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.