Role of the displacement current on Warburg-type behavior

We investigate the role of the displacement current in the analysis of the electric response of an electrolytic cell to an external stimulus. We show that several models proposed to interpret the spectra deduced by means of the impedance spectroscopy technique are questionable. In particular, we demonstrate that even in the frequency range below the Debye frequency the role of the displacement current is fundamental, and its omission leads to incorrect results for the impedance of the cell. In our analysis, the boundary conditions on the bulk current density are of Nernstian and of Ohmic type. The analysis is limited to a fully dissociated electrolyte, and for only one type of mobile ions, as discussed in several papers devoted to the subject. Particular attention is given to the spatial dependence of the current density. We show that Warburg-like behavior is never predicted in the framework of the Poisson-Nernst-Planck model, if the electric impedance of the cell is correctly evaluated. From this conclusion, valid for media with only one type of mobile ions, it follows that if Warburg-like behavior is experimentally observed the theoretical interpretation is still an open problem, and its origin is probably related to the boundary conditions.

Figure 1

Frequency dependence of $R_N/R_0$ (a) and $-X_N/R_0$ (b) and parametric plot of $-X_N/R_0$ vs $R_N/R_0$ (c) for a symmetric cell in the Nernstian approximation, when $a=0$, diffusive current (solid line); $a=1$, Einstein approximation (dashed line); and $a=10$, important drift (dotted line). The curves are drawn for $k=1$. The vertical lines are the circular relaxation frequencies ${\mathrm{\Omega }}_r=a+k$.

Figure 2

Comparison of the frequency dependence of the real (a) and imaginary (b) part of the electric impedance of the cell derived by means of the complete model (solid lines) and of the pure diffusion model (dashed lines), where the total electric current is identified only with the conduction current due to diffusion. The parametric plots are reported in panel (c). The vertical lines are the circular relaxation frequencies determined by means of the two models. The linear part present in the curves derived by means of pure diffusion models has been identified in the past with a Warburg-like impedance.

Figure 3

Left: Total reduced current (blue horizontal line) for zero drift (dashed line) at the frequency $\mathrm{\Omega }=0.01$ and its nonzero components, diffusion (red solid line) and displacement current (black dot-dashed line), for the symmetric case of the Nernstian approximation. Right: Electric field in the cell. Note that the electric field is not zero far from the electrodes, thus neglecting drift seems questionable.

Figure 4

Left: Total reduced current (blue horizontal line) at the frequency $\mathrm{\Omega }=0.01$ and its components, diffusion (red solid line), drift (dashed line), and displacement current (black dot-dashed line), for the symmetric case of the Nernstian approximation. Right: Electric field in the cell.

Figure 5

Frequency dependence of the real (a) and imaginary (b) part of the impedance, and parametric plot of the imaginary vs the real part (c). The cell in the pure diffusive regime is limited by a reflecting electrode, at $\zeta =-1/2$. On the electrode at $\zeta =1/2$ the boundary condition is of Nernstian type characterized by $k=1$. Full expression of the real and imaginary parts of the impedance is shown by the solid line, while approximated expressions valid in the high and low frequency regions are shown by the red dashed and black dotted lines, respectively. In panel (a), the horizontal lines are the plateaus corresponding to $5R_0/\left(3k\right)$ and to $2R_0$. In panel (c) the vertical asymptote corresponds to $5R_0/\left(3k\right)$.

Figure 6

Same as Fig. 5, with $k={10}^{-5}\le k_c$.

Figure 7

Spectra of the real (a), imaginary (b), and parametric (c) plot of the real part with respect to the imaginary part, of $Z(h,h,0)$ (black solid line) and $Z(h,0,0)$ (dashed blue line). In the same figure are also reported the dc values for the resistances deduced by Eqs. (68) and (69). The relaxation frequency, visible in panel (b), is ${\mathrm{\Omega }}_r={\pi }^2/{\left(2M\right)}^2$. In the parametric plot (c), the vertical asymptote (red line) is defined by Eq. (69), and the corresponding frequency is ${\mathrm{\Omega }}_r$. The curves are drawn for $M={10}^3,h=2$.

Figure 8

Frequency dependence of the logarithm of the real part of $Z(h,0,0)$, and the approximated expression given by the real part of Eq. (71). Note that it is practically frequency independent in the low frequency region, dependent on ${\mathrm{\Omega }}^{-2}$ in the intermediate frequency region, and dependent on ${\mathrm{\Omega }}^{-5/2}$ in the high frequency region. The curves are drawn for $M={10}^3,h=2$.