Probabilistic Approach to the Length-Scale Dependence of the Effect of Water Hydrogen Bonding on Hydrophobic Hydration

Towards accurate infrared spectral density of weak H-bonds in absence of relaxation mechanisms

Following the previous theoretical developments to completely reproduce the IR spectra of weak hydrogen bond complexes within the framework of the linear response theory (LRT), the quantum theory of the high stretching mode spectral density (SD) of weak H-bonds is reconsidered. Within the LRT theory, the SD is the one sided Fourier transform of the autocorrelation function (ACF) of the high stretching mode dipole moment operator. In order to provide more accurate theoretical bandshapes, we have explored the equivalence between the SDs given in previous studies with respect to a new quantum one, and revealed that in place of the basic equations used in the precedent works for which the SD I_Old(ω)=2Re∫₀^∞G_Old(t)e^-iωtdt where the ACF G_Old(t) = ⟨μ(0)μ(t)⁺⟩ = tr {ρ {μ(0)} {μ(t)}⁺}, one can use a new expression for the SD, given by I_New(ω)=2ωRe∫₀^∞G_New(t)e^-iωtdt where G_New(t)=μ(0)μ(t)⁺=1βtrρ_B∫₀^βμ(0)μ(t+iλℏ)⁺dλ. Here ρ_B is the Boltzmann density operator, μ(0) the dipole moment operator at initial time and μ(t) the dipole moment operator at time t in the Heisenberg picture, ℏ is the Planck constant, β is the inverse of the Boltzmann factor k_BT where T is the absolute temperature and k_B the Boltzmann constant. Using this formalism, we demonstrated that the new quantum approach gives the same final SD as used by previous models, and reduces to the Franck-Condon progression appearing in the Maréchal and Witkowski's pioneering approach when the relaxation mechanisms are ignored. Results of this approach shed light on the equivalence between the quantum and classical IR SD approaches for weak H-bonds in absence of medium surroundings effect, which has been a subject of debate for decades.

Equivalence between the Classical and Quantum IR Spectral Density Approaches of Weak H-Bonds in the Absence of Damping

The aim of this paper is to overhaul the quantum elucidation of the spectral density (SD) of weak H-bonds treated without taking into account any of the damping mechanisms. The reconsideration of the SD is performed within the framework the linear response theory. Working in the setting of the strong anharmonic coupling theory and the adiabatic approximation, the simplified expression of the classical SD, in the absence of dampings, is equated to be I_Cl(ω) = Re[∫₀^∞G_Cl(t)e^-iΩt dt] in which the classical-like autocorrelation function (ACF), G_Cl(t), is given by G_Cl(t) = tr{ρ(β){μ(0)}{μ(t)}^†}. With this consideration, we have shown that the classical SD is equivalent to the line shape obtained by F(ω) = ΩI_Cl(ω), which in turn is equivalent to the quantum SD given by I_Qu(ω) = Re[∫₀^∞G_Qu(t)e^-iΩt dt], where G_Qu(t) is the corresponding quantum ACF having for expression G_Qu(t) = (1/β) tr{ρ∫₀^β[μ(0)}{μ(t + iλℏ)}^† dλ}. Thus, we have shown that for weak H-bonds dealt without dampings, the SDs obtained by the quantum approaches are equivalent to the SDs geted by the classical approach in which the incepation ACF is, however, of quantum nature and where the line shape is the Fourier transform of the ACF times the angular frequency. It is further shown that the classical approach dealing with the SD of weak H-bonds leads identically to the result found by Maréchal and Witkowski in their pioneering quantum treatment where they ignored the linear response theory and dampings.

Recent developments in the theoretical, simulational, and experimental studies of the role of water hydrogen bonding in hydrophobic phenomena

Hydrophobic effects (hydrophobic hydration and hydrophobic interaction) constitute an important element of a wide variety of phenomena relevant to biological, physical, chemical, environmental, engineering, and pharmaceutical sciences, such as the immiscibility of oil and water, self-assembly of amphiphiles leading to micelle and membrane formation, folding and stability and unfolding of the native structure of a biologically active protein, gating of ion channels, wetting, froth floatation, and adhesion. On the other hand, the hydrogen bonding ability of water plays a major (if not crucial) role in hydrophobic phenomena. We present a review of most important and relatively recent experimental, simulational, and theoretical research on hydrophobic phenomena in various systems. With a particular interest we survey investigations clarifying the role of water hydrogen bonding therein, because it has been the main object of our own recent research. We have developed a probabilistic hydrogen bond (PHB) model that allows one to obtain an analytic expression for the number of bonds per water molecule as a function of its distance to a hydrophobe, hydrophobe radius, and temperature. Knowing that function, one can explicitly identify a water hydrogen bond contribution to the external potential whereto a water molecule is subjected near a hydrophobe. Combining the PHB model with the classical density functional theory (DFT), one can examine the contribution of water hydrogen bonding to the temperature and lengthscale effects on the hydration of particles and on their solvent-mediated interactions over the entire low-to-high temperature and small-to-large lengthscale ranges. We applied the combined DFT/PHB model to study a variety of hydrophobic phenomena such as (liquid) water in contact with a hydrophobic plate, solvation of spherical solutes of various radii in associated and non-associated liquids at various temperatures, the solvent-mediated interaction of spherical solutes and its temperature dependence, interaction of C60 fullerenes in water, temperature effect on the evaporation lengthscale of water confined between two hydrophobes, temperature dependence of the effective width of the solute-solvent transition layer and average density therein. These applications demonstrated that the DFT/PHB model can serve as a valuable tool in studying hydrophobic phenomena because it constitutes a balanced combination of simplicity, accuracy, and detail. The predictions of the combined DFT/PHB approach for the solvent density profiles and thermodynamic aspects of hydrophobic phenomena are generally in good agreement with experiments and simulations. For example, it predicts the small-to-large crossover lengthscale of its mechanism to be approximately in the range from 1nm to 4nm, and decreasing with increasing temperature. It also suggests that, in terms of the average fluid density in the solute-solvent transition layer, the transition layer for small hydrophobes (of radii ≲2 nm) becomes enriched with rather than depleted of fluid when both the solvent-solute affinity and hb-energy alteration ratio become large enough. The boundary values of these parameters, needed for the depletion-to-enrichment crossover, are predicted to decrease with increasing temperature.

Fluid transition layer between rigid solute and liquid solvent: is there depletion or enrichment?

The fluid layer between solute and liquid solvent is studied by combining the density functional theory with the probabilistic hydrogen bond model. This combination allows one to obtain the equilibrium distribution of fluid molecules, taking into account the hydrogen bond contribution to the external potential whereto they are subjected near the solute. One can find the effective width of the fluid solvent-solute transition layer and fluid average density in that layer, and determine their dependence on temperature, solvent-solute affinity, vicinal hydrogen bond (hb) energy alteration ratio, and solute radius. Numerical calculations are performed for the solvation of a plate and spherical solutes of four different radii in two model solvents (associated liquid and non-associated one) in the temperature range from 293 K to 333 K for various solvent-solute affinities and hydrogen bond energy alteration ratios. The predictions of our model for the effective width and average density of the transition layer are consistent with experiments and simulations. The small-to-large crossover lengthscale for hydrophobic hydration is expected to be about 3-5 nm. Remarkably, characterizing the transition layer with the average density, one can observe that for small hydrophobes, the transition layer becomes enriched with rather than depleted of fluid when the solvent-solute affinity and hb-energy alteration ratio become large enough. The boundary values of solvent-solute affinity and hb-energy alteration ratio, needed for the "depletion-to-enrichment" crossover (in the smoothed density sense), are predicted to decrease with increasing temperature.

Temperature dependence of the evaporation lengthscale for water confined between two hydrophobic plates

Liquid water in a hydrophobic confinement is the object of high interest in physicochemical sciences. Confined between two macroscopic hydrophobic surfaces, liquid water transforms into vapor if the distance between surfaces is smaller than a critical separation, referred to as the evaporation lengthscale. To investigate the temperature dependence of the evaporation lengthscale of water confined between two hydrophobic parallel plates, we use the combination of the density functional theory (DFT) with the probabilistic hydrogen bond (PHB) model for water-water hydrogen bonding. The PHB model provides an analytic expression for the average number of hydrogen bonds per water molecule as a function of its distance to a hydrophobic surface and its curvature. Knowing this expression, one can implement the effect of hydrogen bonding between water molecules on their interaction with the hydrophobe into DFT, which is then employed to determine the distribution of water molecules between two macroscopic hydrophobic plates at various interplate distances and various temperatures. For water confined between hydrophobic plates, our results suggest the evaporation lengthscale to be of the order of several nanometers and a linearly increasing function of temperature from T=293 K to T=333 K, qualitatively consistent with previous results.

Effect of Water Hydrogen Bonding on the Solvent-Mediated "Oscillatory" Repulsion of C60 Fullerenes in Water

The solvent-mediated interaction of C60 fullerenes in liquid water is examined by using the combination of the probabilistic hydrogen bond model with the density functional theory. This combination allows one to take into account the effect of hydrogen bonding between water molecules on their interaction with fullerenes and to construct an approximation for the distribution of water molecules in the system, which provides an efficient foundation for studying hydrophobic phenomena. Our numerical evaluations predict the solvent-induced interaction of two C60 fullerenes in water at 293 K to have an oscillatory-repulsive character (previously observed in molecular dynamics simulations) only when the vicinal water-water hydrogen bonds are slightly weaker than bulk ones. Besides indicating the direction of the energetic alteration of water-water hydrogen bonds near C60 fullerenes, our model also suggests that the hydrogen bonding ability of water plays a defining role in the solvent-mediated C60-C60 repulsion.

Order and correlation contributions to the entropy of hydrophobic solvation

The entropy of hydrophobic solvation has been explained as the result of ordered solvation structures, of hydrogen bonds, of the small size of the water molecule, of dispersion forces, and of solvent density fluctuations. We report a new approach to the calculation of the entropy of hydrophobic solvation, along with tests of and comparisons to several other methods. The methods are assessed in the light of the available thermodynamic and spectroscopic information on the effects of temperature on hydrophobic solvation. Five model hydrophobes in SPC/E water give benchmark solvation entropies via Widom's test-particle insertion method, and other methods and models are tested against these particle-insertion results. Entropies associated with distributions of tetrahedral order, of electric field, and of solvent dipole orientations are examined. We find these contributions are small compared to the benchmark particle-insertion entropy. Competitive with or better than other theories in accuracy, but with no free parameters, is the new estimate of the entropy contributed by correlations between dipole moments. Dipole correlations account for most of the hydrophobic solvation entropy for all models studied and capture the distinctive temperature dependence seen in thermodynamic and spectroscopic experiments. Entropies based on pair and many-body correlations in number density approach the correct magnitudes but fail to describe temperature and size dependences, respectively. Hydrogen-bond definitions and free energies that best reproduce entropies from simulations are reported, but it is difficult to choose one hydrogen bond model that fits a variety of experiments. The use of information theory, scaled-particle theory, and related methods is discussed briefly. Our results provide a test of the Frank-Evans hypothesis that the negative solvation entropy is due to structured water near the solute, complement the spectroscopic detection of that solvation structure by identifying the structural feature responsible for the entropy change, and point to a possible explanation for the observed dependence on length scale. Our key results are that the hydrophobic effect, i.e. the signature, temperature-dependent, solvation entropy of nonpolar molecules in water, is largely due to a dispersion force arising from correlations between rotating permanent dipole moments, that the strength of this force depends on the Kirkwood g-factor, and that the strength of this force may be obtained exactly without simulation.

The solvent-induced interaction of spherical solutes in associated and non-associated liquids

We propose an efficient method for studying the solvent-induced interaction of two solvophobic particles immersed in a liquid solvent. The method is based on the combination of the probabilistic hydrogen bond model with the density functional theory. An analytic expression for the number of hydrogen bonds per water molecule near two spherical hydrophobes is derived as a function of the molecule distance to both hydrophobes, distance between hydrophobes, and their radii. Using this expression, one can construct an approximation for the distribution of fluid (liquid water) molecules in the system which provides a reasonably good (much faster and accurate enough) alternative to a standard iteration procedure. Such an approximate density distribution constitutes an efficient foundation for studying the length-scale and temperature dependence of hydrophobic interactions. The model is applied to the interaction of solvophobic solutes in both associated and non-associated liquids. Of these two cases, the model predictions for the solvent-induced potential of mean force between two solutes in associated liquids are closer to the results of molecular dynamics simulation of hydrophobic interactions in the SPC/E model water. Our results suggest that the hydrogen bonding ability of water molecules may play a major role in hydrophobic phenomena.