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

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