  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 the 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 hydropobe 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.
