Introduction of effective dielectric constant to the Poisson-Nernst-Planck model

The Poisson-Nernst-Planck (PNP) model has been widely used for analyzing impedance or dielectric spectra observed for dilute electrolytic cells. In the analysis, the behavior of mobile ions in the cell under an external electric field has been explained by a conductive nature regardless of ionic concentrations. However, if the cell has parallel-plate blocking electrodes, the mobile ions may also play a role as a dielectric medium in the cell by the effect of space-charge polarization when the ionic concentration is sufficiently low. Thus the mobile ions confined between the blocking electrodes can have conductive and dielectric natures simultaneously, and their intensities are affected by the ionic concentration and the adsorption of solvent molecules on the electrodes. The balance of the conductive and dielectric natures is quantitatively determined by introducing an effective dielectric constant to the PNP model in the data analysis. The generalized PNP model with the effective dielectric constant successfully explains the anomalous frequency-dependent dielectric behaviors brought about by the mobile ions in dilute electrolytic cells, for which the conventional PNP model fails in interpretation.

Figure 1

Schematic representation of ionic distribution under an external field.

Figure 2

Equivalent circuits for representing (a) the PNP model, (b) the PB model, and (c) an equivalent parallel circuit for analyzing experimental data.

Figure 3

Frequency dependences of the complex dielectric constant for a dilute electrolytic cell calculated by using the equivalent circuits in Fig. 2. Solid lines represent the values calculated by using the PNP model with the contribution of $P_{\mathrm{scp}}$ in the dielectric constant of Poisson's equation. The dotted lines represent the values by using the PNP model without the contribution of $P_{\mathrm{scp}}$. The open circles represent the values calculated by using the PB model.

Figure 4

Frequency dependences of (a) relative dielectric constant ${\varepsilon }_r^{\prime }$ and (b) relative dielectric loss factor ${\varepsilon }_r^{\prime \prime }$. Filled circles represent observed values for TBATPB-$o$-DCB solutions and solid lines represent calculated values fitted to the observed values.

Figure 5

Frequency dependences of relative dielectric constant ${\varepsilon }_r^{\prime }$ and relative dielectric loss factor ${\varepsilon }_r^{\prime \prime }$. Filled and open circles represent observed values for TBATPB-DMSO solution. Solid and dashed lines represent calculated values fitted to the observed values.

Figure 6

Frequency dependences of relative dielectric constant, ${\varepsilon }_r^{\prime }$, and relative dielectric loss factor, ${\varepsilon }_r^{\prime \prime }$. Filled and open circles represent observed values for a KCl aqueous solution. Solid and dashed lines represent calculated values fitted to the observed values.

Figure 7

Frequency dependences of (a) relative dielectric constant ${\varepsilon }_r^{\prime }$ and (b) relative dielectric loss factor ${\varepsilon }_r^{\prime \prime }$. Filled circles represent observed values for TBATPB-CB solutions and solid lines represent calculated values fitted to the observed values.

Figure 8

Effective dielectric constant as a function of ion density of solution: (a) TBATPB in $o$-DCB; (b) TBATPB in DMSO; (c) KCl in water; (d) TBATPB in CB. Solid and dashed lines represent theoretical maximum and minimum values in ${\varepsilon }_{\mathrm{eff}}$, respectively. Filled circles represent ${\varepsilon }_{\mathrm{eff}}$ values determined by the data analysis shown in Tables 1, 2, 3, 4.