Equilibration Process in Response to a Change in the Anode Gas Using Thick Sm-Doped Ceria Electrolytes in Solid-Oxide Fuel Cells

The electronic conductivity of Sm-doped ceria is very low in air but increases substantially in H 2 gas. Conventional models can explain the equilibration processes of yttria-stabilized zirconia electrolytes and thin mixed ionic-electronic conducting electrolytes in response to variations in the fuel composition. However, the equilibration processes of thick samaria-doped ceria electrolytes have not yet been explained. We measured and attempted to explain the equilibration process of a very thick (6.6 mm) samaria-doped ceria electrolyte in response to a change in the anode gas. The measured open-circuit voltage gradually increased to an equilibrium voltage of 0.80 V within 5 min. However, based on the chemical diffusion coefﬁcient equation for the electron diffusion current, the equilibrium time should have been much longer than 5 min. When we assumed a current-independent constant anode voltage loss (0.35 V), the calculations were substantially improved for determining the experimental results.

Solid-oxide fuel cells (SOFCs) directly convert the chemical energy of fuel gases, such as hydrogen and methane, into electrical energy.SOFCs use a solid-oxide film as the electrolyte, and oxygen ions serve as the main charge carriers.Typically, yttria-stabilized zirconia (YSZ) is used as the electrolyte material in these cells.The current operating temperature of these cells (873-1273 K) should be reduced to extend their lifespans.Therefore, the use of more advanced ionconducting electrolyte materials, such as samaria-doped ceria (SDC) electrolytes, at lower temperatures is preferred.The open-circuit voltage (OCV) of an SDC cell is approximately 0.8 V, which is lower than the Nernst voltage (V th ) of 1.15 V at 1073 K.The low OCV is attributed to the low value of the ionic transference number (t ion ).
The model described by Duncan and Wachsman 1,2 can explain the equilibration processes of thin and thick YSZ electrolytes and thin mixed ionic-electronic conducting (MIEC) electrolytes in response to higher transient voltages or variations in the fuel composition.However, the equilibration process using thick SDC electrolytes in response to a change in the anode gas remains unexplained.In this work, we measured the equilibration process using a very thick (6.6 mm) SDC electrolyte in response to a change in the anode gas and used Wagner's Equations 3, 4 to calculate the oxygen chemical potential profile.Using Rickert's approach, 5 we subsequently analyzed the drift current and electron diffusion current throughout the entire electrolyte under open-circuit conditions.Weppner's method 6,7 was used to calculate the equilibration rate of the electron diffusion current.On the basis of these calculations, we showed that the equilibration time of the electron diffusion current is the limiting factor when using a thick SDC electrolyte.Consequently, the equilibration rates should differ from those obtained using thin and thick YSZ electrolytes and thin MIEC electrolytes.Furthermore, we discussed the problems that occurred and their solutions.When we assumed a current-independent constant anode voltage loss (V const = 0.35 V), the calculations were substantially improved for determining the experimental results.

Theory
Oxygen chemical potential profile.-Theflux of the oxygen ions under open-circuit conditions, as described by Wagner's * Electrochemical Society Member.
z E-mail: tom_miya@ballade.plala.or.jpEquations 3, 4, is are the O 2 partial pressures at the cathode and the anode, respectively; R, T, and F are the gas constant, the absolute temperature, and Faraday's constant, respectively; L is the thickness of the membrane or film; and σ el and σ ion are the conductivities of the electrons and oxygen vacancies, respectively.From Equation 1, Equations 2 and 3 can be deduced, as shown in Appendix A: OC V = V th − R i I i [2]   where R i and I i are the ionic resistances of the electrolyte and ionic current, respectively.

OC V = RT 4F
ln pO cathode 2 ln pO anode 2 t ion d ln pO 2 [3]   Equation 3 is famous and often called "Wagner's equation" when σ ion is constant under various O 2 partial pressures 5 and I i is not apparently expressed.Therefore, Equations 2 and 3 are not empirical equations.When operating under OCV conditions, the electronic leakage current (I e ) can be defined using the external current (I ext ): In general, I i and I e are not zero.Parameter t ion is expressed as follows: However, σ el is a function of the O 2 partial pressure: 8 where pO * 2 corresponds to the oxygen partial pressure at which t ion = 1/2.
According to Gödickemeier and Gauckler, 9 at 1073 K, pO * 2 is 4.8 × 10 −17 atm and σ ion is 0.089 S/cm. 9The O 2 partial pressure of H 2 gas (including 3% H 2 O) at 1073 K is 2.7 × 10 −22 atm, which can be calculated from V th (1.15 V) when the pO cathode 2 is 1 atm.The relationship between the distance from the anode ( ) and the O 2 partial pressure (the oxygen chemical potential profile) and that between and t ion are shown in Fig. 1.The calculation procedure is presented in Appendix B. These values can be calculated using the models described by Duncan and Wachsman, 1,2 Choudhury and Patterson, 10 Yuan and Pal, 11 Näfe, 12 and Riess. 13The model described by Duncan and Wachsman is a nonlinear-type model.According to Singh and Jacob, 14  Electron drift current and electron diffusion current.-Theresults in Fig. 1 are correct after dϕ/dx is determined.The electron drift and the electron diffusion current (Ie_ diffusion ) should then be determined in terms of I e .According to Rickert, 5 where S, z i , η ion , and η el are the cross section, valence of species i, electrochemical potential of the ions, and electrochemical potential of the electrons, respectively.In general, I ext is not zero in Equation 7. Thus, from Ref. 6, The electrochemical potential (η) may be separated into the chemical potential (μ) and the electrostatic potential (ϕ): Equation 8 can then be rewritten in terms of the fluxes of the ions and electrons resulting from diffusion and drift: The μ ion gradient can be neglected because of the high concentration of oxide ions relative to the electrical field (dϕ/dx).
From Equation 12, I e_drift and I e_diffusion can be defined as and In Equation 13, the direction of I e_drift must be the same as that of I i throughout the electrolyte.Therefore, I e_drift should be recycled as I e_diffusion throughout the whole electrolyte.A schematic showing the directions of I i , I e_drift and I e_diffusion is presented in Fig. 2.
Under OCV conditions and based on Equations 4, 11 and 13, Equation 15 has been proven theoretically and is compatible with Equation 3. The proof is shown in Appendix C. From Equations 6 and 13, From Equations 6 and 15, When pO cathode 2 , pO anode 2 , and pO * 2 are 1, 2.7 × 10 −22 , and 4.8 × 10 −17 atm, respectively, the calculated OCV from Equation 3 is 0.85 V. Thus, when R i is 1 , I i is 0.30 A, and I e is −0.30 A. In the case of a 1-mm-thick electrolyte, dϕ/dx is −0.30V/mm.The relationship between and I e_diffusion in the electrolyte is shown in Fig. 3a.The relationship between and I e_drift in the electrolyte is shown in Fig. 3b.
The calculation results shown in Figs.3a and 3b are correct after dϕ/dx is determined because the oxygen chemical potential profile shown in Fig. 1 is used.The absolute values of both I e_diffusion and I e_drift should increase when σ el is increased near the anode.Even when I e is −0.30A, I e_drift can exceed 6 A near the anode.A large energy loss should occur because of the large recycled I e_drift in the SDC electrolyte; this energy loss causes serious problems and is discussed later.

Chemical diffusion coefficient equation for the electron diffusion current.
-According to Liu and Weppner, 6 a partial ion-blocking electrode can be used to measure I e_diffusion experimentally.To calculate the value of I e_diffusion , Liu and Weppner used Rickert's approach.Therefore, the I e_diffusion value calculated in the previous section should be essentially equivalent to that used in their calculation.However, the experimental situation was different.A schematic of the experimentally determined directions of I i , I e_drift , and I e_diffusion is shown in Fig. 4. In the experiment by Liu and Weppner, dϕ/dx was negligible; thus, I e_drift should have also been negligible.Consequently, the equilibration rate shown in Fig. 4 should be higher than that observed under typical SOFC conditions in response to a change in the anode gas for the open-circuit case shown in Fig. 2 because I e_drift does not suppress I e_diffusion .
According to Thangadurai and Weppner, 7 in the case of using the partial ion-blocking electrode shown in Fig. 4, the time delay of the change in I e_diffusion in response to a change in the surrounding gas can be accurately measured and calculated using the chemical diffusion coefficient equation: where τ, L, and D are the equilibrium time, length of the sample, and chemical diffusion coefficient, respectively.
Equilibration process for thin and thick YSZ electrolytes and thin MIEC electrolytes.-Recently,a more refined model was developed by Duncan and Wachsman to describe the dependence of the point defect concentration. 1,2Using this model, they explained the equilibration processes of thin and thick YSZ electrolytes and thin MIEC electrolytes in response to higher transient voltages or variations in the fuel composition (p.28 in Ref. 2).Duncan and Wachsman discovered that, although the boundary concentrations are established instantly, the concentration in the bulk lags behind, ostensibly limited by the electron diffusivity and electrolyte thickness.Therefore, the voltage appears to be generated at the interface between the electrode and the electrolyte before the bulk reaches equilibrium.However, their model does not explain the equilibration process for a thick SDC electrolyte in response to a change in the anode gas.
Equilibration process for a thick SDC electrolyte.-Onthe basis of the theoretical considerations posited by Wagner, Rickert and Weppner, we attempted to explain the equilibration process of a very thick SDC electrolyte in response to a change in the anode gas.
Assumption: The voltage determined using pO cathode 2 and pO anode 2 is generated at the interface between the electrode and electrolyte before the bulk reaches equilibrium.
Even in the case of the aforementioned assumption, the following problem will likely occur because the generated OCV determined using pO cathode 2 and pO anode 2 when using SDC electrolytes should initially be V th (1.15 V) and not 0.80 V at 1073 K.In the case of a very thin (10 μm) SDC electrolyte, the change from 1.15 V to 0.80 V is immediate and cannot be observed.However, when the thickness is sufficiently large, τ in Weppner's method cannot be ignored in response to a change in the anode gas under open-circuit conditions.Thus, although the bulk plays only a secondary role in generating the voltage, in the case of a thick SDC electrolyte, the τ of I e_diffusion should be the limiting factor for achieving an OCV of 0.80 V.This problem will never occur in the case of using YSZ electrolytes because the OCV at equilibrium is V th .Consequently, the equilibration rates should differ from those obtained when using thin and thick YSZ electrolytes and a very thin (10 μm) SDC electrolyte.
Problem: According to Weppner's method, a delay should occur before I e_diffusion reaches its equilibrium value and before dϕ/dx reaches its equilibrium value.
From Equations 2 and 4, [19]   where OCV(t) and I e (t) are functions of t (time) and I e is negative when I i is positive.Therefore, a delay in I e before reaching its equilibrium value should be observed because of an overshooting in the OCV before it reaches its equilibrium value.However, when SDC electrolytes are used, a delay in I e_diffusion in response to a change in the anode gas has never been reported.The delay in the response to a change in the anode gas can be observed by measuring the equilibration process using a very thick (6.6 mm) SDC electrolyte, for which τ is 40 (6.6 × 6.6) times longer than that for a 1-mm-thick electrolyte.The derivation of the necessary equations to calculate the equilibration rate is shown in Appendix D.
To avoid the electrode interactions influenced by the newly supplied gas, a porous Pt electrode should be used as the anode.NiO should not be used because the reduction of NiO influences the measurement of the equilibrium time.In this work, the comparison with cathode gas (from air to O 2 gas) Al 2 O 3 tube cathode (porous Pt) Pyrex glass sealing SDC electrolyte (thickness 6.6 mm) anode (porous Pt) current collector (Pt mesh) Pt wire anode gas (from air to H 2 gas via N 2 gas) Liu and Weppner's experiment is very important.Liu and Weppner used Pt electrodes for the cathode and anode, 6 which is another reason why we selected Pt electrodes for the cathode and anode.In addition, Liu and Weppner performed their tests at 1073 K. Therefore, we also chose an operating temperature of 1073 K. Other interactions can be expected, such as oxygen collisions with the electrolyte surface.Furthermore, the gas diffusion rate of the newly supplied gas to the anode decreases the accuracy of the measurement.The superiority of our experiment lies in its simplicity, which is required to avoid objections based on possible serious technological difficulties.These objections are often endless "indeed or not" arguments, as shown in Appendices E and F. Accuracy is not required when the measured equilibrium time is substantially different from the calculated equilibrium time based on conventional theory.Thus, we chose simplicity over accuracy.

Experimental
CeO 2 (Tosoh) and Sm 2 O 3 (Tosoh) powders were mixed (8:1 mole ratio) and calcined at 1573 K for 10 h before being pressed into disks (10 mm thick) both axially at 100 MPa and then isostatically at 200 MPa.The disks were sintered at 1923 K for 15 h in air.A thick SDC electrolyte was prepared via a solid-state reaction at a high temperature to obtain high-quality products; Liu and Weppner prepared SDC electrolytes at similar temperatures. 6After being polished, the SDC electrolyte employed in this investigation had a diameter of 25 mm and a thickness of 6.6 mm.Porous Pt electrodes (10 mm in diameter) were used as both the cathode and anode and were sintered at 1173 K for 1 h in air.Pt meshes (5 mm in diameter) were used as current collectors.The cell was heated in air from room temperature to the operating temperature of 1073 K. O 2 gas (1 atm) was fed to the cathode, and N 2 gas was supplied to the anode to flush the air and to avoid an explosion resulting from a mixture of air and H 2 gas.At the start of the experiment, the anode gas was replaced with H 2 gas (containing 3% H 2 O).The cathode and anode gas flow rates were 100 sccm.The time dependence of the OCV was measured using a voltmeter.A schematic of the experimental setup is shown in Fig. 5.

Figure 6.
The voltage of the SDC cell as a function of time in the presence of thick electrolytes during the initial stages of the fuel-supply process.When V 0 was 0.80 V and T eq was 1 min, the model was in good agreement with the experimental results.OCV overshooting was not observed.

Results and Discussion
Calculated leakage currents.-TheOCV gradually increased to an equilibrium voltage of 0.80 V at 1073 K within 5 min, as shown in Fig. 6.The Nernst voltage was 1.15 V at 1073 K.According to Gödickemeier and Gauckler, the ionic conductivity of the SDC electrolyte was 0.089 S/cm at 1073 K. 9 The starting OCV was only 21 mV.Therefore, the flushing time of the N 2 gas was sufficiently short to avoid reducing the SDC electrolyte; the initial OCV should have been higher than that observed here.Hence, the anode reference gas was air.The electronic conductivity of the gadolinia-doped ceria (GDC) electrolytes in air was only 8.7 × 10 −6 S/cm at 1073 K (Fig. 10 in Ref. 6).The calculated electronic conductivity of the SDC electrolytes in air using Equation 6 was 1.1 × 10 −5 S/cm at 1073 K. Since the electrical properties of SDC electrolytes are very similar to those of GDC electrolytes, this value was valid.The oxygen pressure of the H 2 gas (containing 3% H 2 O) at 1073 K was 2.7 × 10 −22 atm, as calculated using the Nernst voltage of 1.15 V at 1073 K.The measured electronic conductivity of the GDC electrolyte was 2.5 S/cm at 1073 K (Fig. 10 in Ref. 6).The calculated electronic conductivity of the SDC electrolyte was 1.8 S/cm at 1073 K. Therefore, this value was valid.The leakage currents under OCV conditions at equilibrium were calculated using Equations 2 and 3.The calculation results are shown in Table I.Consequently, I e should decrease from 0 mA to −32 mA at an equilibrium voltage of 0.80 V at 1073 K.   where T eq , V 0 , and t are the equivalent time constant of the whole circuit, the equilibrium voltage ( = 0.80 V) and time (min), respectively.The value of V 0 was lower than the calculated OCV ( = 0.85 V).The difference between the calculated OCV value and the experimentally determined value is discussed later.When V 0 was 0.80 V and T eq was 1 min, the model was in good agreement with the experimental results shown in Fig. 6.The delay (5 min) may have been due to the bulk equilibration that resulted from vacancy diffusion; hence, T eq for a 1mm-thick electrolyte was 1.5 (60 s / (6.6 × 6.6)) s.This time is short compared to the fast response of the YSZ electrolytes upon changes in the oxygen partial pressure.The thickness provides a simple explanation for this delay (5 min).The occurrence of OCV overshooting is not evident in Fig. 6.

Calculated delay in the open-circuit voltage.
-According to Thangadurai and Weppner, 7 using 1-mm-thick SDC electrolytes when D is 2 × 10 −6 cm 2 /s leads to a τ of 83 min at 823 K under oxygen partial pressures from 10 −3 to 1 atm.According to Wang et al. (Equation 13 in Ref. 16), D is 3.2 × 10 −6 cm 2 /s at 1073 K. Therefore, using 1-mm-thick SDC electrolytes, τ is 52 min at 1073 K. Thus, for a 6.6-mm-thick SDC electrolyte, the calculated τ was 2080 (52 × 6.6 × 6.6) min, which is much longer than 5 min.The experiment performed here differed from the experiment by Thangadurai and Weppner as follows: Difference 1: The O 2 partial pressure was not constant in the SDC electrolyte.
Difference 2: The σ el value was not constant in the electrolyte.Difference 3: The O 2− ions could be supplied from the electrode.Difference 4: The dϕ/dx derivate was not zero in the SDC electrolyte.
Difference 5: Both I e_diffusion and I e_drift should have occurred in the SDC electrolyte.
Difference 6: The anode surface of the SDC electrolyte was reduced by the H 2 gas.
Regarding difference 1, the equilibration of the electrolyte after varying the O 2 partial pressure depends on the chemical diffusion process when other processes, such as dissociation or redox processes, are not rate determining.In this experiment, the cell operated under an air atmosphere.Therefore, the electron concentration was very low.After the anode gas was changed, the electrons should have been supplied by the anode.When electrons are supplied within an SDC electrolyte, the following dissociation or redox processes will occur inside the electrolyte: 8 Virkar and Wright investigated the reduction speed using porous SDC electrolytes (∼30% sample porosity). 17Even when they used a porous body, the time required to reduce the SDC electrolytes via a change in the O 2 partial pressure of the surrounding gas was not negligible.The delay described by Equation 21inside the SDC electrolyte should have been much longer than 5 min.Consequently, difference 1 could be ignored.The relationship between and the O 2 partial pressure (the oxygen chemical potential profile) and that between and t ion in the electrolyte when V const is 0.35 V.Because of the anode-shielding effect, t ion in the electrolyte becomes higher than the t ion shown in Fig. 1.
Regarding difference 2, according to Weppner, 6 the electron mobility will be constant when σ el is changed.Therefore, τ should be constant, and difference 2 could be ignored.
Regarding difference 3, an increase in I i led to a decrease in dϕ/dx.Thus, I e_drift was caused by dϕ/dx suppressing I e_diffusion .Consequently, difference 3 could be ignored.
Regarding difference 4, decreasing dϕ/dx cannot increase I e_diffusion because I e_drift is caused by dϕ/dx suppressing I e_diffusion .Thus, difference 4 could be ignored.
Regarding difference 5, I e_drift could increase I e_diffusion because I e_drift was caused by dϕ/dx suppressing I e_diffusion .Therefore, difference 5 could be ignored.
Regarding difference 6, the unexplained voltage loss (approximately 0.35 V) at the reduced MIEC anode surface, which was attributed to the low value of t ion , is discussed later.
Except for difference 6, there was no evidence indicating why OCV overshooting could not be observed before reaching equilibrium.From Equation 19, the calculated OCV changed from 1.15 to 0.85 V over a duration of more than 2080 min.The differences between the calculated and experimental results were both quantitative and qualitative.The errors in the accuracy associated with the interactions and experimental setup were not serious because the measured equilibration time was substantially different from the equilibration time calculated using the conventional theory.
To avoid objections to our results, we proposed another experiment, as shown in Fig. 7, for determining the time dependence of the OCV in response to a change in the anode gas.Using a partial ionblocking cathode, we could measure the time dependence of the OCV in response to a change in the anode gas.When OCV overshooting was not observed before reaching equilibrium, the measured OCV could not be explained using Equation 3 alone.

Calculated results using the current-independent constant anode voltage loss.
-The calculated OCV will always be higher than the experimental results, and additional voltage loss does not disprove Equation 3. The polarization voltage losses (η act ) are frequently used to compensate for these losses: 1,2,9,18-22 However, when I e is negligible at 5 min, I i and η act should also be negligible.To solve this problem, we propose a solution based on V const , which was first mentioned by Gödickemeier and Gauckler. 9hen V const occurs at the reduced MIEC anode surface, [23]   The interfacial O 2 partial pressure of oxygen in the electrolyte at the anode ( pO 2 ) can be calculated using the Nernst Equation 9: RT × pO anode 2 [24]   When V const is 0.35 V, pO anode 2 is 2.7 × 10 −22 atm, pO * 2 is 4.8 × 10 −17 atm, and pO 2 is 9.9 × 10 −16 atm.The relationship between and the O 2 partial pressure (the oxygen chemical potential profile) and that between and t ion are shown in Fig. 8.The calculated OCV from Equation 3 is 0.76 V. Thus, when R i is 1 , I i is 0.040 A ((1.15 V-0.35 V-0.76 V) / 1 ), and I e is −0.040 A. When a 1-mm-thick electrolyte is used, dϕ/dx is −0.040V/mm.The relationship between and I e_diffusion in the electrolyte is shown in Fig. 9a, and the relationship between and I e_drift in the electrolyte is shown in Fig. 9b.
From Fig. 8, t ion in the electrolyte exceeds the t ion shown in Fig. 1.In general, t ion in the electrolyte increases as the anode voltage loss increases.This relationship is called the "anode-shielding effect". 18hen V const is 0.35 V, pO 2 ( = 9.9 × 10 −16 atm) becomes much larger than pO * 2 (4.8 × 10 −17 atm).Therefore, t ion in the electrolyte approaches 1 throughout the electrolyte.As shown in Figs.9a and 9b, the maximum I e_drift becomes much smaller than the value shown in Fig. 2b.The energy loss resulting from the recycled I e_drift in the SDC electrolyte is negligible relative to the energy loss caused by V const × I i .Thus, this problem can be solved.
The effect of V const in this experiment can be explained as follows.When R i is 9.4 , I i is only 4.2 mA.Therefore, R i I i is 40 mV, which is much smaller than 0.35 V.The typical polarization resistance is 0.5 /cm 2 , even at 873 K. 18 Therefore, η act is less than 2.1 mV, which is much smaller than 0.35 V.When R i I i and η act are negligible in Equation 23, [25]   Lai and Haile applied Equation 25 experimentally using electrochemical impedance spectroscopy (EIS). 19However, problems with their interpretation still remain.From Equation 25, the OCV should be constant during equilibration according to Weppner's method.In Equation 25, the OCV calculated from Equation 3 is ambiguously expressed, that is, one can obtain the value of V const (approximately 0.35 V) by measuring the equilibrium OCV ( = 0.80 V) using a sufficiently thick (6.6 mm) SDC electrolyte.
From Equation 19, for an SDC electrolyte with a thickness of 6.6 mm, the calculated OCV changed from 0.80 V to 0.76 V over a period longer than 2080 min.Given V const (approximately 0.35 V), the results of the aforementioned calculations provide substantially improved quantitative and qualitative explanations of the experimental results.A schematic of the explanation for V const (approximately 0.35 V) is shown in Fig. 10.Consequently, both Wagner's equation and Weppner's method can coexist using this method.

Verification of the thinner SDC electrolytes.-This solution is
useful not only for a thickness of 6.6 mm but also for a thickness of 100 μm.In the case of typical thin films (100 μm thick), the calculated equilibration time is 31 s (52 × 60 s / (10 × 10)).The anode gas diffusion rate of the gas newly supplied to the anode strongly depends on the microstructure of the anode.When the porosity is sufficiently large, a delay in I e for reaching its equilibrium value should be observed due to the OCV overshooting within 31 s.In the case of ceria-doped electrolytes with a thickness between 1 mm and 100 μm, studies detailing a substantial number of samples have already been published by numerous scientists.No such delays have been reported in any of these studies.Hence, a substantial number of samples have already verified this experimental result.
In the case of very thin films (10-μm thick), the calculated equilibration time is only 0.3 s (52 × 60 s / (100 × 100)).Therefore, overshooting problems in the OCV will not occur.Because of the large I i , η act in Equation 23 cannot be ignored.However, R i I i should still be negligible because R i becomes very small for very thin films.
The assumption of V const is consistent with conventional models for very thin films and is necessary for improving the analysis of the experimental results.Such an improvement can be attained using cathode (air or O2 gas) SDC electrolyte (higher tion due to the anode-shielding effect) electron diffusion current (small) electron drift current (small) ionic current (small) anode MIEC layer with a fast Vconst response  Equation 26 instead of Equation 22. [26]   Thus, thickness plays a role when using thin SDC electrolytes. 1,2he assumption of V const in the present report is only one of the necessary conditions and is insufficient by itself.Even when Equation 23 holds and equilibrium is established, electrode processes must occur through ion transport, which may result in deviations from the expected OCV.These electrode processes are generally rate determining.When the thickness of the SDC electrolyte is 0.6 mm, the equilibration time is 19 (52 × 0.6 × 0.6) min.However, the OCV did not change, even during electrode degradation when a 0.6-mm-thick electrolyte was used, as described previously 20,21 and explained in Appendix F.
Floating flat potential in the SDC electrolytes.-Wediscuss a hypothesis we developed to explain the current-independent constant voltage loss in the SDC electrolyte.At first, a floating flat potential should exist for ions in the electrolyte, as shown in Fig. 11 and proven experimentally by Lim and Virkar. 22However, problems with their interpretation remain.The floating flat potential in an electrolyte was first mentioned by Duncan and Wachsman. 1 In Fig. 11, the anodeshielding effect is much more effective because of the length of the floating flat potential for the ions because ions cannot drift though a floating flat potential.In this report, Point A in Fig. 11 should be 9.9 × 10 −16 atm.However, Point B in Fig. 11 should be 9.9 × 10 −16 atm when there is a floating flat potential for the ions.
The value of V const was empirically determined to be 23 where E a is the ionic activation energy.The E a is 0.7 eV for the SDC electrolytes.Therefore, V const in Equation 27 is 0.35 V (0.7 eV/2e).V const can be attributed to an unexplained hopping conduction mechanism.
In the Equation 28, the following transformation should be considered during ion hopping: where N, μ i_hopping , μ i_vacancies , ϕ i_hopping , and ϕ i_vacancies are Avogadro's number, the chemical potential of the hopping ions, the chemical potential of the ions in vacancies, the electrical potential of the hopping ions, and the electrical potential of the ions in vacancies, respectively.Using neutron powder diffraction methods, Equation 29 was proven experimentally by Yashima. 24Equation 30 is discussed later.Thus, The electrical potential (0.35 V = 0.7 eV / 2e) in Equation 31 is neutralized by free electrons, resulting in current-independent constant voltage losses at Point A.
Microscopic explanation for the current-independent constant anode voltage loss.-Voltagelosses occur during every hopping process in the MIEC area.Therefore, the voltage loss resulting from only 4 hopping processes (1.4 V = 0.35 × 4 V) should be higher than the Nernst voltage (1.15 V).However, the voltage loss is attributed to only one hopping process.The energy losses ascribed to the remainder of the hopping processes are converted into thermal energy, which can be recycled to excite the ions used in the next hop.Ions in equilibrium state A lose energy 0.7 eV) during hopping in the MIEC electrolyte.The colder ions can immediately become heated in the vacancies.However, after the last hop, the ions cannot be heated, and they exit the electrolyte in equilibrium state B. A schematic of this explanation for the energy loss is shown in Fig. 12.
Thermodynamic explanation for the current-independent constant anode voltage loss.-Inthermodynamics, the Helmholtz energy difference ( F = F B − F A ) between 2 states, A and B, is connected to the work (W) done on the system through the following inequality: The Jarzynski equality 25 is where the overline indicates an average over all possible realizations of an external process that takes the system from equilibrium state A to a new generally nonequilibrium state under the same external conditions as that of equilibrium state B. Using Gibbs energy, the Jarzynski equality 26 is e −( F+PV )/kT = e −W/kT [34]   where P is the pressure from the lattice structure on the hopping ions and V is the volume of the oxygen ions during hopping.Then, PV per mol is equal Ea.The Boltzmann distribution of oxygen ions in the electrolyte at 1073 K is displayed in Fig. 13.The ions with energies exceeding Ea become carriers (hopping ions).Fig. 14 presents an incorrect carrier (hopping ion) distribution.The Boltzmann distribution cannot be separated using passive filters (because of the phenomenon known as "Maxwell's demon"), and an accurate distribution is provided in Fig. 15.The loss of Gibbs energy is illustrated in Fig. 14.Then, the real free energy should be the Helmholtz energy difference when t ion is sufficiently small near the anode.The real free energy should be determined using Equation 35:

Conclusions
The electronic conductivity of SDC is low in air and increases substantially in H 2 gas.We measured and attempted to explain the equilibration process using a thick (6.6 mm) SDC electrolyte in response to a change in the anode gas.For this purpose, we used Wagner's equation and Rickert's approach to analyze the electron drift current and diffusion throughout the electrolyte.The direction of the electron drift current was the same as that of the ionic current.The electron drift current was recycled as an electron diffusion current throughout the electrolyte.The OCV gradually increased to an equilibrium voltage of 0.80 V within 5 min, which appeared to be due to bulk equilibration after the diffusion of the vacancies.The time constant was only 1 min.According to Weppner's method, the equilibrium time should have been much longer than 5 min.The calculated OCV changed from 1.15 V to 0.85 V over more than 2080 min.The OCV during equilibration in response to a change in the oxygen activity of the anode gas was impossible to explain.To avoid objections, we proposed an experiment to determine the time dependence of the OCV.When V const (approximately 0.35 V) was assumed, the calculated OCV changed from 0.80 V to 0.76 V after more than 2080 min because of the anode-shielding effect.Therefore, the calculation results provided a substantially improved explanation of the experimental results, both quantitatively and qualitatively.Consequently, both Weppner's method and Wagner's equation were consistent with this method.The value associated with this voltage loss (approximately 0.35 V) could be obtained by measuring the equilibrium OCV (0.80 V) using sufficiently thick SDC electrolytes.This proposed volt-Figure 14.The forbidden distribution of the hopping ions.This distribution is forbidden according to "Maxwell's demon."age loss could coexist with conventional models for very thin films and was necessary for improving the analysis of the experimental results.
The assumption of this voltage loss was only one of the necessary conditions and was insufficient on its own.Finally, we developed a hypothesis to explain this voltage loss in the SDC electrolyte.

Appendix A
Equations 2 and 3 can be deduced from Equation 1. Equation 1 is Oxygen gas forms oxygen anions in the electrolyte.Therefore, The O 2 flux can be written in terms of I i : Therefore, from Equations A1 and A3, where σ ion is constant in SDC electrolytes under various O 2 partial pressures. 5Therefore, and R i is given as Therefore, from Equations A5 and A6, where t ion is expressed as follows: From Equations A7 and A8, The Nernst voltage (V th ) is expressed as follows: Therefore, from Equations A9 and A10, ) unless CC License in place (see abstract  6 because the direction of I e_diffusion is opposite that of I i , unlike in Weppner's method, a negative sign should be added.The I e_diffusion of each mesh (I e_diffusion_mesh ) is I e di f f usion mesh = −σ el mesh SkT q L mesh e q(OCV mesh −V dri f t mesh ) kT

− 1 [C2]
where σ el_mesh , k, q and V drift_mesh are the σ el of the mesh, Boltzmann constant, ion charge and potential difference resulting from the ionic current of the mesh, respectively.The applied voltage for the electrons in each mesh is OCV mesh .Thus, V drift_mesh is Therefore, I e di f f usion mesh = −σ el mesh SkT q L mesh e q×V th mesh kT in which V th_mesh becomes small when N is sufficiently large.Then, e q×V th mesh kT From Equations C4 and C5, I e di f f usion mesh = −σ el mesh S × V th mesh L mesh [C6]   From Equations B7 and C6, where t ion_mesh is expressed as follows: From Equations C7 and C8, Consequently, Equation C1 is applicable for the SDC electrolyte and is compatible with Equation 3 (Wagner's equation) because dϕ/dx in Equation B7 is also true in Equation C7.

Appendix D
The derivation of the necessary equations to calculate the equilibration rate is shown in this section.According to the model by Riess, 13 where ν e and n are the electron mobility and the electron concentration, respectively.During the transient process, ϕ is a function of time, and n is a function of time and position.Therefore, where ϕ(0) is 0 V, ϕ(∞) is 0.35 V (V th -OCV) and n(x, 0) and n(L, t) are the electron concentrations in air.In addition, n(0, t) is the electron concentration in the hydrogen gas atmosphere, and n(x, t) can be calculated using Fick's second law.

Appendix F
According to the model by Riess, 9,13 the OCV using an SDC electrolyte should be reduced during electrode degradation. 20,21This method was used to disprove the occurrence of large leakage currents in strontium-and magnesium-doped lanthanum gallate (LSGM) electrolytes in 2008. 27However, the change in the OCV using an SDC electrolyte during electrode degradation has never been reported.Here, I i and I e are 9,13 where V a , V c , V cell , β, and σ 0 e are the anode voltage loss, cathode voltage loss, cell voltage, 1/kT, and the electron conductivity near the anode in the SDC electrolyte, respectively.Under open-circuit conditions, I i = -I e .V cell is the OCV.Therefore, where D is both constant and positive.Therefore, the OCV should decrease as (V c + V a ) increases.When the OCV remains unchanged, the numerator of the previously mentioned fraction should be large, and the denominator should be small.However, many scientists have indicated the need for further investigations to elucidate the mechanism of electrode degradation.

2 ln pO anode 2 σ
el σ ion σ el + σ ion d ln pO 2 [1] where J O 2 and pO 2 are the O 2 flux and the O 2 partial pressure, respectively; pO cathode 2 and pO anode 2

Figure 1 .pO 2 )Figure 2 .
Figure 1.The relationship between and the O 2 partial pressure (the oxygen chemical potential profile) and that between and t ion in the electrolyte.The values at equilibrium are determined by pO cathode 2 the analyses of Choudhury and Patterson, Yuan and Pal, and Riess are essentially equivalent; they were developed using Wagner's classical approach for open-circuit conditions.However, Näfe's formulation differs.The calculation results shown in Fig. 1 are compatible with the analyses of Choudhury and Patterson, Yuan and Pal, and Riess (Fig. 3 in Ref. 14) and are thus valid.

Figure 3 . 2 , pO anode 2 ,Figure 4 .
Figure 3. (a) The relationship between and I e_diffusion when R i is 1 in the electrolyte.(b) The relationship between and I e_drift when R i is 1 in the electrolyte.The values at equilibrium are determined by pO cathode 2

Figure 5 .
Figure 5. Schematic of the experimental setup.The time dependence of the OCV in response to a change in the anode gas was measured using a voltmeter.

Figure 7 .
Figure 7. Schematic of the experimental setup used for determining the time dependence of the OCV in response to a change in the anode gas.A partial ionblocking cathode was used, and the time dependence of the OCV in response to a change in the anode gas was measured.

Figure 8 .
Figure8.The relationship between and the O 2 partial pressure (the oxygen chemical potential profile) and that between and t ion in the electrolyte when V const is 0.35 V.Because of the anode-shielding effect, t ion in the electrolyte becomes higher than the t ion shown in Fig.1.

Figure 9 .
Figure 9. (a) The relationship between and I e_diffusion when V const is 0.35 V. (b) The between and I e_drift when V const is 0.35 V.The maximum I e_drift becomes much smaller than the value shown in Fig. 2b.

Figure 11 .
Figure 11.A floating flat potential for ions in SDC electrolytes.A flat potential was observed experimentally, except near the cathode.Point B should correspond to 9.9 × 10 −16 atm.

Figure 12 .
Figure12.A schematic of the explanation for the energy loss.Topologically, a series of hopping processes for one direction should be equivalent to one hopping process for one direction.

Figure 13 .
Figure 13.The Boltzmann distribution at 1073 K. Ions with energies exceeding the ionic activation energy are converted into charge carriers (i.e., hopping ions).

Figure 15 .
Figure 15.The correct distribution of the hopping ions.The shape of the distribution in Fig. 15 should be the same as the shape of the distribution in Fig. 13.

Table I . Calculation results.
Downloaded on 2018-07-20 to IP According to Liu and Weppner, ). ecsdl.org/site/terms_useaddress.Redistribution subject to ECS terms of use (see 207.241.231.83 Figure E1.A thin YSZ film is deposited onto the SDC electrolyte on the cathode side.A thin YSZ film on the cathode side can block electrons: "indeed or not".