Interpreting Electrochemical Impedance Spectra with Physical Models of Mixed-Conducting Protonic Ceramic Electrochemical Cells
Authors: Zhu Huayang , Ricote Sandrine , Weddle Peter , Jackson Gregory S. , Kee Robert J. 
Published: May 2020

As Mogensen and colleagues demonstrated effectively for YSZ-based electrochemical cells [1,2], thoughtful implementation and interpretation of electrochemical impedance spectroscopy (EIS) can evaluate performance aspects of high-temperature electrochemical materials and devices, including fuel cells and electrolyzers. Impedance spectra measurements are typically interpreted with equivalent-circuit models, using variants of resistor-capacitor-inductor networks. Indirect inferences can be drawn between equivalent-circuit elements and physical processes and properties (e.g., charge-transfer rates, porous-media transport, charge-defect mobility, etc.). In addition to and support of measurements, detailed physical models of high-temperature electrochemical materials and devices, based upon deriving and solving dynamic conservation equations of the relevant fluid, thermal, and electrochemical processes, can produce impedance spectra. Such models link critical material properties and rate parameters, which can be measured or estimated independently, to EIS and thereby assist in evaluation of design and operational alternatives of electrochemical systems.
As in experiments, impedance spectra from a physical model are derived by imposing a small harmonic-current actuation and recording the harmonic response of voltage (and possibly other observables such as temperature). By repeating this over a wide range of perturbation frequencies, the complex impedance is constructed from magnitude differences and phase shifts between the actuations and responses. Deriving impedances from physical models requires that the computational algorithm represent transient responses accurately. In principle, physical models can be understood in the context of locally linearized state-space representations. The rate of change of the electrochemical state (in terms of ion and species concentrations, electrostatic potentials, temperatures, etc...) is a linearized function of the state itself and the transient “actuation” (generally a sinusoidal current). The locally linear relationships in a state-space models are represented in terms of Jacobian matrices, which can be evaluated using numerical differentiation of the physical model. Although matrix evaluation may be a difficult and tedious task for an electrochemical system, the evaluation is always possible and producing the complex electrochemical impedance spectra involves inverting the Jacobian matrix. Oftentimes, the Jacobian may be large due to discretization of the physical model and/or poorly conditioned due to a wide range of time-constants of the governing processes. Nevertheless, evaluation of the full impedance spectra is computationally very efficient and thus with the physical model derived EIS, direct relationships can be evaluated between physical properties and the impedance spectra. Calibration of model-derived impedance spectra with measured spectra provides a valuable tool for assessing otherwise difficult, if not impossible, physical parameters to isolate from experimental measurements in high-temperature electrochemical cells.
This paper illustrates the power of using dynamic physical models to formulate impedance spectra as a means for interpreting high-temperature electrochemical processes measured experimentally in protonic-ceramic symmetric-cells as reported by Pers, et al. [3]. This system (Fig. 1) was chosen because the present investigators have independently developed a physical model for BCZYYb (BaCe0·7Zr0·1Y0.1Yb0.1O3−δ) and Ni-BCZY composite electrodes [4] to interpret the performance of this system. Unlike the more well understood YSZ system, the BCZYYb and Ni/BCZY electrochemical cell presents unique modeling challenges due to the multiple ion conduction in BCZYYb and BCZY – notably protons and oxide vacancies as illustrated in Figure 1. The physical model incorporating ambipolar diffusion of the multiple ions in the electrolyte and cermet electrodes has been resolved and reported by Zhu and Kee [5]. The physical model predicts accurately the experimentally measured EIS as illustrated in Figure 1 The present study utilizes these models to develop equivalent-circuit models to evaluate the measured and the physically modeled impedance of the high-temperature proton-conducting electrochemical cells based on BCZYYb electrolytes. The results show the power of models to provide new insights in understanding alternative interpretations of the EIS for these promising materials for next-generation fuel cells and electrolyzers. The physical models provide direct relationships between impedance features and physical properties that can assist quantifying equivalent-circuit elements and physical behaviors. Such quantitative insights may accelerate design and development of electrochemical material systems for fuel cell and electrolysis units.

References:

1. J. Jørgensen, M Mogensen, J. Electrochem. Soc., 148:A433-A442 (2001).
2. H. Jensen, A. Hauch, P.V. Hendriksen, M. Mogensen, N. Bonanos, T. Jacobsen, J. Electrochem. Soc., 154:B1325-1330 (2007).
3. Pers, V. Mao, M. Taillades, G. Taillades, Intl. J. Hydrogen Energy, 43:2402-2409 (2018).
4. Zhu, S. Ricote, C. Duan, R.P. O’Hayre, R.J. Kee, J. Electrochem. Soc., 165:F845-F853 (2018).
5. Zhu and R.J. Kee, J. Electrochem. Soc., 164:F1400-F1411 (2017).

Figure 1. Diagram of a symmetric-cell using a thin BCZYYb protonic-ceramic electrolyte membrane and relatively thick porous Ni/BCZY electrodes. Inset shows example of impedance spectra measured and modeled for such a cell.

Figure 1