Density Depletion and Enhanced Fluctuations in Water near Hydrophobic Solutes: Identifying the Underlying Physics

We investigate the origin of the density depletion and enhanced density fluctuations that occur in water in the vicinity of an extended hydrophobic solute. We argue that both phenomena are remnants of the critical drying surface phase transition that occurs at liquid-vapor coexistence in the macroscopic planar limit, i.e., as the solute radius $R_s\to \infty$. Focusing on the density profile $\rho (r)$ and a sensitive spatial measure of fluctuations, the local compressibility profile $\chi (r)$, we develop a scaling theory which expresses the extent of the density depletion and enhancement in compressibility in terms of $R_s$, the strength of solute-water attraction ${ϵ}_s$, and the deviation from liquid-vapor coexistence $\delta \mu$. Testing the predictions against results of classical density functional theory for a simple solvent and grand canonical Monte Carlo simulations of a popular water model, we find that the theory provides a firm physical basis for understanding how water behaves at a hydrophobe.

Figure 1

Cross sections of a snapshot of mw water molecules (diameter ${\sigma }_{\mathrm{mw}}=0.239\mathrm{nm}$) surrounding a hard spherical solute of radius $R_s=2.15\mathrm{nm}$ (a) and $R_s=4.07\mathrm{nm}$ (b). The shade of the mw water molecules lightens the greater their distance from the cross-sectional plane. The corresponding average local compressibility profiles $\chi (r)$ (c) and density profiles $\rho (r)$ (d), scaled by their bulk values. The profile line color matches that of the corresponding solute. Temperature $T=426\mathrm{K}$, $\beta \delta \mu ={10}^{-3}$.

Figure 2

DFT results for (a),(b) $\chi (r)$ and (c),(d) $\rho (r)$, scaled by their bulk values, for a truncated LJ model liquid in contact with a spherical solute. (a) and (c) show the effect of varying solute radius $R_s$ for constant ${ϵ}_{\mathrm{sf}}=0.0$—the hard sphere solute. (b) and (d) show the effect of increasing attraction ${ϵ}_{\mathrm{sf}}$ for constant $R_s=100\sigma$. In all cases, $T=0.775T_c$ and $\beta \delta \mu ={10}^{-3}$. The inset shows how the contact angle $\theta$, pertaining to bulk coexistence and the planar limit, depends on ${ϵ}_{\mathrm{sf}}$.

Figure 3

GCMC results for (a),(b) $\chi (r)$ and (c),(d) $\rho (r)$, scaled by their bulk values, for monatomic water (diameter ${\sigma }_{\mathrm{mw}}=0.239\mathrm{nm}$) in contact with spherical solutes. (a) and (c) refer to a hard solute for seven solute radii $R_s$, spanning the interval $(1.2:4.06)\mathrm{nm}$. (b) and (d) refer to four values of the substrate-fluid attraction ${ϵ}_{\mathrm{sf}}$ and constant radius $R_s=17{\sigma }_{\mathrm{mw}}$.

Figure 4

(a),(b) DFT results for values of $\beta \delta \mu$ and $R_s$ spanning the intervals $({10}^{-6},{10}^{-3})$ and $(10\sigma ,\infty )$, respectively, and various ${ϵ}_{\mathrm{sf}}/ϵ$ as shown on the color chart, testing scaling predictions (3) and (5). (c),(d) GCMC results for mw at $\beta \delta \mu ={10}^{-3}$ and $T/T_c=0.464$ testing the scaling predictions in the computationally accessible curvature limit. The values of ${ϵ}_{\mathrm{sf}}/{ϵ}_{\mathrm{mw}}$ are given in the legend.