Contribution of Ion Desolvation to Capacitance
Figure
1 (c) depicts a representative nanopore embedded in microporous
electrode. The nanopore is modelled as a structureless slit pore with
pore width , and both surfaces are positively charged. When calculating
the capacitance, the solvated ions are usually described with
coarse-grained model. Here the
restricted
primitive model (RPM) is adopted, in which
both
the cation and anion are modelled as charged hard-sphere particles of
the same size, and the solvent is treated as a dielectric continuum
characterized with a fixed dielectric
constant
. The ionic size often takes the value of solvation diameter,
.20,
32
Similar to the MDFT,
the
grand potential of a simple ionic system in the framework of CDFT can be
formulated as a functional of local density distribution:
, (5)
where
is the chemical potential of ionic
species determined by the given bulk thermodynamic condition. is the
non-electrostatic external
potential, and in the coarse-grained model, unless otherwise specified
elsewhere we adopt the hard-wall potential, i.e ., for and
otherwise with being the perpendicular distance of the ionic particle
within
the pore to either nanoslit surface.
is
the intrinsic Helmholtz free energy,
and it is generally composed of the ideal contribution and the excess
one due to the intermolecular interaction :
. (6)
The ideal Helmholtz free energy of the system can be formulated exactly,
reading:
, (7)
where , and
is
the thermal wavelength which is
immaterial to the final calculation results. By minimizing the grand
potential functional, i.e ., , we can obtain the local density
distribution of ionic particle at thermodynamic equilibrium:
, (8)
where is the excess part of . For the RPM model system, three
contributions should be included in the excess Helmholtz free energy
functional, including the
excluded
volume effect , the electrostatic
correlation
, and the direct Coulomb interaction .50 The
excluded volume effect is calculated by the modified
fundamental
measure theory (MFMT),51-52 extended from the
original FMT proposed by
Rosenfeld.53By ignoring the higher-order correlation, the electrostatic correlation
is derived from
the
second-order functional expansion of the excess Helmholtz free energy
functional with respect to the bulk system.37Following the mean-spherical
approximation,54the
excess Helmholtz free energy functional due to
the
contribution from direct Coulomb interaction can be formulated, and this
contribution can be numerically solved by integrating the Poisson
equation with the applied voltage, , on both slit surfaces. The detailed
expressions of three contributions are given
in
the SI .
Because the structural and thermodynamic properties vary along the
normal direction of nanoslit, we only consider the one-dimensional local
density distribution of ionic particle, , in eq.(8). For a like-charged
nanoslit, the surface charge density, , can be determined upon by
neutralizing the net charge accumulation in the half-slit, i.e.,
, (9)
where
is
elementary
charge, and
is the valence carried on the ionic particle of i -th species.
With
the relation between the applied surface voltage and the resultant
surface charge density,
the
integral capacitance of each like-charged nanoslit with pore width ,
hereafter designated as , can be calculated by .
The
overall capacitance of a microporous electrode with the PSD can be
calculated as:
, (10)
where the low limit in the integral is the smallest size of enterable
pore, and it should be determined by the bare ion diameter, i.e .,
. The upper limit, , should be the maximum pore size in the microporous
electrode.
In experiment, the microporous electrode is often characterized with the
average pore size, , in addition to the PSD. The averaged pore size is
determined by:55
, (11)
Here, is the minimum measured pore size of nanopore in the electrode
material. In this work,
is
directly extracted from the experimental data, which was determined
through the argon adsorption isotherms.56