Physical Interpretations of Nyquist Plots for EDLC Electrodes and Devices

Abstract

This study aims to provide unequivocal physical interpretations of electrochemical impedance spectroscopy (EIS) results for electric double layer capacitor (EDLC) electrodes and devices. To do so, a physicochemical transport model was used for numerically reproducing Nyquist plots accounting for (i) electric double layer (EDL) formation at the electrode/electrolyte interface, (ii) charge transport in the electrode, and (iii) ion electrodiffusion in binary and symmetric electrolytes. Typical Nyquist plots of planar EDLC electrodes were reproduced and interpreted numerically for different electrode conductivity and thickness, electrolyte domain thickness, as well as ion diameter, diffusion coefficient, and concentrations.

Analysis

Electrochemical impedance spectroscopy (EIS) consists of imposing a time harmonic oscillating electric potential ψ_s(t) of small oscillation amplitude (e.g., less than 10 mV) around a time-independent “DC potential” at the electrode surfaces and measuring the resulting harmonic current density j_s(t) [1-3]. Using complex notations, the imposed electric potential ψ_s(t) and the resulting current density j_s(t) can be expressed as [1,4-6],

where ψ_dc is the time-independent DC potential, ψ₀ is the amplitude of the oscillating potential at frequency f, j_dc is the time-independent DC current density, j₀ is the amplitude of the oscillating current density, and ϕ(f) is the frequency-dependent phase angle between the imposed potential ψ_s(t) and the current density j_s(t). Then, the electrochemical impedance Z can be defined as [1,4-6]

where Z_re and Z_im (expressed in m²) are the real and imaginary parts of the complex impedance, respectively. A typical Nyquist plot presents the imaginary part - Z_im as a function of the real part Z_re of the complex impedance of an EDLC electrode or an EDLC device.

Simulations reported in this study were based on the modified Poisson-Nernst-Planck(MPNP) model for binary and symmetric electrolyte for EDLC electrodes or devices. The governing equations, initial and boundary conditions, and method of solution were described in detail in Refs. [7,8]. Note that the electrical conductivity and thickness of the electrode, the effective ion diameter, ion diffusion coefficient, bulk ion concentration in the electrolyte, the length of the electrolyte domain were treated as variables to assess the effects of electrode and electrolyte resistances and electrode capacitance on the Nyquist plots of a single EDLC electrode.

Fig. 1 shows (a) the simulated one-dimensional domain consisting of a planar current collector supporting a working electrode of thickness L_e and an electrolyte domain of thickness L corresponding to a three-electrode setup and (b) an EDLC device consisting of two identical electrodes of thickness L_e separated by an electrolyte domain of thickness 2L. The electrolyte near each electrode consisted of a Stern layer of thickness H and a diffuse layer of mobile ions of thickness L_D. Simulations reported in this study were based on the modified Poisson-Nernst-Planck (MPNP) model for binary and symmetric electrolyte for EDLC electrodes or devices [7-9]. The governing equations, initial and boundary conditions, and method of solution were described in detail in Refs. [9,10]. Note that the electrical conductivity and thickness of the electrode, the effective ion diameter, ion diffusion coefficient, bulk ion concentration in the electrolyte, the length of the electrolyte domain were treated as variables to assess the effects of electrode and electrolyte resistances and electrode capacitance on the Nyquist plots of a single EDLC electrode.

Fig. 1. Schematics of (a) the simulated one-dimensional EDLC electrode in a three-electrode setup and (b) EDLC devices consisting of two identical electrodes. The dashed line encloses the computational domain simulated.

Results

Fig. 2 summarizes the physical interpretation of the Nyquist plots computed numerically for a wide range of cases with different electrode conductivity and thickness, electrolyte thickness, and ion diameters, concentrations, and diffusion coefficients.

Fig. 3 shows the Nyquist plots obtained expetrimentally for EDLC devices consisitng of activated carbon electrodes and electrolytes of (1) 1M LiPF₆ in EC:DMC(1:1), (2) 1M citric acid in DI water, and (3) 1M TEATFB in Acetonitrile. It indicates that the electrode resistances R_e^exp was small and did not vary significantly among the three devices. This is consistent with the fact that the electrodes of all three devices were nearly identical and made of activated carbon [11]. In addition, the electrode resistance R_e^exp was small compared with that of the bulk electrolyte resistance R_∞^exp, which contributed the most to the internal resistance. Moreover, the bulk electrolyte resistance R_∞^exp for Device 2 was much larger than that for Devices 1 and 3. This can be attributed to the fact that citric acid is a weak electrolyte featuring low ionic conductivity. Furthermore, the internal resistance of the devices retrieved from the Nyquist plot was in good agreement with the resistance retrieved from “IR drop” in galvanostatic cycling. These results were consistent with the physical interpretation derived numerical simulations and illustrated in Fig. 2.

Fig. 2. Numerically generated Nyquist plots for planar EDLC electrodes and associated physical interpretations.

Figure 3. Experimental Nyquist plots for EDLC devices.

Conclusion

The electrode resistance, bulk electrolyte resistance, diffuse layer resistance, and equilibrium differential capacitance can be retrieved directly from Nyquist plots.
The internal resistance retrieved from the sum of electrode and bulk electrolyte resistances in EIS simulations showed good agreement with the internal resistance retrieved from the so-called ``IR drop'' in galvanostatic cycling.
The interpretations were confirmed experimentally for EDLC devices with electrodes made of activated carbon and various electrolytes.

References

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Publications

B.M. Mei, O. Munteshari, J. Lau, B. Dunn, and L. Pilon, 2018. “Physical Interpretations of Nyquist Plots for EDLC Electrodes and Devices”, Journal of Physical Chemistry C, Vol. 122, No. 1, pp. 194-206. doi:10.1021/acs.jpcc.7b10582 pdf