Insights into strontium zirconate-induced interface pressures in solid oxide electrolysis cells

Modeling the oxygen partial pressure at the interfaces of solid oxide electrolysis cells is essential to understand their degradation. Here we present a bilayer electrolyte equivalent circuit model that does not assume continuity of the oxygen chemical potential at the electrolyte interface, in line with experimental results. We use the model to assess the assumption that strontium zirconate growth is responsible for oxygen electrode interface pressure increases and delamination. We find that 5% Y-doped strontium zirconate does not drive fracture for accumulations up to 250 nm except at 800 ° C and 1.6V. With a low electronic conductivity, 5% La-doped strontium zirconate shows a region of stability where increased thickness actually stabilizes the interface pressure. We also highlight the importance of modeling interface stability with the proper electronic conductivity. Finally, we present interface stability requirements and contextualize the strontium zirconate composition results in terms of interface stabilization strategies.

• An equivalent circuit model for solid oxide cell bilayer electrolytes is presented.• Refined stability criteria for solid oxide electrolysis cells are presented.• The oxygen electrode interface pressure induced by strontium zirconate formation is quantified.

A B S T R A C T
Modeling the oxygen partial pressure at the interfaces of solid oxide electrolysis cells is essential to understand their degradation.Here we present a bilayer electrolyte equivalent circuit model that does not assume continuity of the oxygen chemical potential at the electrolyte interface, in line with experimental results.We use the model to assess the assumption that strontium zirconate growth is responsible for oxygen electrode interface pressure increases and delamination.We find that 5% Y-doped strontium zirconate does not drive fracture for accumulations up to 250 nm except at 800 °C and 1.6 V.With a low electronic conductivity, 5% La-doped strontium zirconate shows a region of stability where increased thickness actually stabilizes the interface pressure.We also highlight the importance of modeling interface stability with the proper electronic conductivity.Finally, we present interface stability requirements and contextualize the strontium zirconate composition results in terms of interface stabilization strategies.

Introduction
The durability of solid oxide electrolysis cells (SOECs) currently limits their large-scale adoption for efficient clean hydrogen production.The severity of SOECs' electrolyte cracking and oxygen electrode delamination has been directly linked to the applied current density [1][2][3].This cracking is a direct consequence of local equilibrium; the conversion of oxygen ions to gaseous oxygen (O 2 ) in defects, voids, and grain boundaries can generate pressures sufficient to crack even a perfect lattice [4][5][6].Modeling the oxygen chemical potential/partial pressure across a SOEC is essential to identify and ameliorate the operation conditions, microstructures, and cell architectures that generate the most severe internal pressure conditions.
Industrial steam electrode-supported SOECs typically use a lanthanum strontium cobalt ferrite oxygen electrode (La 1-x Sr x Co y Fe 1-y O 3- , LSCF) and a supporting nickel and yttriastabilized zirconia (usually Y 0.16 Zr 0.84 O 2.92 , YSZ) steam/fuel electrode.A dense bilayer electrolyte composed of ≈10 μm YSZ and 1-2 μm gadolinia-doped ceria (Gd x Ce 1-x O 2- , GDC) separates the electrodes.The GDC layer primarily serves to minimize the reaction between LSCF and YSZ to form strontium zirconate (SrZrO 3 , SZO).Severe crackingup to complete delamination-is observed most frequently near the interfaces of these layers during long term SOEC operation [1,3,7].Using bulk fracture toughness values, the Barnett group has estimated the cracking threshold pressures to be 7200 atm for LSCF, 13,900 atm for GDC, and 27,500 atm for YSZ [2].Using an embedded Pt probe, Virkar et al. observed a fracture event at a 5% CeO 2 +YSZ -YSZ interface with an ≈28,100 atm critical pressure [6], which matches well with the above prediction.Such accumulated pressures are possible due to interface effects that decouple the electrolyte's oxygen chemical potential from the cell's thermodynamic bounds at the electrodes [4].
In 2010, Virkar synthesized his prior work on equivalent circuit modeling of fuel cells to explain the delamination of SOECs in terms of local equilibrium-driven oxygen accumulation and electrode interface effects [5].Subsequently, Zhang, Zhu, and Virkar used the boundary conditions resulting from this model to numerically calculate the oxygen chemical potential distribution across a single layer YSZ electrolyte and predicted pressures exceeding 10 5 atm across approximately 40% of the electrolyte for various operating conditions [8].Likewise, Chatzichristodoulou et al. numerically simulated the oxygen chemical potential distribution across a bilayer YSZ-GDC electrolyte assuming an abrupt interface with internal continuity of oxygen chemical potential and comparable electrode interface effects, yielding similar profiles and pressures [9].Conversely, if continuity of the oxygen chemical potential is assumed at the electrodes and bilayer interface using Choudhury and Patterson's extension of the Wagner model [10,11], no pressure exceeding that of the oxygen electrode is predicted for a YSZ-GDC electrolyte [12].Recently, the Barnett and Voorhees groups have incrementally developed a model that accounts for critical current density [2], a low conductivity YSZ-GDC interface layer [13], and a diffuse YSZ-GDC interface coupled with varying oxygen vacancy concentrations [14].Notably, the latter two refinements predict pressure maxima at locations that correspond well with cracks observed at the YSZ side of the inter-diffusion zone in aged SOECs [3,7,13,14].
While these models predict many of the observed SOEC degradation phenomena, most assume that the oxygen chemical potential is continuous across the bilayer electrolyte interface.However, Kwon, Lee, and Yoo studied the ionic and partial electronic conductivities of bilayer YSZ-GDC electrolytes using EIS and Hebb-Wagner polarization and found that the resulting currents cannot be explained by theories assuming continuous oxygen chemical potentials across the bilayer interface [15,16].
Electrolyte sintering and aging of the oxygen electrode can also give rise to nanoscale interfaces with abrupt changes in oxygen chemical potential and conductivity.The polarization resistance, as measured by electrochemical impedance spectroscopy (EIS), combines the effects of many processes between ad/desorption of the oxygen molecule and ionic charge transfer to/from the electrolyte.The latter charge transfer is relevant for this study and can be deconvoluted using physically-reasoned distribution of relaxation times [17].The polarization resistance's growth over time in electrolysis mode has been attributed to the filling of LSCF oxygen vacancies, which drives Sr surface segregation and diffusion via surface boundaries or Sr(OH) 2 (g) species [18,19].The rate/quantity of SZO formation has been linked with the porosity of the GDC barrier layer and LSCF contact layer: the denser the layers, the less SZO is formed [19,20].Unlike the YSZ-GDC interface, which displays a roughly 1 μm inter-diffusion zone, the interface of SZO layers/inclusions with GDC or YSZ is abrupt (<100 nm) and has sharp changes in ionic conductivity [21][22][23].For microstructural observations of these phenomena, the reader is referred to the studies by Rinaldi et [7,19,20,23].The cation diffusion and SZO formation scenario is schematized in Fig. 1.
Develos-Bagarinao et al. investigated the origin of the ionic transfer resistance induced by SZO formation using oxygen isotope exchange profiles and thermodynamic calculations [21].Their results indicate that oxygen ions can flow through SZO with sufficient freedom to establish a uniform 18  8 O penetration distribution; however, the ionic flux is reduced by ≈1/3 at GDC-SZO interfaces and 1-2 orders of magnitude at YSZ-SZO interfaces.They also found that diffusion-driven Y-doping of SZO creates favorable thermodynamic conditions for the adjacent precipitation of yttrium oxides.Given yttria's very low ionic conductivity [24], even a few nanometers could account for this conductivity drop.Oxygen structural disorder has been indicated at these interfaces by micro-Raman and transmission electron microscopy electron energy loss spectroscopy [21,25], which suggests a change in crystal structure to relax an unstable interface.Micro X-ray fluorescence analysis of an aged SOEC reveals a Gd enhancement at the YSZ-GDC interface [23], which may indicate that a similar precipitation situation can occur at the barrier layer/electrolyte interface during long term operation.Investigating the pressure effect of these evolved nanoscale interfaces is best accomplished using an abrupt interface boundary condition.
In light of the aforementioned results, we extend Virkar's original equivalent circuit model to include bilayer electrolytes with abrupt interfaces.We then investigate the effect of SZO accumulation and composition on interface pressure to determine if an yttrium oxide interface layer is required to explain fracture conditions.Finally, we identify critical resistance ratio thresholds to prevent delamination conditions and propose a new strategy for interface pressure stabilization based on the SZO composition results.

Model and fundamentals
Virkar's equivalent circuit model, and the current bilayer extension, rely on assumptions of local microscopic equilibrium, steady state conditions, and a dilute solution approximation for the conducting species [5].The cation lattice is assumed to be immobile on the short time scale of electronic and ionic conduction; however, cation migration is significant when considering operando-evolved interfaces.For a detailed exploration of these assumptions, their validity, and their implications, the reader is referred to Virkar's original article [5].The dilute solution approximation assumes that the externally applied electrostatic potential () renders the electrostatic interactions between carriers and defects negligible [26], allowing the definition of the electrochemical potential ( μ) in terms of the chemical potential () as follows.
is the charge of a particle. is not an experimentally measurable parameter.Conversely, the electric potential, , is measurable as a voltage ( = ).
Local equilibrium implies the following.Applying Eqs. ( 1) and ( 2) to Eq. ( 3) gives the fundamental relation of the Virkar model in a measurable form (see Virkar's initial article [5]).
This relation enables a change in ionic electrochemical potential at an interface to be treated as a voltage source similar to a junction potential in equivalent circuit modeling.The sum of these voltage sources across the electrolyte must equal the thermodynamic Nernst potential (  ).For an arbitrary interface, , the potential across that interface is proportional to the change in oxygen chemical potential.
For the equivalent circuit shown in Fig. 2, these relations take the following forms, where  H 2 ∶H 2 O and   are the respective bounding fuel electrode and oxygen electrode oxygen partial pressures. ) Applying Eqs. ( 4)-( 6) and using Ohm's and Kirchoff's laws to cancel the ionic and electronic currents in favor of the total electronic (  ) and ionic (  ) resistances yields Eq. ( 7) for any interface  in terms of interface electronic and ionic charge transfer resistances (  and   ) and the applied electric potential (  ).
The expanded bilayer electrolyte circuit model presented in Fig. 2 can then be algebraically solved using simple circuit techniques and yields Eqs. ( 8)- (10).Eqs. ( 8) and (9) were previously demonstrated by Virkar [5].Further algebraic details of the derivation are given in Supplementary Section 1.
Here    and   are the oxygen partial pressures just across the respective interface from the fuel electrode atmosphere,  H 2 ∶H 2 O , and oxygen electrode atmosphere,   .Representative values of 10 −24 and 0.21 atm were used for the bounding electrode oxygen partial pressures.  is the pressure in the barrier layer just before the interface on the GDC side and   is the pressure just on the YSZ side of the interface.The fuel electrode interface charge transfer resistances are intractably small and usually assumed to be 0 [27,28].The remaining resistances used to evaluate the model are closely matched to industrial cells and are collected in Table 1.Information on relevant extrapolations can be found in Supplementary Section 2.1.

Pristine cell model evaluation
Zhang et al. derived an analytical approximation of the pressure at the GDC-YSZ interface based on their numerical simulations [14].We present it here adapted only to be consistent with our notation, where  is the external current density (assumed to equal a dominantly ionic internal current),   is the GDC thickness, and  is the respective partial conductivity.

𝑙𝑛(𝑃 𝑏𝑎𝑟
Eq. ( 11) was used to calculate was calculated using Eq. ( 9) and the appropriate resistances.YSZ conductivities were taken from Park and Blumenthal [33] and GDC conductivities were taken from Nam et al. [34] and Lee et al. [32].We assume the following customary relationships for oxygen ion/vacancy (  ), electronic (  ), electron (  ), and hole (  ) conductivities, with  ⋆ incorporating the concentration and mobility constant terms and   designating the activation energy. ) Due to a dearth of sufficiently resolved electronic conductivity information for the YSZ-GDC interface in the literature, we approximate the inter-diffusion zone as the interface (  ) in order to demonstrate the features of the model.Two cases were examined for the electronic conductivity of this region: (A) that of single crystalline (Ce 0.5 Zr 0.5 ) 0.8 Y 0.2 O 1.9 as measured by Eufinger et al. [31] extrapolated from 700 to 800 °C (see Supplementary Section 2.1.2),and (B) that of Gd 0.1 Ce 0.9 O 1.95 (following the assumption of Zhang et al. [13,14]) as measured by Lee et al. [32].Both cases assume the oxygen partial pressure within this region to be equal to   O 2 as given by Eq. ( 11).The corresponding resistance values,    () and    (), are shown in Table 1.In case A, these resistances decrease with increasing temperature as the interface pressure is near the intrinsic region.In case B, the resistances for the hole-dominated GDC case atypically increase with temperature due to a decrease in the oxygen partial pressure.
In order not to propagate any error associated with this rough approximation, the effect of SZO accumulation was investigated only at the GDC-LSCF oxygen electrode interface.Given the similar form of the equations for the oxygen electrode and YSZ-GDC interfaces (Eqs.( 9) and ( 10)), the trends and qualitative conclusions should hold equally true, albeit with different critical magnitudes.

SZO accumulation evaluation
To investigate the effect of SZO accumulation and composition, the interface pressure was calculated for accumulations up to 250 nm of pure SZO, 5% Y-doped SZO (SrZr 0.95 Y 0.05 O 2.975 , 5SZYO), 16% Y-doped SZO (SrZr 0.84 Y 0.16 O 2.92 , 16SZYO), and 5% La-doped SZO (Sr 0.95 La 0.05 ZrO 3.025 , 5SLZO).The effect of an added thickness of SZO on the oxygen electrode interface pressure is given by Eq. ( 16), where  is the accumulated SZO thickness, a prime indicates the resistance value prior to the accumulation, and the conductivities () correspond to the respective SZO composition.
The conductivities used to determine the added interface charge transfer resistances due to the SZO accumulation are shown in Table 2.
Information about relevant extrapolations for these values can be found in Supplementary Section 2.2.Due to a lack of thin-film conductivity data for these materials in the literature, we have assumed that the bulk values hold (or are at least representative) for the conductivity on the nanoscale.

Sensitivity analysis
We examined the sensitivity of the model using local partial derivatives and variance-based global sensitivity analysis.To determine the relative parameter importance local to our simulated regimes, we took the partial derivative of   O 2 (Eq.( 9)) with respect to all its input parameters.Each was locally evaluated at 700 and 800 °C for 1.3 and 1.6 V with the appropriate resistance values shown in Table 1.Variancebased global sensitivity analysis was conducted to provide another measure of parameter importance using the SobolGSA software [37].We focused our analysis on the total sensitivity index (  ), which can be used to determine the qualitative contribution of each parameter range to the total output range by quantifying the relative anticipated reduction in the model output variance by fixing a variable/narrowing its range [38].Further details are presented in Supplementary Section 3.

Results
For the resistances given in Table 1 and an applied potential of 1.3 V, Eq. ( 9) gives oxygen electrode interface pressures of 11, 3.6, and 1.8 atm at 700, 750, and 800 °C respectively.Using these values and representative external current densities of −0.6, −0.7, and −0.8A cm −2 , Eq. ( 11) returns barrier layer interface pressures of 12, 3.9, and 1.9 atm at 700, 750, and 800 °C.For case A, where (Ce 0.5 Zr 0.5 ) 0.8 Y 0.2 O 1.9 electronic conductivity is assumed, our interface approximation yields electrolyte interface pressures of 3.7E−4, 1.6E−4, and 3.0E−5 atm at 700, 750, and 800 °C via Eq.(10).For case B, where GDC electronic conductivity is assumed, Eq. ( 10) returns electrolyte interface pressures of 40, 20, and 18 atm at 700, 750, and 800 °C.These results indicate that pristine and well optimized SOECs are stable when operating near the thermoneutral potential, but that the electronic conductivity of the YSZ-GDC interface has a determining effect on the interface pressure.
The effect of applied potential on the oxygen electrode interface pressure in the pristine cell is shown in Fig. 3A.LSCF fracture is predicted around 1.78 V at 700 °C in this case.Given that fractures and delamination are observed in aged cells for voltages below this value, the effect of changing the relative interfacial resistance ratio (   ∕  ) with constant electronic resistances is shown in Fig. 3B.Fractures are expected for ratios of 0.25, 0.50, and 0.66 at 800, 750, and 700 °C respectively.

SZO accumulation and composition
Oxygen electrode interface pressure (  O 2 ) vs. accumulated thickness curves representative of the results for all SZO compositions and temperatures studied are shown in Fig. 4. Individual pressure graphs for all four SZO compositions (at 1.3 and 1.6 V, 700, 750, and 800 °C) can be found in Supplementary Section 4. SZO and 16SZYO have similar effects on the interface pressure due to their similar conductivities.

Fig. 3. (A)
For resistance values typical of industrial SOECs, severe pressures and fracture occur for applied potentials above 1.6 V at 700 °C (black), while a wider voltage range is accessible at 750 (red) and 800 °C (blue).(B) For constant electronic resistances, increasing the ionic interfacial resistance ratio and applied voltage (solid = 1.6 V, segmented = 1.3 V) generate the conditions for LSCF fracture (dashed dark green).(For color references, the reader is referred to the web version of this article.)Fig. 4. The accumulated SZO and its doped varieties do not drive the oxygen electrode interface to the critical pressure at 1.3 V and 750 °C (A).5SZYO (light blue) is shown in the inset.Modest accumulations of 5SLZO (goldenrod), 16SZYO (green), and pure SZO (magenta) are able to drive fracture at 1.6 V (B), while 5SZYO remains subcritical.(For color references, the reader is referred to the web version of this article.).
Both generate LSCF fracture pressures for 160-190 nm of accumulated thickness at 1.3 V and 800 °C as well as 29-50 nm for all tested temperatures at 1.6 V (Figs.S2 & S3).5SZYO has the highest tested ionic and electronic conductivity (Table 2) and shows the mildest pressures, only reaching a critical level at 800 °C and 1.6 V (Figs. 4 & S4).5SLZO displays an intriguing subcritical peak and subsequent pressure stabilization at 1.3 V (Figs. 4A & S5 A); yet, 5SLZO shows rapid pressure increases and fracture for only 11-16 nm at 1.6 V (Figs. 4B & S5B).
More generally, the initial interface pressure decreases with increasing temperature (Fig. 3A).However, as seen in Fig. 3B, the pressure increases to critical levels for lower accumulated resistances at higher temperatures.Both trends are attributable to the ionic resistance of the cell decreasing with rising temperature.Thus, the interfacial ionic resistance ratio (  ∕  ) is much more reactive to accumulated resistance at higher temperatures.Note that the interfacial electronic resistance ratio (  ∕  ) is less sensitive to accumulated resistance due to the order of magnitude higher total electronic resistance.

Stability limits
Eqs. ( 9) and (10) demonstrate that the pressure will increase across an interface when their exponents are positive.Incorporating the proposed critical pressures,   , the following delamination thresholds can be refined [2,5].The total term on the left should be kept below these temperature-dependent thresholds for stable interface pressures.
As it is the ratio of an interface's charge transfer resistance to the total resistance that determines the pressure increase/decrease, optimizing for cell performance by minimizing the total ohmic resistance can be detrimental to the stability of the cell.As seen in Fig. 3B,    need not be large itself if   is small (≤0.1 Ω cm 2 ) to cause delamination, especially at high applied voltage.Differentiating Eq. ( 16) with respect to the accumulated thickness () to find the thickness at which the maximum pressure occurs yields Eqs. ( 19) and (20).
A physically real oxygen interface pressure maximum, and therefore a subsequent stabilization, is predicted for  values where the numerator and denominator of Eq. ( 20) are both positive or negative.The value and importance of  is primarily dictated by the ionic to electronic conductivity ratio and applied potential.While the maximum for 5SLZO at 1.3 V is subcritical (Fig. 4A), the mathematical maximum at 1.6 V lies beyond the fracture regime (Fig. 4B).

Parameter sensitivities
The local partial derivative (LPD) numerical values can be found in Supplementary Table 2. Qualitatively, the magnitudes of the LPDs (|  ∕|) follow the inequality trend of the parameters below.
The LPD magnitude for    is consistently 4 orders of magnitude larger than that for T. Each LPD magnitude increases by around an order of magnitude between 1.3 and 1.6 V for the same temperature, except for those of    and   at 700 °C, which increase by two orders of magnitude.
Our variance-based global sensitivity analysis indicates that the first order sensitivity indices ( 1 ), which quantify the direct impact of a parameter on the output variance, are all < 0.062.This means that around 85% of the model output variance is attributable to correlation effects (i.e.simultaneous high   , low   , high    , and low   ).Given the form of the function ( > 0), higher output variance directly correlates with a greater number of high pressures.The total sensitivity index,   , sums both direct and correlation effects.Representative values of   are shown in Supplementary Table 3. Qualitatively,   () follows the inequality trends of the parameters below.
(, 800 In both regimes,   (  ) is three orders of magnitude larger than   ( ) and   (  O 2 ), but only one order of magnitude larger than   (  ).

Discussion
We have used the 7200 atm LSCF fracture pressure benchmark throughout this work with the aim of making comparable predictions.However, we note that there is some unquantified uncertainty associated with this value.This pressure was obtained with bulk material values in the following equation [2,39], where   is the fracture toughness,  is the Poisson ratio, and  is the radius of the coin-shaped crack.
Given that the fracture toughness is a scale-dependent property [40], cracks and pores are observed preferentially along the grain boundaries orthogonal to the electrochemical potential gradient [41], and that the pressure is internally developed, knowledge of the in-situ mechanical properties of the grain boundaries themselves would enable refinement of these delamination conditions.Despite best efforts, no data for the single grain boundary fracture toughness of 8YSZ, 10GDC, LSCF, nor their respective interfaces/diffusion zones could be found in the literature.Yet, the fracture toughness for a single grain boundary of alumina has been measured using a nanoindenter and a micro-cantilever crafted by focused ion beam milling [42].This approach is promising to refine the knowledge of ceramics' grain boundary fracture mechanics required for more precise modeling.Regardless of the exact fracture threshold pressures, our interdiffusion zone interface approximation results indicate the crucial importance of a proper electronic characterization of this region.A switch from near-intrinsic ((Ce 0.5 Zr 0.5 ) 0.8 Y 0.2 O 1.9 , case A) to hole dominant electronic conduction (GDC, case B) results in a 5-6 order of magnitude change in the electrolyte interface pressure near the thermoneutral potential.This switch also changes the behavior from a pressure decrease across the interface to an increase.We have assumed a thick interface with relatively high electronic resistance.Decreasing the modeled thickness towards the real aged nanoscale interface will increase the predicted pressures.Higher applied potentials will also exacerbate the difference between these two possible conductivity regimes.While our approach is an extreme case, it highlights the impact of the electronic conductivity in this region on device stability.Fully understanding the evolution of the inter-diffusion zone electronic conductivity is crucial for predicting degradation conditions.
According to our LPD analysis, the oxygen electrode ionic charge transfer resistance (   ) is the most reactive parameter.This is perfectly in accordance with the Barnett group's proposal of the oxygen electrode polarization resistance (which includes    ) as a degradation indicator [2].However, our global sensitivity analysis finds that the vast majority of the interface pressure variance is attributable to correlation effects.This indicates that a single parameter optimization approach may yield limited success.Rather, managing an ensemble of parameters is more promising.Overall, it is clear from both sensitivity analysis methods that   ,    , and   are the most critical variables to manage collectively.
Moreover, our 5SLZO results suggest that it is possible, at least theoretically, to engineer a ''sacrificial'' layer that would direct the evolution of the charge transfer resistance and interface pressure.In essence, one would deposit a layer at the interface that would scavenge the diffused and migrated cations to obtain the desired stabilizing conductivity ratio.By depositing a layer with a much lower electronic than ionic conductivity and a thickness given by Eq. ( 20), further sintering/operando accumulation would actually stabilize the interface pressure.However, this strategy is not universal.As shown in Fig. 4B, 5SLZO drives LSCF fracture with the lowest accumulation at high voltage.Thus, this strategy would work best for cells designed to operate in a narrow and fairly constant voltage window near the thermoneutral potential.
On the other hand, our 5SZYO results (Fig. 4 & SF4) support a more general interface stabilization strategy.5SZYO has the highest electronic conductivity of the tested compositions and shows the slowest pressure increase with respect to thickness accumulation.These results are consistent with the strategy of electronic conductivity doping that has been proposed and implemented by Virkar et al. [43][44][45] Increasing the local electronic conductivity decreases the oxygen chemical potential and partial pressure.
Given that approximately 2-5% Y has been observed at the location of SZO inclusions [7], we expect that 5SZYO is the studied composition closest to the reality in aged SOECs.Because our 5SZYO results do not predict critical LSCF fracture conditions except at 800 °C and 1.6 V (SF6), it is plausible that an adjacent yttrium oxide interface layer, like that proposed by Develos-Bagarinao et al. [21], is the true culprit when delamination is observed.Assuming bulk conductivity for an order of magnitude estimate, 1 nm of yttria (Y 2 O 3 ) would increase the ionic resistance by 5.8, 2.8, and 1.5 Ω cm 2 at 700, 750, and 800 °C respectively [24].This would certainly elevate the interfacial resistance ratio to critical levels.Further nanoscale characterization of SZO interfaces in aged SOECs is required to confirm this proposed fracture mechanism.
Irrespective of the mechanism responsible for resistance increases, our conservative calculations in Fig. 3B-which do not take into account the reduction in pressure by evolved electronic resistance-suggest that the stability requirements shown in Eq. ( 17) are satisfied when the oxygen electrode ionic interfacial resistance ratio is <0.2 for 1.6 V at 800 °C and <∼0.35 for all other conditions evaluated.As demonstrated by Bertei et al., oxygen electrode charge transfer resistance can be deconvoluted from the polarization resistance measured by EIS using physically-reasoned distribution of relaxation times [17].Automating this deconvolution could lead to real-time detection of resistances evolving into critical regimes using only EIS measurements.For stacks, these EIS spectra may need to be simulated for individual cells from stack/cell relationships generated by intentional failure tests.
Previous criticism of the equivalent circuit model has largely focused on the ≈1 μm inter-diffusion zone that forms between SOEC electrolytes and barrier layers.Although we have made a first approximation of this inter-diffusion zone as an interface with a junction potential (solely due to lack of sufficiently detailed electronic conductivity data), we emphasize that the model is agnostic about the mathematical location of the interface.In future studies, the interface may be more appropriately defined within the inter-diffusion zone where there are local conductivity minima and/or abrupt changes in chemical potential.Probable conditions include the 50% YSZ-GDC ionic conductivity minimum [30,31] and the electronic conductivity drop off between ≈2% Ce in YSZ and pure YSZ, where cracks have been observed [20].The latter location corresponds well with the YSZ peak pressure condition predicted by Zhang et al. using their diffuse interface model [13].Thus, Eq. ( 10) may be useful as a boundary condition for numerical models to include the effects of evolved nanoscale interfaces.However, further characterization of the conductivity-especially the electronic conductivity-of the YSZ-GDC inter-diffusion zone and the aged interface(s) is required.

Conclusion
Based on our extension of Virkar's solid oxide cell equivalent circuit model to include bilayer electrolytes and investigations into the oxygen electrode interface pressures induced by SZO accumulation, we reach the following conclusions.
1. SZO accumulation may not be sufficient to explain the onset of delamination conditions in electrolysis mode.Reasonable accumulation of 5% Y-doped SZO, does not drive LSCF fracture pressures (∼7200 atm) except at the most extreme conditions tested (800 °C and 1.6 V).This is consistent with the hypothesis of Develos-Bagarinao et al. [21] that the true ion blocking effect of Y-doped SZO originates from a thermodynamically unstable interface, which relaxes by adjacent precipitation of YO x .The added resistance of such a layer would likely push our results into the fracture regime.Further nanoscale characterization is required to confirm this hypothesis.2. Keeping the oxygen electrode interfacial resistance ratio (   ∕  ) below threshold values (0.2 for 800 °C and ∼0.35 for 750 and 700 °C) can prevent the most severe pressure accumulations.The ionic interfacial resistance ratio has a much greater proportional incremental change than the electronic counterpart.Thus, efforts to manage the growth of this parameter will yield greater stabilization results.3. Our SZO composition results are in agreement with previous studies suggesting greater electronic conductivity stabilizes interfaces under electrolysis conditions.However, a very low electronic to ionic interface conductivity ratio (∼10E-6) can also stabilize the interface pressure by overpowering the pressure increase due to the ionic resistance increase.We propose that depositing a layer designed to scavenge the inevitable diffused/ migrated cations to achieve the desired conductivity ratio could stabilize the interface of cells designed to operate near the thermoneutral potential.4. High fidelity modeling of the interface pressure requires an accurate understanding of the electronic conductivity of the bilayer electrolyte interface.The literature currently lacks sufficiently detailed data on this region to avoid making semiarbitrary assumptions.Changing the assumed electronic conductivity of the inter-diffusion zone from the near-intrinsic behavior of (Ce 0.5 Zr 0.5 ) 0.8 Y 0.2 O 1.9 to the hole dominant behavior of GDCwhich would make a negligible difference on cell performance due to the low electronic current-increases the predicted interface pressure by at least 5 orders of magnitude.It is essential to increase our understanding of the proper electronic conductivity and under what conditions (if any) continuity of oxygen chemical potential can be assumed at the bilayer electrolyte interface in future studies.

Fig. 1 .
Fig. 1.Scheme of the relevant SOEC cation diffusion phenomena.During sintering and operation, Zr and Y diffuse/migrate (white arrows) from the YSZ electrolyte (blue) into the GDC barrier layer (red) to form the inter-diffusion zone (purple) and SZO at the electrolyte and oxygen electrode interfaces (green).Sr (black arrows) as well as Ce and Gd (gray arrows) diffuse towards the YSZ.(For color references, the reader is referred to the web version of this article.)

Fig. 2 .
Fig. 2. Equivalent circuit model for transport through the SOEC.The voltage sources represent both junction potentials (   ,   ,   ) and potential drops across the thin films due to changes in electrochemical potential (  ,   ).Interface charge transfer (   ,   ,   ; red arrows) and ohmic resistances (  ,   ; blue arrows) are distinguished by ionic (O 2− /  ) and electronic ( − or ℎ + /  ) carriers.(For color references, the reader is referred to the web version of this article.) al., De Vero et al., Bernadet et al., and Monaco et al.