂xsol<−κT0$ so that $νsol∞$ grows with increasing pressure. Examining the curve $∂κT*/∂xsol=−κT0$ (Figure 2(b)), we find that below 10 °C for the alcohols and 20 °C for the alkanes, the solutes swell with increasing pressure.
Kirkwood-Buff (KB) theory connects $∂κT/∂xsol$ with the pressure dependence of solute-water correlations.2121. J. G. Kirkwood and F. B. Buff, J. Chem. Phys. 19, 774 (1951). https://doi.org/10.1063/1.1748352 Specifically, $νsol∞$ is determined as an integral over the solute-water radial distribution function ($gsw(r)$ or RDF)
 $νsol∞=kTκT0+∫[1−gsw(r)]4πr2dr,$ (3a)
whose pressure derivative is (see the supplementary material for derivation)
 $∂νsol∞∂P|T=kT∂κT0∂P|T+1kT∫νsw(r)gsw(r)4πr2dr.$ (3b)
Here $kT$ is the product of Boltzmann’s constant and the temperature, while $νsw(r)(=−kT∂lngsw(r)/∂P|T)$ is the solute-water association volume as a function of separation. In liquids $kTκT0$ and its pressure derivative, evaluated from simulations at elevated pressures, are smaller than the integral terms. The RDF between water’s oxygen and any solute site, like methane’s carbon, exhibits long-range correlations that challenge evaluation of the KB integrals. Recognizing that specific sites are not required to measure correlations in KB theory, packing oscillations can be minimized. Subsequently, the water oxygen and solute carbon nuclei can be uniformly smeared over a radius λ.22,2322. A. V. Sangwai and H. S. Ashbaugh, Ind. Eng. Chem. Res. 47, 5169 (2008). https://doi.org/10.1021/ie071444823. D. M. Lockwood, P. J. Rossky, and R. M. Levy, J. Phys. Chem. B 104, 4210 (2000). https://doi.org/10.1021/jp994197x For λ’s comparable in size to water, oscillations are suppressed and the integrals rapidly converge. Methane-water RDF smearing at 5 °C with λ = $2.0Å$ is demonstrated in Figure 3(a), reducing correlations to a comparatively featureless function whose integrated value is unaffected.2222. A. V. Sangwai and H. S. Ashbaugh, Ind. Eng. Chem. Res. 47, 5169 (2008). https://doi.org/10.1021/ie0714448 Smearing has been used to evaluate solute volumes, compressibilities, and osmotic virial coefficients.22–2422. A. V. Sangwai and H. S. Ashbaugh, Ind. Eng. Chem. Res. 47, 5169 (2008). https://doi.org/10.1021/ie071444823. D. M. Lockwood, P. J. Rossky, and R. M. Levy, J. Phys. Chem. B 104, 4210 (2000). https://doi.org/10.1021/jp994197x24. H. S. Ashbaugh, K. Weiss, S. M. Williams, B. Meng, and L. N. Surampudi, J. Phys. Chem. B 119, 6280 (2015). https://doi.org/10.1021/acs.jpcb.5b02056
The methane-water association volume, evaluated from the pressure derivative of the smeared RDF at 5 °C and 75 °C, shows distinct features which contribute to the large variation in $κsol∞$ as a function of temperature (Figure 3(b)). At overlapping, smeared separations $νsw(r<3Å)$ is negative, indicative of closer water packing against methane with increasing pressure that reduces methane’s volume. The association volume in this regime decreases with increasing temperature, indicating that the compressibility will increase with temperature. Notably, at intermediate separations, $νsw(r∼5Å)$ is positive at 5 °C and nearly zero at 75 °C, indicating that waters just beyond the first shell on average swell the solute with increasing pressure at lower temperatures and are unresponsive at higher temperatures. While the magnitude of the positive association volume at intermediate separations is not as large as the negative volume at overlap, the integration volume in Eq. (3) grows as $r2$, magnifying the positive association volume’s contribution. Moreover, the decreasing negative values of $νsw(r)$ (Figure 3(b)) are generally observed for r < 3 Å where the smeared correlation function drops to values well below one (Figure 3(a)). As a result, the product $νsw(r)gsw(r)$ for r < 3 Å contributes less to the integral in Eq. (3b) than might be anticipated. These contributions work to reduce and potentially reverse the sign of $κsol∞$.
Combining the smeared distributions and association volumes, $∂κT/∂xsol$ for methane can be evaluated via Eq. (2) (Figure 3(c)). The agreement between $∂κT/∂xsol$ obtained from pressure derivative of $νsol∞$ determined via KB theory and that evaluated directly from the concentration dependence of $κT$ is excellent. We conclude that the differences in $νsw(r)$ with temperature (Figure 3(b)) are the controlling factor for the variation of $κsol∞$. Moreover, the agreement between KB theory and direct measurements of $∂κT/∂xsol$ at low temperatures supports the inferred negative partial compressibility in this regime.
To gain structural insights, we have examined methane’s hydration shell using a hierarchical set of measures: the number of hydrogen-bonds between waters; the tetrahedral order of water with its 4 nearest neighbors quantified using the Errington-Debenedetti parameter q;2525. J. R. Errington and P. G. Debenedetti, Nature 409, 318 (2001). https://doi.org/10.1038/35053024 and larger-scale coordination amongst up to 8 waters quantified by Molinero’s CHILL+ bond order parameter c(i,j).2626. A. H. Nguyen and V. Molinero, J. Phys. Chem. B 119, 9369 (2015). https://doi.org/10.1021/jp510289t In the bulk, each water participates in just under 4 hydrogen-bonds (donors + acceptors) with the number of bonds decreasing with temperature (Figure 4(a)). The number of bonds per water in the hydration shell (r < 5.4 Å) tends to be slightly enhanced relative to the bulk at 5 °C. Bond enhancement progressively decreases with heating so that hydration shell bonding is depleted below that in the bulk at 75 °C. We note that while bonding for waters closer than 3.5 Å is lower than the bulk at 5 °C, these waters are pressed into methane’s excluded volume (Figure 3(a)) and may be distorted. Similarly, water’s tetrahedral order in the bulk decreases with increasing temperature from qbulk = 0.70 (5 °C) to 0.61 (75 °C) (Figure 4(b)) (q = 1 indicates perfect tetrahedral order, while 0 indicates a disordered arrangement). Relative to the bulk, however, water’s tetrahedral order in the hydration shell is increased. The tetrahedral enhancement of the hydration shell is greatest at 5 °C and decreases with temperature, reminiscent of melting. Galamba observed similar temperature dependent structural changes.99. N. Galamba, J. Phys. Chem. B 117, 2153 (2013). https://doi.org/10.1021/jp310649n
The CHILL+ bond order parameter c(i,j) discriminates between cubic ice, hexagonal ice, clathrate, and liquid states based on the distribution of staggered, eclipsed, and disordered bonds between neighboring waters.2626. A. H. Nguyen and V. Molinero, J. Phys. Chem. B 119, 9369 (2015). https://doi.org/10.1021/jp510289t This parameter, which varies between −1 and 1, describes the relative orientation of water’s neighboring two central water molecules i and j evaluated by the dot product between their bond order vectors. These vectors are determined by a rank-3 spherical harmonic projection of the distribution of the 4 closest waters about i or j. Values of c between −0.35 and 0.25 correspond to eclipsed orientations, while values less than −0.8 are staggered (Figure 5). The inset to Figure 5 shows the normalized probability distribution of observing specified values of c(i,j), P(c), for waters within methane’s hydration shell (ri > 5.4 Å) and outside the shell at 5 °C. Waters beyond the hydration shell exhibit a broad distribution consistent with liquid water.2626. A. H. Nguyen and V. Molinero, J. Phys. Chem. B 119, 9369 (2015). https://doi.org/10.1021/jp510289t The hydration shell’s distribution exhibits only minor perturbations. Taking the difference between the hydration shell and bulk distributions (Figure 5), however, we find that the hydration shell waters are enriched in eclipsed orientations and depleted in staggered orientations at 5 °C. Eclipsed enrichment is consistent with a preference toward clathrate-like hydration shell structures, while the depletion of staggered orientations signifies that ice-like structures are disfavored. Bond orientation preferences progressively decrease with temperature, such that at 75 °C the eclipsed enrichment is half that observed at 5 °C, while the depletion of staggered orientations is only weakly perturbed.
While the waters hydrating methane are liquid, the increased hydrogen-bonding, tetrahedral order, and eclipsed bond orientations signify enhanced clathrate-like structuring between hydration shell waters at low temperatures that dissipate with heating. The negative association volumes reported in Figure 3(b), however, indicate that even though cold hydration waters are more structured, they contribute to reduction of the solute volume with increasing pressure. The magnitude of the reduction is lowest at 5 °C, suggesting that the more structured hydration shell waters are more compression resilient. The waters that contribute positive association volumes to further resist compression, however, lie on average between the structured shell and the bulk. We surmise that the positive association volumes observed at low temperature result from incongruous packing between hydration shell waters and those in the bulk, slightly expanding the volumes of waters at the hydration shell boundary. As temperature increases and structure is lost, the hydration waters more readily mesh with the bulk, deflating the positive association volume. So while it may be tempting to ascribe the anomalous solution stiffening simply to “iceberg” formation, a full accounting of $κsol∞$ necessitates characterization of the interaction of hydration waters with the bulk as encapsulated by the association volume.
We have shown that aqueous solution stiffening results from differences between $κsol∞$ and $κT0$. The significant change in the solute’s compressibility with temperature, varying from negative to positive with increasing temperature, is tied to the solute-water association volume. These results, in turn, correlate with hydration shell structural changes that lean toward clathrate-like structures at low temperature and dissipate with heating. Not unexpectedly the hydration shell structure about the –OH group is distinct from that about a carbon unit, as suggested by the crossover differences observed in Figure 2. Nevertheless, we observe similar enhanced structuring to methane about the distal carbon units from the –OH group (e.g., the methyl unit of 1-propanol). More intriguingly, Ben-Amotz and co-workers88. J. G. Davis, K. P. Gierszal, P. Wang, and D. Ben-Amotz, Nature 491, 582 (2012). https://doi.org/10.1038/nature11570 observed from Raman scattering experiments that waters in the shells of long chain alcohols appear to transition from more tetrahedrally ordered to more disordered relative to bulk water at temperatures of ∼60 °C. This transition appears to coincide with the crossover temperature we observe in Figure 2 (albeit for shorter alcohols), suggesting these phenomena may be connected.
See supplementary material for details regarding the molecular simulations performed, evaluation of the partial molar volume via smeared distribution functions, evaluation of the solute-water association volumes, and derivations of key mathematical expressions.
We thank financial support from the NSF-DMR (No. 1460637), NSF-OIA (No. 1430280), and Louisiana Board of Regents Graduate Research Fellowship program (J.W.B.). Computational support from the Louisiana Optical Network Initiative is gratefully acknowledged. H.S.A. also thanks Amish Patel for invaluable discussions improving this work.
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