where is the volume of the solvated ion, i.e ., , with being the solvation diameter. is the volume of the bare ion, and its diameter, , is evaluated from the Barker-Henderson theory.46represents the occupied volume of contributing MeCN molecules, and being estimated through . Here is the number density of bulk MeCN solvent, and is the solvation number of MeCN. For an anion, the solvation number is calculated through the following expression:
, (2)
where with , and is the critical orientation angle in the first solvation shell to weight the contribution of MeCN molecules to the solvation diameter.36is the local orientation-dependent distribution function of MeCN solvent surrounding the anion. represents the radius of the first valley in . In general, the value of varies at different orientations. In other words, can be expressed as a function of ,i.e .,. The solvation diameter can be solved by combining eqs. (1) and (2) once the critical orientation angle is determined.
We determine with the help of MDFT. Within the framework of MDFT, the ion solvation in MeCN solvent can be described in the grand canonical ensemble, in which the solvent chemical potential , the system volume , and the system temperature are fixed. A single ion is immersed in MeCN under given thermodynamic condition, and by minimizing the grand potential functional of the inhomogeneous solvent system, the local density distribution of MeCN, can be obtained.47The grand potential functional is associated with the solvation free energy functional, , through:
, (3)
where represents the grand potential of the reference bulk solvent. In general, includes three contributions: the ideal term , the external contribution, and the excess term due to the intermolecular interactions .37 The expression of the solvation free energy functional follows:
. (4)
Here, denotes the deviation of local density with respect to the bulk one, and is the Boltzmann constant. is the angular-dependent direct correlation function (DCF) of the bulk solvent. Here DCF is prepared in prior by solving the molecular Orenstein-Zernike equation with the help of exact pair correlation function extracted from simulation.48 represents the external potential consisting the solute-solvent interaction and the interfacial interaction. The former includes the pairwise Lennard-Jones (LJ) potential energy and direct Coulomb interaction between the solute ion and MeCN. The latter is the non-electrostatic interaction between the MeCN solvent and flat walls, and it can be described by the 10-4 potential.49For ion dissolved in the solvent free of confinement, the latter interaction vanishes and only the solute-solvent interaction is involved. The detailed expressions of are given in Supporting Information (SI) .
2.2.