Osmotic second virial coefficients for hydrophobic interactions as a function of solute size

To gain quantitative insight into how the overall strength of the hydrophobic interaction varies with the molecular size, we calculate osmotic second virial coefficients B for hydrophobic spherical molecules of different diameters sigma in water based on molecular simulation with corrections to the finite-size and finite-concentration effects. It is shown that B (< 0) changes by two orders of magnitude greater as sigma increases twofold and its solute-size dependence is best fit by a power law B proportional to sigma(alpha) with the exponent alpha ? 6, which contrasts with the cubic power law that the second virial coefficients of gases obey. It is also found that values of B for the solutes in a nonpolar solvent are positive but they obey the same power law as in water. A thermodynamic identity for B derived earlier [K. Koga, V. Holten, and B. Widom, J. Phys. Chem. B 119, 13391 (2015)] indicates that if B is asymptotically proportional to a power of sigma, the exponent alpha must be equal to or greater than 6. Published under an exclusive license by AIP Publishing.