Analytical theory of the hydrophobic effect of solutes in water

Abstract

We develop an analytical statistical-mechanical model for hydrophobic solvation in water. In this three-dimensional Mercedes-Benz–like model, two neighboring waters have three possible interaction states: a radial van der Waals interaction, a tetrahedral orientation-dependent hydrogen-bonding interaction, or no interaction. Nonpolar solutes are modeled as van der Waals particles of different radii. The model is sufficiently simple that we can calculate the partition function and thermal and volumetric properties of solvation versus temperature, pressure, and solute radius. Predictions are in good agreement with results of Monte Carlo simulations. And their trends agree with experiments on hydrophobic solute insertion. The theory shows that first-shell waters are more highly structured than bulk waters, because of hydrogen bonding, and that that structure melts out faster with temperature than it does in bulk waters. Because the theory is analytical, it can explore a broad range of solvation properties and anomalies of water, at minimal computational expense.

I. INTRODUCTION

The solvation properties of the nonpolar molecules in water are sufficiently peculiar that they are given their own name, the hydrophobic effect [1–5]. These unusual behaviors are particularly reflected in the enthalpies, entropies, heat capacities, and volumes of transfer of such solutes into water [1–3]. It has long been of interest to interpret these observations in terms of the structures and energies of individual water molecules: their hydrogen bonding, tetrahedrality, etc. So a substantial body of work has sought structure-property relationships using atomistic potentials with explicit-solvent models sampled by molecular dynamics (MD) or Monte Carlo (MC) computer simulations [6–8].

An alternative route to insights is through coarser-grained analytical theory [9–11]. The evident downside is that coarse graining throws away some details of molecular structure. Those details might matter. However, analytical theories also offer some advantages: (1) They can be much faster to compute. So a single study (such as the present one) can explore how solvation depends on temperature, pressure, solute radius, and solute energy parameters all at the same time, without large computing resources. (2) Analytical models are particularly good for exploring principles. In analytical theories, there can be tight logical linkages from molecular structure to observable properties. Parameters can be varied—much more readily than in atomistic simulations—to test how the macroscopic properties arise from microscopic structures. (3) Analytical models often give the best basis for engineering models, where there is a need to express observed behaviors fairly accurately with a minimum number of physical parameters. (4) And, even the disadvantage—that some details of molecular structure must be left out of the model—can actually be an advantage, because they provide an opportunity to explore which details matter and which don’t. Here we give an analytical model of the hydrophobic effect; it builds upon earlier work by Luksic et al. [12].

Hydrophobic hydration has some interesting aspects. Solvation thermodynamics in simpler systems is determined by an unfavorable enthalpy change of dissolution. But dissolving nonpolar solutes in water has a favorable enthalpy of transfer (ΔH_TR < 0) (i.e., enthalpy for the process of transferring nonpolar particle from vacuum to bulk water) at low temperatures. Dissolving oil in water is unfavorable (ΔG_TR > 0) due to the negative entropy of transfer ΔS_TR < 0 [1,2]. Also unusual is the observation that both the enthalpy and entropy change strongly with temperature. This fact is summarized by stating that the heat capacity of transfer is high and positive (ΔC_{p,T R} ≫ 0), since

$$\mathrm{\Delta }{C}_{p,TR}=\frac{d\mathrm{\Delta }{H}_{TR}}{dT}=T\frac{d\mathrm{\Delta }{S}_{TR}}{dT},$$ (1)

where T is temperature. These features are known as the hydrophobic effect [1–3,13–19]. There have been many models for how hydrophobic solvation depends on solute size. Among the first, scaled particle theory (SPT) estimates the work necessary to create a spherical cavity in water [5,20]. It successfully predicts the free energy of small cavity formation and was constructed to give the surface tension of water in the planar limit. However, SPT has been criticized for predicting a monotonic increase in the entropy penalty of transfer with increasing cavity size and an incorrect temperature dependence in the surface tension [5]. Stillinger was perhaps the first to suggest that water solvates large nonpolar molecules differently than small molecules [5]. Pratt and Chandler developed an integral equation method that used pair correlations of bulk waters to predict the solubilities of small solutes [21]. The theory of Lum et al. [17] reduces to Pratt-Chandler theory for small solutes but predicts large-scale drying as predicted by Stillinger near larger nonpolar surfaces [5]. The hydrophobic effect plays an important role in many common processes in nature (e.g., protein folding, ligand binding) and technology (e.g., micelle formation). So it has been extensively studied experimentally [22–26]. Hydrophobic solvation as a function of solute size has also been studied in computer simulations using explicit and implicit water models. A key conclusion from such simulations is that at planar nonpolar surfaces water will waste a hydrogen bond by pointing the bond directly at the surface, in contrast to small nonpolar surfaces, where water conserves hydrogen bonds by pointing those bonds in directions that straddle the solute. Explicit models (such as TIP or SPC) can be computationally expensive, precluding the calculation of full temperature or pressure dependencies. And computer simulation results are, in general, susceptible to large statistical errors, causing difficulties in determining certain quantities accurately, such as the heat capacity [27,28]. Implicit models are faster, but the trade-off is sometimes physical inaccuracies. There exist also many other waterlike models that were used for treating hydrophbic effect. Core softened [29] or Jagla fluids [30] possess waterlike structural, dynamic, and thermodynamic anomalies, and these fluids also display waterlike solvation thermodynamics. Another aspect of structure-based coarse graining relies on matching the pair correlation functions of a reference (atomistic) and coarse-grained system [31,32]. It was demonstrated that it can be generalized for inhomogeneous systems as well as solvation. Coarse gaining performed in inhomogeneous systems improves thermodynamic properties and the structure of interfaces without significant alterations to the local structure of the bulk liquid.

Here we adapt a Mercedes-Benz (MB)-like model of water, which has previously been studied in two and three dimensions [9–11] to study the three-dimensional (3D) hydration of a nonpolar solute. The idea behind the model originated with Ben-Naim in the 1970s [33–36]. Recently, it has been developed further by Bizjak et al. [27,37] and Dias et al. [38,39], and studied using computer simulations [27,37–39] and integral equation theory [37]. The MB models of water are toy models that have the advantage of explaining in a simple way interplay of thermodynamic properties and the angle-dependent potential, but cannot be used for qualitative prediction of properties. The analytical theories for MB-like models allow the inclusion of orientation-dependent hydrogen bonding within a framework that is simple and nearly analytical. According to the 3D MB model, each water molecule is a Lennard-Jones (LJ) sphere with four arms, oriented tetrahedrally to mimic formation of hydrogen bonds. In a statistical mechanical model, which is based on the two-dimensional (2D) Urbic and Dill (UD) model being directly descended from a treatment of Truskett and Dill (TD), who developed a nearly analytical version of the 2D MB model [40,41], each water molecule interacts with its neighboring waters through a van derWaals interaction and an orientation-dependent interaction that models hydrogen bonds. A related analytical treatment was developed by Coronas et al. [42] and was also studied by Monte Carlo simulations. It is a coarse-grained model for bulk water that includes many-body interactions associated with water cooperativity. It possesses waterlike anomalies and the liquid-liquid phase transition also present in our analytical treatment of water. The main difference between the two models is that in ours, energy is a continuous function of the relative orientation θ of two water molecules, while in the other the hydrogen-bonding (HB) energy is a discontinuous function of the relative configurations, but both models possess similar features. For hydrophobic hydration, Xu and Dill [43] proposed a very simple analytical theory of the hydrophobic effect in two dimensions which builds on a 2D MB model of water. Starting from the statistical partition functions for a water molecule in the bulk and a water in the first solvation shell around a hydrophobe, the theory reproduces the main characteristics of the hydrophobic effect and accounts for different solute size effects. The theory in Ref. [43] required the results of a reference Monte Carlo simulation of pure bulk water phase. That approach was improved by Luksic et al. [12], which is simpler and circumvents any computer simulation steps by using an analytical model of the pure phase of water. Both solvation methods were for 2D cases. In this work, we apply theory to a 3D MB model of water. In addition to moving the theory to three dimensions, we implement additional improvements by assuming that water properties in the first solvation shell change due to higher density because of interaction between water and solute. The new version of the theory can be used in all liquid regions of the 3D MB model, including the supercooled region where computer simulations cannot obtain solvation properties due to crystallization and convergence problems.

Here we start from an analytical 3D UD theory of water [10]. A partition function for a water molecule in the bulk and the first hydration shell of a hydrophobic solute is then built using the expressions for average energies of different states of the water molecule (hydrogen-bonded, van der Waals, and open), upon considering the geometric restrictions through which a solute dictates the formation or breakage of the hydrogen bonds between water molecules in the first solvation shell. Finally, from statistical mechanical and thermodynamical relations, we calculate the ΔG, ΔH, TΔS, and ΔC_p. In order to explore the performance of the analytical theory in describing the hydrophobic hydration, we used existing computer simulation data [44,45] for the simplified water model, a 3D version of the Ben-Naim water model (3D MB) [27]. The MB model has previously been shown to capture the essential physics of water, namely, van der Waals interaction and hydrogen bonding, which are essential for hydrophobic hydration [14,16–18,28]. Here we explore the performance of the analytical theory for the dependence of the hydrophobic effect on temperature, pressure, and solute size.

II. 3D MB WATER MODEL

We applied analytical theory to the 3D MB water model where each water molecule is represented as a Lennard-Jones sphere with an additional tetrahedrally dependent potential that mimics the hydrogen bonding of true water [27]. The interaction potential between two 3D MB particles is

$$U({\mathbf{X}}_{\mathbf{i}},{\mathbf{X}}_{\mathbf{j}})=U_{L,J}^{11}(r_{ij})+U_{HB}({\mathbf{X}}_{\mathbf{i}},{\mathbf{X}}_{\mathbf{j}}),$$ (2)

where r_ij is the distance between centres of particles i and j and X_i is a vector denoting the position and orientation of particle i. $U_{LJ}^{mn}$ is the standard Lennard-Jones potential

$$U_{LJ}^{mn}(r_{ij})=4{\epsilon }_{LJ}^{mn}\left[{\left(\frac{{\sigma }_{LJ}^{mn}}{r_{ij}}\right)}^{12}-{\left(\frac{{\sigma }_{LJ}^{mn}}{r_{ij}}\right)}^6\right],$$ (3)

where ${\epsilon }_{LJ}^{mn}$ denotes the well depth and ${\sigma }_{LJ}^{mn}$ is the contact parameter. m is 1 if the ith particle is water and 2 the ith particle is a solute, and the same goes for n regarding the j th particle. The HB term is the sum of interactions over all possible pairs of HB arms

$$U_{HB}({\mathbf{X}}_{\mathbf{i}},{\mathbf{X}}_{\mathbf{j}})=\sum _{k,l}^4{U}_{HB}^{kl}(r_{ij},{\mathrm{\Omega }}_{\mathbf{i}},{\mathrm{\Omega }}_{\mathbf{j}}),$$ (4)

where $U_{HB}^{k,l}$ is the interaction between HB arms k and l on two particles and vector Ω_i denotes the orientation of particle i. The interaction between two HB arms of different particles is

$$U_{HB}^{kl}(r_{ij},{\mathrm{\Omega }}_{\mathbf{i}},{\mathrm{\Omega }}_{\mathbf{j}})={\epsilon }_{HB}G(r_{ij}-r_{HB})G({\mathbf{i}}_{\mathbf{k}}{\mathbf{u}}_{\mathbf{ij}}-1)G({\mathbf{j}}_{\mathbf{l}}{\mathbf{u}}_{\mathbf{ij}}+1).$$ (5)

Here u_ij is the unit vector pointing from particle i to particle j, and i_k is the unit vector representing arm k on particle i. Between vector u_ij and vectors representing the orientations of arms there is dot product. G(x) is the unnormalized Gaussian function [a requirement that G(0) = 1]

$$G(x)=e^{-x^2/2{\sigma }^2}.$$ (6)

The model does not distinguish between HB donors and acceptors. The width of the Gaussian function for distance and angles is the same (σ = 0.085) and was chosen to be small enough that a direct hydrogen bond is more favorable than a bifurcated one [27,37]. It regards two waters as being hydrogen bonded when their HB arms are collinear with each other. The strongest hydrogen bond is formed when two arms are pointing towards each other particles’ centers when the centers are separated by r_HB [27,38,39]. In the following section we will first summarize the theory for pure 3D MB and explain how parameters of the model for analytical theory are extracted from continuous potential presented in this section.

III. THEORY

A. The model for the pure water reference state

The structure of the liquid state is modeled as a variation of the cell model theory and is a perturbation from a hexagonal (ice) lattice (see Fig. 1). One grid point is occupied by only one molecule. Our focus is on a single water molecule and its interactions with the neighboring molecule. The interaction between a pair of molecules can be one of the three possibilities. Note that in our theory we limit interactions from the continuous one to only the three most probable interactions, minimas in energy as function of distance for HB and LJ interactions and no interaction. We say that each water molecule can be in one of three possible orientational states relative to its clockwiselike positioned neighbor on the lattice: (1) hydrogen-bonded (HB) state, (2) van derWaals (LJ) state, or (3) nonbonded (NB) state. This is presented in Fig. 2. First, we compute the isothermal-isobaric statistical weights, Δ_i, of the states as a functions of temperature, pressure, and interaction energies [9–11].

FIG. 1.

The test water molecule in the context of its water lattice. It shows the principal pair neighbor interaction, taken clockwise, to avoid triple counting.

FIG. 2.

The three states that define pairwise water-water interactions: (1) hydrogen-bonded, (2) Lennard-Jones-bonded (LJ) pairs (no hydrogen bond), or (3) nonbonded (having neither a hydrogen bond nor an LJ bond).

In the HB state the test water molecule can point one of its four HB arms at an angle θ to within π/3 of the center of its neighbor water. This is an approximation and was determined by the condition that one quarter of the total solid angle is occupied per hydrogen bond and to keep calculations of integrals analytical. In this case it forms a hydrogen bond [10,11] (see Fig. 2), and the interaction energy of the test water with its neighbor is then described by this equation

$$u_{HB}(\theta )=-{\epsilon }_{HB}+k_s{(1-cos\theta )}^2,0<\theta <\pi /3,$$ (7)

where ε_HB is an HB energy constant of the maximal strength of a hydrogen bond and k_s is the angular spring constant that describes the weakening of the hydrogen bond with angle. This is a type of hydrogen bond that is called a weak bond [9] because it does not cooperate with neighboring hydrogen bonds. The isothermal-isobaric partition function, Δ_HB, of this state is calculated by integrating this Boltzmann factor over all angles ϕ, θ, and ψ and over all the separations x, y, and z of the test molecule relative to its clockwise neighbor. In the vdW state, the test water molecule forms a contact with its clockwiselike positioned water, but it does not form any hydrogen bond. Energy of this state can be written as

$$u_{LJ}(\theta )=-{\epsilon }_{LJ}.$$ (8)

The isothermal-isobaric partition function, Δ_LJ, of this state is obtained using the same integration as for the HB state. In the last NB state, the test water molecule does not interact with its neighbor, so the energy is equal to zero:

$$u_0(\theta )=0.$$ (9)

By knowing the isobaric-isothermal ensemble partition functions for each state we can write the partition function Q₁ for a full single hexagon of six waters as

$$Q_1={({\mathrm{\Delta }}_{HB}+{\mathrm{\Delta }}_{LJ}+{\mathrm{\Delta }}_o)}^6.$$ (10)

If we include also higher cooperativity in ice [9–11] we can write the total partition function for each hexagon as

$$Q_1={({\mathrm{\Delta }}_{HB}+{\mathrm{\Delta }}_{LJ}+{\mathrm{\Delta }}_o)}^6-{\mathrm{\Delta }}_{HB}^6+\delta {\mathrm{\Delta }}_s^6.$$ (11)

δ = exp(−βε_c) is the Boltzmann factor for the cooperativity energy, ε_c, that applies only when six water molecules all collect together into a full hexagonal cage. The terms on the right-hand side of this expression simply replace the statistical weight for each weakly HB full hexagonal cage with the statistical weight for a cooperative strongly HB hexagonal cage. Δ_s is the Boltzmann factor for a cooperative hexagonal cage. It differs from Δ_HB only in the volume per molecule, v_s, instead of v_HB [9–11]. Now we combine the Boltzmann factors for the individual water molecules to get the partition function for the whole system of N particles. The population of different states f_j can be calculated [9–11] and all the other thermodynamic properties from simple derivations of the partition function as described previously [9–11,40,41]. The attraction beyond pair terms is treated in the mean-field attractive level with energy [46], −Na/v, among hexagons, where a is the van der Waals dispersion parameter [9,40,41] and v is the average molar volume. Parameters needed for calculations can be obtained directly from the interaction pair potential between two 3DMBwater particles (ε_HB=1, r_HB=1, ε_LJ=0.1, σ_LJ=0.7) [27,37] or from analyzing the angle and distance dependence of the 3D MB potential in comparison with the potential used in analytical theory (k_s = 80, a = 0.045, ε_c = 0.18).

For modeling the solvation of a nonpolar solute, it is necessary to summarize volumes and to calculate additional quantities. Volumes of the states are [10,11]

$$v_s=\frac{8\sqrt{3}{r}_{HB}^3}{9},$$ (12)

$$v_{HB}=\frac{v_s}{x_v},$$ (13)

$$v_{LJ}={\sigma }_{LJ}^3,$$ (14)

$$v_o=\frac{k_{B}T}{p}+v_{LJ}.$$ (15)

The x_v = 2.5 is chosen empirically to get the proper behavior of the density dependence in the original water papers [10,11]. The ensemble average energy, 〈u_j 〉, for each of the three types of water molecule structures, can be calculated as

$$\langle u_j\rangle =\frac{{\int }_0^{\pi /3}{u}_{j}sin\theta d\theta exp[-(u_j+p{v}_j/2)/k_{B}T]}{{\int }_0^{\pi /3}sin\theta d\theta exp[-(u_j+p{v}_j/2)/k_{B}T]}.$$ (16)

Integration gives us

$$\langle u_{\text{HB}}\rangle =-{\epsilon }_{\text{HB}}+\frac{kT}{2}-\frac{\sqrt{\frac{k_{\mathrm{s}}kT}{4\pi }}exp(-\frac{k_{\mathrm{s}}}{4kT})}{\text{erf}\left(\sqrt{\frac{k_{\mathrm{s}}}{4kT}}\right)},$$ (17)

$$\langle u_{\mathrm{S}}\rangle =\langle u_{\text{HB}}\rangle +\frac{{\epsilon }_{\mathrm{c}}}{6},$$ (18)

$$\langle u_{\text{LJ}}\rangle =-{\epsilon }_{\text{LJ}},$$ (19)

$$\langle u_0\rangle =0.$$ (20)

The average energy 〈u_S〉 is obtained by adding 1/6 of the correlation energy to 〈u_HB〉. The average energy of a water molecule, summed over the four different water states, can be expressed as

$${\langle \epsilon \rangle }_{\mathrm{b}}=2(\langle u_{\text{HB}}\rangle f_{\text{HB}}+\langle u_{\mathrm{S}}\rangle f_{\mathrm{S}}+\langle u_{\text{LJ}}\rangle f_{\text{LJ}}).$$ (21)

B. The model for inserting a nonpolar solute into water

To develop the theory for the solvation of nonpolar solutes, we followed the same steps as in Luksic et al. [12]. A nonpolar solute molecule of diameter σ_s is inserted into water. Now we consider what happens with two water molecules in the first hydration shell of the solute (see Fig. 3). The presence of the solute imposes a geometric restriction since a solvation-shell water molecule may be unable to form all four hydrogen bonds with its neighboring waters as in the bulk. Let ζ (ϕ,θ,ψ) be the maximum number of hydrogen bonds the water in the first solvation shell can form for a specific orientation. ϕ,θ,ψ are Euler angles describing the orientation of a water molecule in the first solvation shell, and ζ (ϕ,θ,ψ) is also a function of the solute radius. There are different possibilities depending on the size of the solute molecule. For smaller solutes, water molecules in the first solvation shell can form a maximum of either three or four hydrogen bonds depending on the orientation. For bigger solutes, first-shell water molecules can form a maximum of only one, two, or three hydrogen bonds. A critical angle ϕ_c is defined as angle at which a HB arm points along a tangent to the solute (see Fig. 3), and both waters are in the Lennard-Jones minimum

FIG. 3.

Definition of the critical angle, ϕ_c.

$${\varphi }_{\mathrm{c}}=arccos\frac{r_{\text{HB}}}{({\sigma }_{\mathrm{s}}+{\sigma }_{\text{LJ}})\sqrt[6]{2}}.$$ (22)

This is the angle where water can still form a hydrogen bond. A similar definition of this angle was used by Chaimovich and Shell in their work [47,48]. Integration over all possible orientations of water (over all three Euler angles) is equal to 8π². The next step is to determine the ratio of orientations of water molecules that can form four, three, two, or one hydrogen bond. Note that water can always forms at least one hydrogen bond in the first solvation shell. Ratios can be determined by sampling over orientations of water in first solvation shell and counting the number of hydrogen bonds water can form. This gives us ratios of orientations ζ_i where water can form i hydrogen bonds. A solute molecule does not impose just geometric restrictions on first-shell water molecules, but it also perturbs the energetics of water-water interactions in the first shell. Water’s density in the first solvation shell is higher than in the bulk, depending on the value of the attraction between the solute and the water

$${\rho }_h={\rho }_b\text{exp}(\beta {\epsilon }_{SW}),$$ (23)

where ε_SW = ε_LJ is the minimum of the Lennard-Jones interaction between solute and water. Since we do not have density as input to the analytical theory, we have to calculate properties of water in the first solvation shell by increasing pressure for

$$\mathrm{\Delta }p=\frac{\beta {\epsilon }_{SW}}{\kappa },$$ (24)

where κ is compressibility of bulk water. The populations of waters with different states in the first solvation shell are now different comparing to the bulk and are equal to $f_j^{\mathrm{h}}$. Note that the average energies of different states stay the same, only populations change. The average energy of a water molecule in the first solvation shell, summed over the four different water states, can be expressed as

$${\langle \epsilon (\varphi ,\theta ,\psi )\rangle }_{\mathrm{h}}=\frac{1}{2}\{\zeta (\varphi ,\theta ,\psi )[\langle u_{\text{HB}}\rangle f_{\text{HB}}^{\mathrm{h}}+\langle u_{\mathrm{S}}\rangle f_{\mathrm{S}}^{\mathrm{h}}]+4\langle u_{\text{LJ}}\rangle f_{\text{LJ}}^{\mathrm{h}}-{\epsilon }_{\text{SW}}\},$$ (25)

Now we can calculate the partition function of water in the bulk by treating interactions in averaged way, namely,

$$\begin{gathered} q_{\mathrm{b}}=\iint d\varphi d\psi {\int }_0^{\pi /3}sin\theta d\theta exp\left(-\frac{{\langle \epsilon \rangle }_{\mathrm{b}}+p{v}_{\text{mol}}^{\mathrm{b}}}{kT}\right) \\ =8{\pi }^{2}exp\left(-\frac{{\langle \epsilon \rangle }_{\mathrm{b}}+p{v}_{\text{mol}}^{\mathrm{b}}}{kT}\right), \end{gathered}$$ (26)

where $v_{\text{mol}}^{\mathrm{b}}$ is molar volume of bulk water. The partition function for a water molecule in the first shell around a solute molecule can be written as

$$q_{\mathrm{h}}=\iint d\varphi d\psi {\int }_0^{\pi /3}sin\theta d\theta exp\left(-\frac{{\langle \epsilon \rangle }_{\mathrm{h}}+p{v}_{\text{mol}}^{\mathrm{h}}}{kT}\right),$$ (27)

where $v_{\text{mol}}^{\mathrm{h}}$ denotes the molar volume of water in first solvation shell. It is smaller than the $v_{\text{mol}}^{\mathrm{b}}$ by the overlap volume $\mathrm{\Delta }v(v_{\text{mol}}^{\mathrm{h}}=v_{\text{mol}}^{\mathrm{b}}-\mathrm{\Delta }v)$ (see Fig. 4) [12]. We compute the Gibbs free energy of transferring a hydrophobic solute into water using [43]

FIG. 4.

Definition of the overlap volume, Δv.

$$\mathrm{\Delta }G=-n({\sigma }_{\mathrm{s}})\mathit{kT}ln\left(\frac{q_{\mathrm{h}}}{q_{\mathrm{b}}}\right),$$ (28)

where n(σ_s) is the average number of water molecules in the first solvation shell. In this theory we assumed that n(σ_s) is proportional to the solvent surface accessible area of the solute [12]. Standard thermodynamic relations give the enthalpy and the entropy of transfer as

$$\mathrm{\Delta }H=n({\sigma }_{\mathrm{s}})k{T}^2\frac{\partial ln(q_{\mathrm{h}}/q_{\mathrm{b}})}{\partial T},$$ (29)

$$T\mathrm{\Delta }S=\mathrm{\Delta }H-\mathrm{\Delta }G.$$ (30)

IV. PREDICTIONS OF THE THEORY AND COMPARISONS TO CORRESPONDING MONTE CARLO SIMULATIONS

In this section, we give the theory’s predictions for how the hydrophobic effect depends on temperature, pressure, and solute size. The analytical results are compared with the Monte Carlo simulation results of 3D MB water by Mohoric et al. [44,45]. As has been done previously [9–12], we present our results in dimensionless units, normalized to the strength of the optimal hydrogen bond ε_HB and HB separation $r_{HB}(T^{\ast }=k_{B}T/{\epsilon }_{HB},u^{ex\ast }=u^{ex}/{\epsilon }_{HB},V^{\ast }=V/r_{HB}^2,\text{and}{p}^{\ast }=p{r}_{HB}^2/{\epsilon }_{HB})$.

A. Inserting the solute changes the hydrogen bonding of waters in the first solvation shell relative waters in the bulk

First, we show the four types of water populations (six water cages, two water H bonded, vdW, and nonbonded) of pure bulk water, with no solute, as a function of temperature; see Fig. 5. This figure allows us to establish the approximate ranges over which the model acts like ice, cold liquid water, hot liquid water, and vapor. Our main interest here is in exploring solvation across the range from cold liquid to hot liquid, where solvation anomalies are most pronounced. This range of anomalies is best represented in this model by the supercooled region of the pT phase diagram reported in previous work [11] where we reported that the 3D MB model exhibits two critical points; the liquid-gas critical point (C1) at $T_{C1}^{\ast }=0.1166,p_{C1}^{\ast }=0.0115,{\rho }_{C1}^{\ast }=0.467$ and the liquid-liquid critical point (C2) at $T_{C2}^{\ast }=0.0779,p_{C2}^{\ast }=0.167,{\rho }_{C2}^{\ast }=1.295$. There exists also a region of pressures between both critical points where we have only one fluid phase, at higher pressures we have two liquid phases, and at lower pressures the liquid and the gas phases. So, while we cannot draw a precise correspondence with true ambient p and T values, Fig. 5 shows that we can approximately regard the model as having ice like behavior below about T^* = 0.1, cold liquid water up to about T^* = 0.1–0.15, hot liquid water up to about T^* = 0.15–0.2, and vapor above about T^* = 0.25.

FIG. 5.

The computed pairwise bulk populations f_j vs temperature, at p^* = 0.12: (black) HB water population, (orange) LJ population, (red) NB population, (pink) HB cage population. It shows the melting out of HB structure with temperature in the bulk, and its replacement by NB.

Then, in the following sections below, we show that the analytical model reproduces fairly well the temperature dependences of the free energy, enthalpy, entropy, and volumes from the underlying model, as determined by Monte Carlo simulations. This is just a validation of the analytical theory against the Monte Carlo simulations. Then, in the last section, we show how the model’s thermal and volumetric behaviors are explained by the underlying microscopic water populations.

B. The thermal and volumetric properties of solvation are captured by the theory

Figure 6 shows the transfer free energy, ΔG^*, as a function of temperature for different sizes of hydrophobic particles ( ${\sigma }_S^{\ast }={\sigma }_{LJ}^{22}/r_{HB}$). For large solutes see Fig. 6(a); for small ones, see Fig. 6(b). The symbols represent the results of the computer simulations, and the lines are the results of the analytical theory. In general, the analytical theory gives good agreement with the simulations. We observe a qualitative difference between small and large hydrophobes. Bigger positive transfer free energies on these plots indicates increasing unfavorability for inserting the solute from vapor into water. The difference is only for very small solutes (smaller than size of a water molecule); in that case, inserting a small solute into hot water is favorable (ΔG^* is negative). These results are in agreement with the experimental observation for the thermodynamics of hydration of argon [2] and show qualitative differences from the behavior of ΔG^* for larger hydrophobes. Increasing the solute size increases the unfavorability of dissolving in water. The analytical theory correctly captures these subtle differences.

FIG. 6.

Transfer free energy, ΔG^*, into water of solutes of different radii. (a) Large solutes: ${\sigma }_S^{\ast }=0.7$ (red), ${\sigma }_S^{\ast }=1.5$ (blue), ${\sigma }_S^{\ast }=2.0$ (green). (b) Smaller solutes: ${\sigma }_S^{\ast }=0.3$ (black), ${\sigma }_S^{\ast }=0.7$ (red). Lines show the theory and points show the Monte Carlo results. The larger solutes are most opposed to dissolving, and this effect weakens upon heating.

Figure 7 shows the enthalpy and entropy of solute transfer from vacuum into water, ΔH^*, and transfer entropy, T^*ΔS^*, for different solute sizes. Both functions first increase, and then decrease with temperature, as observed experimentally [2]. The agreement between the computer simulation results (symbols) and analytical theory (lines) is quite good in the whole temperature range studied, even for the large solutes. Transfer enthalpies are positive for large solutes since solute insertion into water requires breaking water-water bonds, and since the interaction with the solute does not contribute sufficient energy to compensate. For small solutes, transfer enthalpies are negative since small molecules are inserted into empty spaces within the water structure and there is no HB breaking required for insertion. The transfer entropy is most positive when waters in the first solvation shell have the highest density. Analytical theory also predict unexpected behavior at low temperatures where transfer enthalpy and entropy increase upon decreasing the temperature. Simulation data are not available in this range due to convergence problems, so we cannot verify if this is a failure of the theory or not.

FIG. 7.

Transfer enthalpy ΔH^* and entropy T^*ΔS^* into water of large solutes of different radii, for ${\sigma }_S^{\ast }=0.7$ (red), ${\sigma }_S^{\ast }=1.5$ (blue), ${\sigma }_S^{\ast }=2.0$ (green). Lines show the theory and points give the Monte Carlo results. There is substantial enthalpy-entropy compensation.

Figure 8 shows how the volume ΔV^* of the whole system (solute plus solvent) changes as a solute is inserted into water as a function of temperature and for different solute sizes. The analytical theory gives trends that are consistent with experiments, showing that for relatively small solutes, ΔV^* increases with temperature [23]. The main contribution to the transfer volume is the size of the nonpolar solute, then upon increase of temperature water becomes more gaslike and there is more empty space around solutes, which gives an additional increase of transfer volume.

FIG. 8.

Transfer volume, ΔV^*, as a function of temperature, for ${\sigma }_S^{\ast }=0.7$ (red), ${\sigma }_S^{\ast }=1.5$ (blue), ${\sigma }_S^{\ast }=2.0$ (green). Not surprisingly, larger solutes have larger excess volumes of insertion. Lines show theoretical results, and points Monte Carlo results.

C. The theory gives correct trends of solvation thermodynamics versus solute size

Figure 9 summarizes the dependence of the solvation free energy, enthalpy, and entropy on the solute radius. Again, there is good general consistency with the simulations for most solute sizes. The disagreement is bigger for larger solutes, especially for entropy transfer. The reason might be in problems within the theory or bad computer simulation data which authors calculated by Widom’s insertion method, which is problematic for insertion of large particles. The solvation free energy for large solutes increases linearly with area, while for small solutes it increases linearly with volume [14,17,19]. For small solutes, the entropy contribution to the change in free energy dominates (TΔS > ΔH), but for larger solutes, TΔS < ΔH. Figure 9 shows the transfer free energy, ΔG^*, transfer enthalpy, ΔH^*, and transfer entropy, TΔS^*, at two different temperatures, T^* = 0.2 (red) and T^* = 0.3 (green), as a function of solute size. For all three thermodynamic functions describing transfer of a hydrophobic particle, there are clearly two distinct areas of behavior, as observed experimentally [1,2,22–26,49,50]. In Fig. 10 we plotted the dependence of the solvation free energy, enthalpy, and entropy on solute radius and temperature. We can see from the figure the equivalent behavior of the free energy of transfer upon increasing the temperature or decreasing the size of the nonpolar solute. Figure 11 shows 3D plots with the following temperature- and size-dependent fits through the points:

$$\mathrm{\Delta }{G}^{\ast }=-7.47906\ast T^{\ast }+1.20461\ast {\sigma }_s^{\ast }+1.45645,$$ (31)

$$\mathrm{\Delta }{H}^{\ast }=-4.1216\ast T^{\ast }+1.02603\ast {\sigma }_s^{\ast }+0.4797,$$ (32)

$$T\mathrm{\Delta }{S}^{\ast }=-15.1371\ast T^{\ast }-0.102591\ast {\sigma }_s^{\ast }+\mathrm{5.11875.}$$ (33)

FIG. 9.

How the solute insertion thermodynamics depends on solute radius. (a) Transfer free energy, ΔG^*, (b) transfer free energy per unit surface area of the nonpolar solute, (c) transfer enthalpy, ΔH^*, and (d) transfer entropy, T^*ΔS^*, as a function of hydrophobe size ( ${\sigma }_S^{\ast }$) at two different temperatures: T^* = 0.2 (red) and T^* = 0.3 (blue) at p^* = 0.12. Larger solutes are more expensive to insert for enthalpy reasons, but are entropically favored in cold water and disfavored in hot water. Points are Monte Carlo results, and lines are theoretical results.

FIG. 10.

Solute insertion thermal properties vs both temperature and solute radius.

FIG. 11.

Same as Fig. 10, but also showing a plane indicating linear fits in both variables in warm water.

Figure 12 shows 3D plots with the temperature and size dependences of solute insertion thermal properties per unit surface area of the nonpolar solute. Increasing temperature makes solute insertion easier. There are two mechanisms of solvation for small and big solutes. The figure clearly shows that the solvation free energy for large solutes increases linearly with area, while for small solutes it increases linearly with volume at all temperatures.

FIG. 12.

Solute insertion thermal properties per unit surface area of the nonpolar solute vs both temperature and solute radius. Increasing temperature makes solute insertion easier. There are two mechanisms of solvation for small and big solutes.

D. The theory predicts how nonpolar solvation depends on pressure

Figure 13 shows (a) the predicted pressure dependences of the transfer free energy, ΔG^*, (b) the transfer enthalpy, ΔH^*, and (c) the transfer entropy, T^*ΔS^*, for different sizes of hydrophobes. Such quantities are notoriously difficult to obtain from simulations of most atomistic models, particularly at high pressures, where it is difficult to obtain reliable results from the Wisdom insertion method, while our theory can easily calculate these properties. The theory predicts that the free energy of solvation for small solutes becomes linearly more unfavorable with pressure, while the enthalpy and entropy of solvation exhibit very little pressure dependence for small solutes, consistent with experiments [51–53] and theory [54]. For bigger solutes, the situation is different. The free energy of solvation increases more strongly with increased pressure, while the entropy and enthalpy show no monotonic behavior, but both effects compensate.

FIG. 13.

Pressure dependences of solute transfer thermal quantities, for solute sizes ${\sigma }_S^{\ast }=0.7$ (red), ${\sigma }_S^{\ast }=1.5$ (blue), ${\sigma }_S^{\ast }=2.0$ (green). Applying pressure squeezes the space available, making solute insertion more difficult. There is enthalpy-entropy compensation.

The results above show that the present theory reproduces the thermal, volumetric, and solute-size dependences of the solvation properties of nonpolar solutes rather well, compared to Monte Carlo simulations of the same model [44,45], and both the theory and simulations give the experimentally observed trends. Disagreement becomes bigger at large solute sizes and low temperatures, which might be either a problem of the theory or incomplete convergence of the simulation data.

E. The solvation properties can be interpreted in terms of first-shell and bulk water bonding fractions

In this section, we give a more microscopic explanation of these properties based on the different bonding populations. Figure 14(a) shows the temperature dependence of the ratio, $f_j^{\mathrm{h}}/f_j$, of the population of waters forming hydrogen bonds in the first shell around a solute, relative to the population of waters forming hydrogen bonds in the bulk. Note that we omitted ratios of caged populations from the figure since, in most of the ranges, the individual cage population is very close to zero and numerical uncertainty is high when dividing with numbers close to zero. In the following, we interpret these results with increasing temperature, starting from very cold water, to cold water, to hot water.

FIG. 14.

Molecular structure properties vs temperature and pressure for (a) p^* = 0.12 and (b) T^* = 0.2. (Top row) Ratio $f_j^{\mathrm{h}}/f_j$ of populations of first-shell to bulk waters: (black) HB waters, (orange) LJ waters, (red) NB waters. This shows that the shell has more HB and LJ bonds than the bulk, and that shell hydrogen bonds melt out faster with increasing temperature than bulk ones do. It shows that the first shell is better packed (fewer NB waters). (Middle row) 〈r〉 is the average separation between waters. Heating warm water increases the water-water separations. At all temperatures, pressure pushes waters closer together. (Bottom row) 〈(1 − cos θ)²〉 is the variance in HB angles, a measure of “bending flexibility.” Heating increases this variance, while pressure decreases it.

(1) Very cold water

The theory shows that in very cold water, the first-shell solvating waters differ from bulk waters in having more hydrogen bonds, more vdW bonds, and less empty space, and those waters are well-packed (〈r〉 ≈ 1) and have low-variance, well-defined HB angles [Fig. 14(e)]. These first-shell waters are better structured than waters in the bulk. We are seeing a sort of stochastic version of the “iceberg” idea first proposed by Frank and Evans in 1945 [55]. The reason for this structuring is clear from the model. Hydrogen bonding is a driver of water structure. More hydrogen bonds are formed in first shells than in the bulk because the solute restricts the HB angle options more than in the bulk.

(2) Cold water

Increasing the temperature from very cold (T^* = 0.16) to just cold water (T^* = 0.2) melts out vdW interactions, slightly loosening up the water structure, supporting an increase in hydrogen bonds.

(3) Hot water

Increasing the temperature further leads to melting out first-shell water hydrogen bonds and vdW interactions, increasing the average water spacings and increasing the variance in HB angles. Correspondingly, Fig. 14(b) shows how applied pressure affects water molecules in the solvation shell relative to the bulk. Applying pressure has the following effects on solvation-shell waters, relative to bulk waters: pressure squeezes vdW-bonded water molecules together in the first shell, it squeezes out empty spaces, and it reduces the average water-water spacings in the first shell. At the same time, applying pressure increasingly breaks first-shell hydrogen bonds, reducing the excess structure there, and reduces the HB angle variance.

V. CONCLUSIONS

In this work, we have developed a theory for the hydration thermodynamics of a spherical hydrophobe in 3D MB-like water. The results for transfer free energy, transfer enthalpy, transfer entropy, and transfer volume obtained by the analytical theory show good agreement with the computer simulation results for the same model for all parameters studied. The results under these conditions are consistent with the existing experimental and theoretical results. In addition, the analytical theory enabled us to study the thermodynamics of solvation under the conditions where the computer simulation results were unreliable due to the large statistical uncertainty, namely, at high pressures, low temperatures, and large solutes. Theory can easily calculate solvation properties in the supercooled region of phase space since we do not have problems with crystallization. This theory describes only liquid properties of the 3D MB water model. The present work demonstrates that that general anomalies of the hydrophobic effect—which are regarded as arising from the cagelike or networklike properties of water—can be captured in a simple theory in which water-water interactions are only treated up through two-body nearest-neighbor effects and cooperative effects.