Size dependence of the ion pairing preferences investigated by free energy calculations

We calculate pairing free energies for a series of model monovalent ion pairs with equal sizes spanning a large radius range in the aqueous solution. Thermodynamic analysis reveals that the ion pairing strength displays an initial decrease in a nearly linear fashion followed by a significant enhancement at a relatively slower rate as the ion size gradually increases, resulting in the weakest pairing preference for ions with intermediate size. The free energy decomposition illustrates that the ion pairing is jointly governed by a delicate balance of the favorable ion–ion interaction term and the repulsive solvent-induced contribution, with their compensation leading to a small pairing free energy. The dependence of this substantial compensation on the ion size actually dominates the ion pairing strength and the magnitude of the free energy, which accounts for the occurrence of the less association between the intermediate-size ions. Estimation of van der Waals and electrostatic free energies highlights the importance of attractive dispersion forces in determining the shape of the potential of mean force, specifically for large ions with less favorable interaction with water molecules than the water–water interactions. In addition, the effect of varying the cation size on the decreased pairing preference surpasses that of altering the anion size. Our study significantly enhances our understanding of the empirical rule of matching ion size for predicting ion pairing preferences in aqueous solutions.