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