Effects of finite ionic size and solvent polarization on the dynamics of electrolytes probed through harmonic disturbances

We address the implications of finite ionic size and solvent polarization on the response of the electric double layer (EDL) at two cation-selective electrodes in nonequilibrium conditions. The current between the electrodes is driven by a steady-state dc bias in conjunction with a probing high-frequency ac voltage. We report that the finite ionic size (steric) effect is prominent at high voltages near the electrodes where the ion densities are high, while the solvent polarization dramatically alters impedance characteristics for thick EDLs owing to the alteration of solvent permittivity in regions with a high electric field. Depending on the magnitude of the dc bias, our results show that the steric effects and solvent polarization lead to dramatic alterations in the net impedance for moderately thick electric double layers as compared to an extremely thin one. We also highlight that the solvent polarization suppresses the anomalous growth of dc current (anomalous rectification effect) for applied high-frequency ac voltages.

Figure 1

Nyquist plot of the impedance for a thick EDL $\left(\lambda =0.1\right)$ for various applied potentials. (a) Only the steric factor is accounted for, and (b) both the steric factor and polarization effect is accounted for.

Figure 2

Bode plot of the magnitude of impedance for a thick EDL $\left(\lambda =0.1\right)$ for various applied potentials. Again, subplot (a) denotes the case with only the steric factor accounted for, while (b) represents the case with both steric factor and solvent polarization.

Figure 3

Nyquist plot of the impedance for a thin EDL $\left(\lambda ={10}^{-3}\right)$ for various applied potentials. (a) $V=15$ and 25, and only the steric factor is accounted for; (b) $V=15$ and 25, where both the steric factor and polarization effect are accounted for. (c,d) depict the same trends as (a,b), only for $V=0$ and 5 instead.

Figure 4

Bode plot of the magnitude of the impedance for a thin EDL for various applied potentials. Again, subplots (a,c) denote the case with only the steric factor accounted for, while (b,d) represent the case with both steric factor and solvent polarization.

Figure 5

The dc component of $O\left({\alpha }^2\right)$ current, $I_{20}$, as a function of the frequency for an applied voltage of $V=4$. The first row entries, (a,b), depict the case for a thin EDL, $\lambda =0.001$, while the second row entries, (c,d), depict the case for a thick EDL, $\lambda =0.1$. The first column, (a,c), represents the case with only the steric factor accounted for. The second column entries, (b,d), represent the case where the solvent polarization is accounted for.

Figure 6

The dc component of $O\left({\alpha }^2\right)$ current, $I_{20}$, as a function of the frequency for an applied voltage of $V=18$. The various rows and columns correspond to the scheme described in Fig. 5.

Figure 7

Variation of the dc component of $O\left({\alpha }^2\right)$ current, $I_{20}$, as a function the frequency for an applied voltage of $V=18$. The λ is fixed for both the cases at 0.1. For the line represented by the marker, solvent polarization effects are neglected, while the solid line represents the case with solvent polarization. The ionic concentration is $0.001M$ and the length is $0.13\mu \mathrm{m}$.

Figure 8

Spatial variation of the base potential $\left({\psi }_0\right)$ for (a) $\lambda ={10}^{-1}$ and (b) $\lambda ={10}^{-3}$. The insets are magnified portions of the graphs indicated by the arrows. The blue curves depict the characteristics for an applied potential $V=25$, while the green curves indicate the curves for $V=5$.

Figure 9

Spatial variation of the base permittivity $\left({\varepsilon }_0\right)$ for (a) $\lambda ={10}^{-1}$ and (b) $\lambda ={10}^{-3}$. The blue curves depict the characteristics for an applied potential $V=25$, while the green curves indicate the characteristics for $V=5$. For (b), we have depicted only the variation in the small region near the wall since there is not much variation in the base permittivity (${\varepsilon }_0\to 1$ in the bulk) across the channel for an extremely thin EDL.

Figure 10

Spatial variation of the first-order effective permittivity $\left({\varepsilon }_{1,\mathrm{eff}}\right)$ for (a) $\lambda ={10}^{-1}$ and (b) $\lambda ={10}^{-3}$. The insets are the magnified portions of the graphs indicated by the arrows. The blue curves depict the characteristics for an applied potential $V=25$, while the green curves indicate the characteristics for $V=5$. Once again, for (b), we have depicted only the variation in the small region near the wall since there is not much variation in the base permittivity across the channel for an extremely thin EDL.

Figure 11

Spatial variation of the $O\left(\alpha \right)$ potential $\left({\psi }_1\right)$ for (a) $\lambda ={10}^{-1}$ and (b) $\lambda ={10}^{-3}$. The insets are the magnified portions of the graphs indicated by the arrows. The blue curves depict the characteristics for an applied potential $V=25$, while the green curves indicate the characteristics for $V=5$. The characteristics are plotted for $ln\omega /ln\lambda =-1$ as a representative value.