Colloquium: Scaled particle theory and the length scales of hydrophobicity

Hydrophobic hydration plays a crucial role in self-assembly processes over multiple length scales, from the microscopic origins of inert gas solubility in water, to the mesoscopic organization of proteins and surfactant structures, to macroscopic phase separation. Many theoretical studies focus on the molecularly detailed interactions between oil and water, but the extrapolation of molecular-scale models to larger-length-scale hydration phenomena is sometimes not warranted. Scaled particle theories are based upon an interpolative view of that $\text{microscopic}\leftrightarrow \text{macroscopic}$ issue. This Colloquium revisits the scaled particle theory proposed 30 years ago by Stillinger [J. Solution Chem. 2, 141 (1973)], adopts a practical generalization, and considers the implications for hydrophobic hydration in light of our current understanding. The generalization is based upon identifying a molecular length, implicit in previous applications of scaled particle models, which provides an effective radius for joining microscopic and macroscopic descriptions. It will be demonstrated that the generalized theory correctly reproduces many of the anomalous thermodynamic properties of hydrophobic hydration for molecularly sized solutes, including solubility minima and entropy convergence, successfully interpolates between the microscopic and macroscopic extremes, and provides new insights into the underlying molecular mechanisms. The model considered here serves as a reference for theories that bridge microscopic and macroscopic hydrophobic effects. The results are discussed in terms of length scales associated with component phenomena. In particular, first there is a discussion of the microscopic-macroscopic joining radius identified by the theory; then follows a discussion of the Tolman length that describes curvature corrections to a surface area model of hydrophobic hydration free energies and the length scales on which entropy convergence of hydration free energies are expected.

Figure 1

Solubility of several inert gases in water and cyclohexane relative to the dilute gas phase expressed as the Ostwald partition coefficient following from Eq. (1), as a function of temperature at one atmosphere total pressure. The circles and interpolating solid lines are the data for liquid water as a solvent; the stars and the fitted dotted lines are for cyclohexane. Note that in the low-temperature region, these results for the two solvents have slopes of opposite sign. These solubility ratios are expected to increase in higher-temperature regions where the density of either solvent is decreasing and the distinctions between gas and liquid phase diminish.

Figure 2

(Color online) Product of the liquid-vapor surface tension $\gamma$ and the isothermal compressibility ${\kappa }_T\equiv -(1∕V){(\partial V∕\partial p)}_T$ for several liquid solvents at low temperatures so that the density of the coexisting vapor is low. Even though the individual factors differ substantially in magnitude, this product is a length characteristic of the liquid, accessible on the basis of macroscopic measurements, and can be taken as proportional to a molecular correlation length. The interesting observation here is that the temperature dependence of this correlation length is qualitatively different for liquid water than for the organic solvents.

Figure 3

Examples of the entropy-convergence phenomenon for noble gases dissolved in water. Results obtained from Ostwald coefficient results as a function of temperature, following Eq. (1), at one atmosphere total pressure. The bold solid curves are the data plotted up to 70 °C (26, 27, 28); the dashed lines are fitted linear models. Results are available over a broader range for many cases, but this display adopts a common temperature range and gives an indication of the accuracy of the linear behavior. Radon (Rn) is clearly a special case, perhaps because of the strength of water-Rn dispersion interactions and the greater solubility of Rn. It is clear here that there is some variation of intersection (“convergence”) temperatures. This variation is probably less evident in solubilities of small hydrocarbons that are more common as biophysical model systems, perhaps because of the better commonality of fundamental sizes and interaction strengths among those solutes.

Figure 4

(Color) The box on the left shows a configuration of water molecules taken from a simulation of liquid water at the density of the liquid in coexistence with vapor at 300 K. Oxygen atoms are red and hydrogen atoms are silver. The box on the right shows hard spheres of diameter 2.8 Å that can be successfully placed into the configuration on the right without overlap of the van der Waals volume of the water molecules. The insertion probability $p_0$ is determined as the volume accessible to a center of a purple sphere divided by the geometric volume of the box. See also 86.

Figure 5

Cavity-water oxygen radial distribution function for a $R=3\mathrm{\AA }$ cavity at $T=300\mathrm{K}$. The contact value, defined as $G\left(R\right)={\lim }_{r\to R_{+}}g\left(r\right)$, is about 2.3. The thin solid curve indicates the radial distribution function, while the dotted curve indicates the radial integral $N\left(r\right)={\int }_0^r{\rho }_{\mathrm{W}}g\left(\lambda \right)4\pi {\lambda }^{2}d\lambda$. The occupation of the first hydration shell, corresponding to the first minimum in $g\left(r\right)$ at 5.1 Å, is 17.8 water molecules, as indicated by the thick horizontal line. Note that the first minimum, which physically discriminates between first and succeeding hydration shells, is mild and structuring of outer hydration shells is weak (90). These features are in qualitative agreement with the predictions of the Pratt-Chandler theory (88), though that theory has been substantially amended (87).

Figure 6

Cavity contact values for water at $T=300\mathrm{K}$ at liquid saturation conditions. The points are obtained by differentiation of the simulated cavity insertion probabilities. The dashed lines are obtained from Reiss’s original SPT predictions for a hard-sphere solvent, Eq. (9), using effective hard-sphere diameters for water over the range from ${\sigma }_{\mathrm{W}\mathrm{W}}=2.6$ to 3.0 Å in 0.1 Å increments. The solid curve is obtained by differentiation of the revised SPT predictions, Eq. (14), fitted to the simulation results between $R_{\mathrm{sim}}=2.5\mathrm{\AA }$ and $R_{\text{macro}}=3.5\mathrm{\AA }$.

Figure 7

Cavity contact function for water along the liquid saturation curve for several temperatures, as determined by the revised SPT, Eq. (13), with $R_{\mathrm{sim}}=2.5\mathrm{\AA }$ and $R_{\text{macro}}=3.5\mathrm{\AA }$. Results are shown over the range from $T=260$ to 460 K in 20 K increments. Notice that the length defined by the maximum of this curve decreases with decreasing density (following increasing temperature) along the saturation curve. This is qualitatively consistent with the data of Fig. 2.

Figure 8

Excess chemical potential, enthalpy, and $\text{temperature}\times \text{entropy}$ of a methane-sized hard-sphere solute $(R=3.3\mathrm{\AA })$ in water as a function of temperature along the saturation curve. The subscript $\mathrm{A}$ indicates the solute component, which in this case is ${\mathrm{CH}}_4$ (methane). The points are the explicit simulation results for the chemical potential. Error bars are comparable in size to the points. The curves for the excess chemical potential, enthalpy, and entropy are labeled. The curves were determined under the assumption that the heat capacity is independent of temperature [Eqs. (15)].

Figure 9

Ostwald solubility, $-\ln K_{\mathrm{eq}}={\mu }_{\mathrm{A}}^{\mathrm{ex}}∕kT$ [cf. Eq. (1) and following], modeled as in the van der Waals equation of state, ${\mu }_{\mathrm{A}}^{\mathrm{ex}}={\left[{\mu }_{\mathrm{A}}^{\mathrm{ex}}\right]}_{\mathrm{HS}}-2a_{\mathrm{A}\mathrm{W}}{\rho }_{\mathrm{W}}$ for a 3.3 Å solute, as a function of temperature with increasing strength of attractive interactions. The points are simulation results from particle insertion probabilities. The solid curves are the solubilities for increasingly attractive interactions, $a_{SW}⩾0$, proceeding from the top curve to the bottom curve. The dashed curve indicates the maxima in ${\mu }_{\mathrm{A}}^{\mathrm{ex}}∕kT$.

Figure 10

Excess solute chemical potential as a function of temperature for solutes of varying size. The solid lines correspond to the revised SPT model. The solid circles correspond to molecular simulation results for the 2, 3, and 4 Å radius solutes. Estimated statistical errors are smaller than the plotting symbols. The dashed curve indicates the locus of chemical-potential maxima, where $s_{\mathrm{A}}^{\mathrm{ex}}=0$, with changing cavity size.

Figure 11

Solvent-accessible surface (SAS) area derivatives of the hard-sphere solute hydration thermodynamics as a function of solute radius at $T=300\mathrm{K}$. The thin solid, long-short dashed, short dashed, and long dashed curves correspond to $\partial {\mu }_{\mathrm{A}}^{\mathrm{ex}}∕\partial A_{\mathrm{SAS}}$, $\partial h_{\mathrm{A}}^{\mathrm{ex}}∕\partial A_{\mathrm{SAS}}$, $T\partial s_{\mathrm{A}}^{\mathrm{ex}}∕\partial A_{\mathrm{SAS}}$, and $T\partial c_{\mathrm{A}}^{\mathrm{ex}}∕\partial A_{\mathrm{SAS}}$, respectively. The horizontal curve indicates the macroscopic surface tension for a flat surface.

Figure 12

(Color online) Alternative definitions of the surface for evaluation of the surface tension. The solvent-accessible surface (SAS) is defined by the distance of closest approach between the center of the cavity and the water-oxygen center $R$. $R-\Delta R$ locates a neighboring surface that might provide a curvature-corrected surface tension. The radius $R-{\sigma }_{\mathrm{W}\mathrm{W}}∕2$ gives the van der Waals boundary at ${\sigma }_{\mathrm{A}\mathrm{A}}∕2$.

Figure 13

Surface tensions referred to defined surfaces, displaced with respect to the solvent-accessible surface located at the radius $R$, for hydration of hard-sphere solutes as a function of solute size at $T=300\mathrm{K}$. The thick solid curve corresponds to the surface tension determined by the derivative with respect to the solvent-accessible surface (SAS) area [Eq. (16)]. The curves above the baseline SAS tension indicate the effect of increasing $\Delta R$ in 0.25 Å increments from 0.25 to 1.25 Å [Eq. (17)]. $\Delta R>0$ moves the surface inward. The thick dashed curve corresponds to $\Delta R=1\mathrm{\AA }$, for which the surface tension is relatively insensitive to curvature.

Figure 14

The curvature correction $\delta$ as a function of temperature along the saturation curve of water. The points correspond to the values determined by the fit of Eq. (13) to the simulation free energies. The solid curve is a guide to the eye. The dashed curve corresponds to the classic SPT prediction for the Tolman length, Eq. (10) in 108.

Figure 15

Entropy of hydrophobic hydration as a function of temperature for solutes in the size range $2<R<10\mathrm{\AA }$ in 1 Å increments. The dashed lines are excess entropies of hydration. The open circles are convergence temperatures for consecutive solutes, i.e., $s_{\mathrm{A}}^{\mathrm{ex}}\left(R\right)=s_{\mathrm{A}}^{*}(R+1\mathrm{\AA })$. The thick solid line indicates the entropy-convergence temperature in the limit of infinitesimal perturbations in $R$.

Figure 16

Variation of the entropy-convergence temperature with increasing hard-sphere radius. The thin solid curve is the convergence temperature determined under the assumption that the heat capacity is independent of temperature. The thick solid curve is the exact entropy-convergence temperature for $R<{\sigma }_{\mathrm{W}\mathrm{W}}∕2$ from Eq. (22). The dashed curve smoothly interpolates between the exact and constant heat-capacity curves at 1.25 and 3.3 Å, respectively. The filled circle indicates the entropy-convergence temperature of a methane-sized solute $(T_c=382\mathrm{K})$. The open circle indicates the entropy-convergence temperature based on the information theoretic (IT) model criterion $(T_c=420\mathrm{K})$.