(Digital Presentation) Improvement of the Compatibility between the Current-Independent Constant Voltage Loss in Wagner’s Equation for SOFCs Using Sm-Doped Ceria Electrolytes and Jarzynski’s Equality
Authors: Miyashita Tomofumi 
Published: September 2023

A solid oxide fuel cell (SOFC) converts chemical energy from a fuel gas, such as hydrogen or methane, to electrical energy. Yttria-stabilized zirconia (YSZ) films are often used as electrolytes, in which only oxygen ions are carriers. In such cases, the open-circuit voltage (OCV = 1.15 V at 1073 K) is equal to the Nernst voltage (Vth = 1.15 V at 1073 K).
Samaria-doped ceria (SDC) has higher ionic conductivity than YSZ. Therefore, SDC films are possible electrolyte candidates. However, when SDC electrolytes are used in SOFCs, the OCV at 1073 K is only 0.80 V. The voltage loss has been explained by Wagner’s equation.
Numerous subsequent models have been created based on Wagner’s equation. The current-voltage relationship can be calculated with the cathode and anode polarization voltage losses using the Riess model [1]. Furthermore, Duncan and Wachsman created a nonlinear model [2]. This model can also be used to explain the equilibration process.
According to the Riess model, the OCV should decrease during electrode degradation. However, experimentally, the OCV does not change during electrode degradation [3]. The change in the equilibration of thick SDC electrolytes in response to a change in the anode gas has never been explained using the model of Duncan and Wachsman. Experimentally, when a very thick (6.6 mm) SDC electrolyte is used, the OCV can reach 0.80 V in only 5 minutes [4]. According to Weppner, the corresponding delay in the electron diffusion current should be more than 2080 minutes [5].
We proposed a current-independent anode voltage loss (0.35 V = 1.15 V- 0.80 V) [6]. Since we disproved the existence of large leakage currents in SDC electrolytes, our explanations seem to disregard and disprove Wagner’s equation. However, we noted that the dismissal of the idea of large leakage currents in SDC electrolytes is only a side effect. When SDC electrolytes experience a large anode voltage loss (0.35 V), the leakage currents are very small. This phenomenon is called the “anode shielding effect” [7].
The ionic activation energy (Ea) of SDC electrolytes is 0.7 eV (= 0.35 V ×2e). During ion hopping processes in SDC electrolytes, the work done by ions on the surrounding lattice structure is 0.7 eV. The ions should regain 0.7 eV after hopping. However, when there are many electrons in the hopping path, the hopping processes are very complex, and calculating the work loss is challenging [8].
Jarzynski’s equality is very useful for calculating the Gibbs energy loss at the nanoscale. One ion hopping process in SDC electrolytes is at the nanoscale. However, the number of oxygen ions in SDC electrolytes is on the order of 1021, which is near Avogadro’s number. The number of oxygen ions that can participate in the hopping process is on the order of 1016 at 1073 K. Furthermore, the average frequency of the oxygen ions is 1013 Hz. Consequently, we are discussing an average Gibbs energy loss on the order of 1029 in one second. This means that our model is much more statistical as follows.
Jarzynski’s equality ⊇ Our model (OCV = Vth -Ea /2e) (1)
Fortunately, the canonical ensemble for the ion hopping process follows Boltzmann's distribution, which is an exponential distribution and determined only by the temperature of the environment.
We try to improve the compatibility between the current-independent constant voltage loss in Wagner’s equation for SOFCs using SDC electrolytes and Jarzynski’s equality.
References:
[1] I. Riess, J. Phys. Chem. Solids., 47(2), 129 (1986). doi: 10.1016/0022-3697(86)90121-6.
[2] K. L. Duncan and E. D. Wachsman, J. Electrochem. Soc., 156, B1030 (2009). doi:
10.1149/1.315851.
[3] T. Miyashita, J. Mater. Sci., 41(10), 3183 (2006). doi: 10.1007/s10853-006-6371-8.
[4] T. Miyashita, J. Electrochem. Soc., 164(11) 3190 (2017). doi: 10.1149/2.0251711jes
[5] J. Liu and W. Weppner, Ionics, 5(1–2), 115 (1999). doi: 10.1007/BF02375914.
[6] T. Miyashita, J. Mater. Sci., 40(22), 6027 (2005). doi: 10.1007/s10855-005-4560-2.
[7] B. Dalslet, P. Blennow, P. V. Hendriksen, N. Bonanos, D. Lybye, and M. Mogensen, J. Solid State Electrochem., 10(8), 547 (2006). doi: 10.1007/s10008-006-0135-x.
[8] T. Miyashita, Miyashita, ECSarXiv (2020) “Open-Circuit Voltage Anomalies in Yttria-Stabilized Zirconia and Samaria-Doped Ceria Bilayered Electrolytes”,