Derivation of the solvation effect-incorporated Poisson-Boltzmann equation

[Display omitted] •Potential energy of solvated ion was denoted by terms of the ion and solvation shell.•There exists dynamic balance of the formation and decay of ionic atmosphere.•There also exists dynamic balance of the formation and decay of solvation shell.•Solvation effect incorporated Poison-Boltzmann (SEI-PB) equation was derived.•Ion-solvent molecule dipole moment interaction potential energy is derived. Based on molecular kinetic consideration relating to dynamic properties for both the ionic atmosphere and ion solvation shell, solvation effect-incorporated PB (SEI-PB) equation was derived. Consequently, there appeared a solvated ionic screening thickness containing not only the ionic atmosphere thickness (Debye length), but also a new screening thickness of the solvation shell in both the linearized SEI-PB equation and its solution. The thickness of solvated ionic atmosphere has physical meaning of the screening thickness taking into account the volumetric effect of ions together with ionic solvation effect. As a result of the solution of linear SEI-PB equation, the total electric potential was divided into two terms: the first one was derived from the ions, which were affected by the solvation shells and the second one was derived from the solvation shells which were affected by the ions. An analytical equation for ion–solvent molecular dipole moment interaction (ISMDI) potential energy was established based on the solution of linear SEI-PB equation. Also by using the linear SEI-PB equation, analytical expressions of the activity coefficients for ions and electrolytes as well as solvent were derived and an equation for estimating the mean lifetimes of solvation shells according to ionic species was proposed and applied to estimate those of ionic hydration shells. Estimated mean lifetimes for ionic hydration shells according to ionic species were compatible with the previous XRD, NMR data and MD simulations. We confirmed that the SEI-PB equation has peculiarity containing not only solvation effect, but also dielectric saturation and that the dielectric saturation is connected implicitly with ionic solvation effect.