Calculation of the PMF

By lowering the repulsive power of classical force-field potentials, we can test the effects of a softer repulsive force, which resembles better to quantum-mechanical interactions. For different forms of repulsive part in Lennard-Jones potentials (see Fig. 19), we calculated the corresponding PMF in Fig. 20. It's found that the contact configuration becomes less stable when the repulsive part is softer. That means the system turns to be more hydrophilic. The use of $r^{-8}$ and $r^{-9}$ has obvious effects and gives different PMFs. The contact configurations become unfavorable, and the energy barrier is reduced. The potential depths of contact pair and solvent-separated pair become comparable for softer repulsive interactions.

Figure 19: Systematic varying the repulsive part of the Lennard-Jones potential.

Figure 20: There is obviously difference between PMFs when $r^{-8}$ and $r^{-9}$ as the repulsive part compared to $r^{-12}$.

Referring to Fig. 19, the attractive force between methane and water means water near the cap regions of methane-pair would reduce the (effective) attractive hydrophobic interaction between two methane molecules. As the separation between two methane molecules increases and enough room is created for water to form a layer near the mid-region, it acts like a medium to mediate an attractive force between two far-apart methane molecules. So the net effect of (DFT) water-methane interactions is a weakened hydrophobic attraction at short distance and a shallow solvent-separated potential minimum at large separation. The difference in the inter-molecular forces in Fig. 19, therefore, provides a rationale for the different features of the PMF in the quantum-mechanical simulations and the PMF based on Lennard-Jones potentials.

Another argument to call for a shallow second potential minimum is provided by fitting the form of the PMF to the available measurement. Thompson [27] studied an interaction site model to fit experimental second osmotic virial coefficients of eight aqueous alcohol systems. The PMF between two carbon atoms is represented by simple square-well potentials, and an almost perfect fitting can be found assuming only a narrow, contact square well. If a solvent-separated well is added, either stand-alone or together with a contact well, all models give very poor fits to the measurement. With a rather shallow solvent-separated potential well, the PMF in Fig. 18 provides the feature required in the fitting of the experiment.