Probabilistic approach to the length-scale dependence of the effect of water hydrogen bonding on hydrophobic hydration

We present a probabilistic approach to water-water hydrogen bonding that allows one to obtain an analytic expression for the number of bonds per water molecule as a function of both its distance to a hydrophobic particle and hydrophobe radius. This approach can be used in density functional theory (DFT) and computer simulations to examine particle size effects on the hydration of particles and on their solvent-mediated interaction. For example, it allows one to explicitly identify a water hydrogen bond contribution to the external potential, whereto a water molecule is subjected near a hydrophobe. The DFT implementation of the model predicts the hydration free energy per unit area of a spherical hydrophobe to be sharply sensitive to the hydrophobe radius for small radii and weakly sensitive thereto for large ones; this corroborates the vision of the hydration of small and large length-scale particles as occurring via different mechanisms. On the other hand, the model predicts that the hydration of even apolar particles of small enough radii may become thermodynamically favorable owing to the interplay of the energies of pairwise (dispersion) water-water and water-hydrophobe interactions. This sheds light on previous counterintuitive observations (both theoretical and simulational) that two inert gas molecules would prefer to form a solvent-separated pair rather than a contact one.