A probabilistic approach to the effect of hydrogen bonding on the hydrophobic attraction

Water molecules, belonging to the first hydration shell around a hydrophobic particle, form fewer hydrogen bonds than bulk molecules. On the other hand, the former (boundary) bonds may be slightly stronger than the latter. When two hydrophobic particles are sufficiently close to each other, the disruption of water-water hydrogen bonds in their first hydration layers can give rise to an additional contribution to their overall interaction potential. Here we present a probabilistic approach to studying this phenomenon. The proposed method allows one to determine the average number of hydrogen bonds per water molecule in the first hydration shell. Numerical evaluations show that in the interplay between a decrease in the number of boundary bonds per water molecule and the enhancement of such a bond the former effect is clearly predominant. As a result, the disruption of boundary hydrogen bonds, which occurs when the first two hydration shells of two particles overlap, leads to an attractive contribution to the overall particle interaction. This contribution is naturally short range, appearing only when the separation between the two particles becomes smaller than four lengths of a hydrogen bond. It is greater than the overall van der Waals interaction potential of the same hydrophobic particles (with typical Hamaker constants) by at least an order of magnitude.