Gas leak diffusion induced polarization in submicro/nanoscale non-tight electrolytes of solid oxide fuel cells

Abstract

Solid oxide fuel cells with submicro/nanoscale electrolytes (μSOFCs) are attracting increasing attention since the ohmic energy loss arising from an ion-resistive electrolyte decreases significantly with decreasing thickness of the electrolyte interlayers. However, gas leak diffusion can be induced due to increasing microstructural flaws such as cracks and pinholes in thin electrolytes. Evaluation of the effects of gas leak diffusion through electrolyte on cell performance is thus an urgent demand. In this work, the effect of gas leak diffusion on concentration polarizations (CPs) is investigated quantitatively for both anodes and cathodes of SOFCs under various operating conditions. The results show that gas leak diffusion through electrolyte typically induces dominant cathode CP. The direct reaction of leaked H₂ and O₂ correlates has a large impact on both anode and cathode CP induced by gas leak diffusion. Lowering the operating temperature decreases CP induced by gas leak diffusion. Our work provides a quantitative model to evaluate the impact of gas leak diffusion in electrolytes on SOFC performance and facilitates the rational design of high performance μSOFCs.

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1. Introduction

Solid oxide fuel cells (SOFCs) utilize hydrogen and fossil gas fuels to generate electricity on a large scale, and are considered to be a promising solution to meet future sustainable energy demands of our society.^1–4 In SOFCs, superior solid oxygen ionic conductors are typically employed as electrolytes.^5–8 Although a large variety of oxygen ionic conductors have been developed, yttria-stabilized zirconia (YSZ) has been the most common choice of SOFC electrolyte due to its excellent chemical stability and mechanical strength.^9,10 YSZ exhibits poor ionic conductivity unless it works at high enough temperatures, which are typically above 650 °C.^11,12 Such high operation temperatures cause severe thermal degeneration of other components in SOFCs and influence largely the cost-effectiveness and long-term performance.¹² Decreasing the thickness of YSZ is an effective and feasible way to reduce the ohmic resistance of the cells and to achieve low operation temperatures. Previous reports show that the total cell area-specific residence (ASR) attributed to electrolytes should be less than 0.15 Ω cm², to achieve a target cell power density of 1 W cm⁻².¹² As shown in Fig. S1 and S2,† to satisfy this requirement, YSZ with a thickness of ∼150 μm is feasible at an operating temperature of 950 °C, and decreasing the thickness of YSZ to less than 1 μm allows SOFCs to operate at 500 °C.^13,14 SOFCs with nanoscale/sub-microscale electrolytes (μSOFCs, electrolyte thickness < 1 μm) are thus attracting increasing attention.^15–18 Extensive methods have been developed to fabricate μSOFCs. For instance, by combing sputtering, lithography and etching, μSOFCs with 50–150 nm YSZ or YSZ/CGO were fabricated and a peak power density of 400 mW cm⁻² at 400 °C and 200 mW cm⁻² at 350 °C were obtained.¹⁹ Kerman et al. fabricated SOFC devices with YSZ/Gd doped CeO₂ (CGO) of 85 nm by a co-sputtering method, achieving a peak power density of 1175 mW cm⁻² at 520 °C.²⁰ Other methods such as atomic layer deposition,^21,22 plasma enhanced atomic layer deposition,²³ pulsed laser deposition,^24–26 sputter deposition,²⁷ and nano-powder slurry spin coating (NSC)²⁸ have also been developed to fabricate μSOFCs. Although substantial ohmic loss reduction has been achieved by decreasing the electrolyte thickness, these thinner electrolytes exhibit typically reduced mechanical strength and thermal stability, and experience increased probability to crack.^1,29 Thermal shock during continuous work cycles from the room temperature to high operation temperatures accelerates greatly such crack when the electrolyte thickness decreases.^30–33 As a consequence, gas leak diffusion through electrolyte occurs, in which hydrogen gas diffuses through the electrolyte to the cathode and oxygen gas diffuses to the anode. A large gas leak diffusion leads to inefficient cell performance or even explosion since hydrogen and oxygen are mixed together at high temperatures. Although a relatively small gas leak diffusion does not damage devices, it causes severe energy loss. The open circuit voltages (OCV) are reported frequently to be much lower than the theoretical value.^34–36 Materials degradation due to direct oxidation/reduction, as induced by electrolyte gas diffusion, reduce the triple phase boundary (TPB) and create mechanical stress, which impairs the cell integrity. In particular, electrolyte gas leak diffusion induces significant concentration polarization (CP) since the concentrations of reactive H₂ and O₂ at the electrolyte/electrode interfaces decreases due to gas leak diffusion in electrolyte. Up to this date, effect of the gas leak diffusion through electrolyte to the overall performance of SOFCs is still elusive. In this work, an electrolyte gas leak diffusion model is built for SOFCs. The effects of the gas leak diffusion rate on the concentration polarizations of both anodes and cathodes are studied quantitatively under operation current densities and temperatures when the electrolyte thickness varies from nanoscale to micro-scale. Our work facilitates the rational design of μSOFCs with improved performance.

2. Models and calculation methods

Fig. 1 shows schematics of electrolyte gas diffusion induced concentration polarization in SOFCs. Fig. 1a shows an SOFC with a porous anode, a porous cathode and a cracked electrolyte. When gas leak diffusion occurs, H₂ and H₂O in the anode diffuse along crack channels in the electrolyte to the cathode, whereas O₂ diffuses directly to the anode. The partial pressures of H₂, H₂O and O₂ at the electrode/electrolyte interfaces differ from those without leak diffusion, which are demonstrated in Fig. 1b. and are the partial pressures of the gas H₂(H₂O) in the anode-side flow field and at the anode/electrolyte interface, respectively. and are the partial pressures of O₂ in the cathode-side flow field and at the cathode/electrolyte interface, respectively. , and are the partial pressures of H₂, H₂O and O₂, which are induced by electrolyte gas leak diffusion. and represent the partial pressures of H₂(H₂O) and O₂ in the cathode-side and anode-side flow fields, respectively. The dash lines show the distributions of the partial pressures of H₂, H₂O and O₂ with no leak diffusion in the electrolyte, whereas the solid lines correspond to the gas leak diffusion scenario. , , and decrease when leak diffusion through the electrolyte occurs. As a consequence, the gas leak diffusion induces concentration polarization according to the Nernst equation. The anode and cathode concentration polarization can be written as eqn (1) and (2), respectively,where η^t_a(η^t_c) is the total concentration polarization of the anode (cathode), F is the Faraday constant, R is the ideal gas constant, and T is the working temperature of SOFCs. To calculate η^t_a(η^t_c), with known , and as determined by the composition of the supplied fuel/air gases, , and must be calculated based on the gas diffusion in both porous electrodes and cracked electrolyte. As shown in Fig. 1, at a steady state, gas flux at the anode (cathode) is required, while gas leak diffusion flux occurs at the anode (cathode)/electrolyte interface due to the cracked electrolyte. The pressures of H₂ and H₂O in the cathode-side flow field ( and ) are assumed to be zero since the leaked H₂ and H₂O can be taken away by the O₂–N₂ flow. Likewise, the pressures of O₂ and N₂ in the anode-side flow field ( and ) are also assumed to be zero since the leaked O₂ and N₂ can be taken away by the H₂–H₂O flow. By assuming that the leaked O₂ only reacts with H₂ at the anode/electrolyte interface where a catalytic layer exists, the following equations can be obtained according to the mass conservation law,where is the effective diffusivity of H₂ and H₂O in the cathode, and is the effective diffusivity of O₂ and N₂ in the anode. L_a(L_c) is the thickness of the anode (cathode). D_m and L_m are the effective diffusivity and thickness of the electrolyte. x is the ratio of O₂ flux reacting with H₂ at the anode/electrolyte interface to the total .

Fig. 1 Schematics of electrolyte gas leak diffusion induced concentration polarization in SOFCs. (a) A schematic shows a porous anode, a porous cathode and a crack electrolyte. A crack electrolyte not only transports O²⁻ from the cathode to the anode, but also induces leak diffusion of H₂ and H₂O from the anode to the cathode and O₂ from the cathode to the anode along the crack channels in the electrolyte. (b) The distribution of partial pressures of H₂, H₂O and O₂ along the cell components. The dash lines show the case without gas leak diffusion in the electrolyte, whereas the solid lines denote the case as gas leak diffusion in the electrolyte occurs.

Since the gas leak diffusion fluxes and the direct reaction between H₂ and O₂ at the anode/electrolyte interface do not contribute to the output current of SOFCs, the correlations between the output current density and the fluxes can be presented as eqn (6)–(8) according to Fick's law,^37,38

where is the effective binary gas diffusivity of the anode (cathode). According to the Chapman–Enskog model,^39,40 the effective binary gas diffusivity is given as,where A–B is either H₂–H₂O or O₂–N₂, M_A and M_B are the molecule weight of the components A and B, respectively, p is the total pressure, Ω is the collision integer, σ_A–B is the collision diameter, ϕ is the porosity of the porous electrode, and τ is the tortuosity of the porous electrode. Based on the data of Ω and σ_A–B from Cussler,³⁹ and . By combing eqn (3)–(8), all parameters can be obtained to calculate the concentration polarization of SOFCs, as listed in Table 1.

η^l_a = η^t_a − η^nl_a (30)

η^l_c = η^t_c − η^nl_c (31)

Table 1 Gas partial pressures and concentration polarizations inside SOFCs^a

Item	No gas leak at electrolyte	Eqn no.	Electrolyte with gas leak diffusion	Eqn no.
a Where ias(ics) is the anode (cathode) limiting current density (LCD). When Dm = 0, the pressure parameters and anode (cathode) concentration polarization without gas leak diffusion in the electrolyte, η^nla and η^nlc can be calculated, as shown in the 1^st column of Table 1. The contribution of the electrolyte gas leak diffusion to the anode (cathode) CP, η^la(η^lc), can be obtained by eqn (30) and (31).
Partial pressure		(10)		(11)
		(12)		(13)
		(14)		(15)
		(16)		(17)
		(18)		(19)
		(20)		(21)
LCD		(22)		(23)
		(24)		(25)
CP		(26)		(27)
		(28)		(29)

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In this work the parameters in the calculations are shown in Table 2.

Table 2 Parameters used in the determination of concentration polarizations of SOFCs

Parameter	Value (anode)	Value (cathode)	Value (electrolyte)	Unit
a The structure parameters and the effective diffusivities are adopted from ref. 40.
Thickness^a	La = 0.75	Lc = 0.2	Lm = 0.01–100 × 10^−3	mm
Effective diffusivity^a	= 0.070 (800 °C), 0.060 (750 °C), 0.055 (700 °C), 0.046 (650 °C)	= 0.072 (800 °C), 0.067 (750 °C), 0.062 (700 °C), 0.056 (650 °C)	Dm = or	cm^2 s^−1
H2 partial pressure			—	Pa
H2O partial pressure			—	Pa
O2 partial pressure			—	Pa
N2 partial pressure			—	Pa

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3. Results and discussion

When gas leak diffusion occurs in the electrolyte, the mixed H₂ and O₂ react directly to produce H₂O due to the high working temperatures and the catalyst layer at the anode/electrolyte interface. One possibility is that the leaked H₂ and O₂ do not react (x = 0), and the leaked H₂ and O₂ can also possibly react completely (x = 1). Partial reaction between H₂ and O₂ occurs as 0 < x < 1.

3.1 CP of SOFCs with no reaction between leaked H₂ and O₂ (x = 0)

When x = 0, i_as and η^t_a are independent of D_m and L_m, indicating that anode CP is not influenced by the electrolyte gas leak diffusion. The expressions of i_as and η^t_a are identical to those for SOFCs with no electrolyte gas leak. Thus, the electrolyte gas leak diffusion induced anode CP, η^l_a, is zero. This is because the diffusion flux of H₂O through the electrolyte results in a decrease of , which is in proportional to the decrease of . Therefore, in the following discussion, only the cathode CP is investigated.

The effect of gas leak diffusion rate of electrolyte on cathode polarization is first investigated. The gas diffusion rate of the electrolyte is dependent on both D_m and L_m. Fig. 2a–b shows the plots of η^l_c versus and the proportion of η^l_c to total cathode CP, respectively, when L_m varies from micro-scale to nano-scale with current density output fixed at 0.1 A cm⁻² η^l_c increases as increases regardless of change in L_m. At fixed , both η^l_c and η^l_c/η^t_c increases as L_m decreases. Since η^nl_c is independent of according to eqn (28), η^l_c/η^t_c increases with increasing for electrolytes with all thicknesses. Both η^l_c and η^l_c/η^t_c reach larger plateaus when L_m is smaller. For instance, η^l_c and η^l_c/η^t_c increase from 0 to 1.08 mV and 60%, respectively, with increasing from 0 to 0.1% when L_m is 100 nm. In comparison, η^l_c and η^l_c/η^t_c increase from 0 to only 0.25 mV and 30%, respectively with increasing from 0 to 0.1% when L_m is 10 μm. The results indicate that that even with a small electrolyte gas leak diffusion of = 0.5%, more than 4 times of gas leak induced cathode CP for a 100 nm electrolyte is induced as compared to a 10 μm electrolyte. Fig. 2c–d shows the effect of electrolyte gas leak diffusion on cell polarization as SOFCs work at different current workloads. Two typical electrolyte thicknesses of 100 nm and 10 μm, are chosen, which represent frequently reported μSOFCs and conventional SOFCs (cSOFCs). According to eqn (28), (29) and (31), η^l_c is independent of i. When i is zero, η^nl_c is zero, and only η^l_c contributes to the total concentration polarization at such open circuit circumstance. Therefore, η^l_c reflects the decrease of OCV of SOFCs resulted from electrolyte gas leak diffusion. OCV for μSOFCs is typically lower than that of cSOFCs. This is a drawback of μSOFCs to be overcome. η^l_c/η^t_c as a function of i is studied further for both μSOFCs and cSOFCs to better understand the contribution of η^l_c to the total concentration polarization. η^l_c/η^t_c decreases with increasing output current for both μSOFCs and cSOFCs. η^l_c/η^t_c of μSOFCs is larger than that of cSOFCs as the electrolyte has the same crack extent, but the difference becomes smaller with larger output currents. For instance, at i = 0.1 A cm⁻² and = 0.1%, η^l_c/η^t_c is 62% and 31% for μSOFCs and cSOFCs, respectively. When i increases to 1 A cm⁻², the corresponding values are ∼12% and ∼4%, respectively. The result indicates that, if the leaked H₂ and O₂ do not react directly, the contribution of gas leak diffusion induced cathode CP for μSOFCs is more significant than that of cSOFCs at small output currents. Nevertheless, such a contribution becomes small at larger SOFC output currents for both μSOFCs and cSOFCs.

Fig. 2 (a and b) Plots of (a) η^l_c, and (b) the proportion of η^l_c to total cathode CP versus . The electrolyte thickness varies from 100 nm to 100 μm. SOFCs work at a output current density of 0.1 A cm⁻² and T = 700 °C. (c and d) The proportion of η^l_c to the total cathode CP versus i when electrolyte thickness is (c) 100 nm and (d) 10 μm. SOFCs operate at 700 °C. It is assumed that the leaked H₂ and O₂ do not react (x = 0).

3.2 CP of SOFCs with reaction between leaked H₂ and O₂ (x = 1)

Since SOFCs typically operate at temperatures above 600 °C and the catalytic layer at electrode/electrolyte interface is active to the reaction between H₂ and O₂, the reaction between the mixed H₂ and O₂ due to electrolyte gas leak diffusion cannot be neglected in certain cases. Here, it is assumed that the leaked O₂ reacted completely with H₂ at anode/electrolyte interfaces. In such a case, the electrolyte gas leak diffusion induced CP is investigated. η^l_a is no longer zero when x is zero according to eqn (27). Fig. 3 shows the correlations between η^l_a(η^l_a/η^t_a) and as well as i. As shown in Fig. 3a and b, both η^l_a and η^l_a/η^t_a increase with , and the slopes of the curves become larger when L_m decreases from 100 μm to 10 nm. For instance, at L_m = 10 μm, η^l_a increases from 0 to ∼16 mV and η^l_a/η^t_a from 0 to ∼25% when increases from 0 to 0.1%. In comparison, at L_m = 100 nm, η^l_a increases from 0 to >200 mV and η^l_a/η^t_a from 0 to >80% when increases from 0 to 0.1%. In addition, the values of electrolyte gas leak diffusion induced CP with x = 1 are much larger than those with x = 0, and such trends are more obvious as the electrolyte thickness decreases. The results indicate that the direct reaction between the leaked H₂ and O₂ is detrimental to SOFCs. Fig. 3c and d further shows the variation of η^l_a/η^t_a with SOFC output current density for both μSOFCs and cSOFCs, respectively. η^l_a/η^t_a decreases with increasing i firstly and then increases when i is above 1 A cm⁻². This is because η^l_a and η^t_a are dependent of both i and the amount of leaked O₂ according to eqn (23) and (27). At the same , η^l_a/η^t_a for μSOFCs is larger than that of cSOFCs. For instance, at = 0.01%, and i = 1 A cm⁻², η^l_a/η^t_a is ∼20% for μSOFCs and only 0.3% for cSOFCs. When is larger than 0.1%, H₂ at the anode/electrolyte interface is depleted by the leaked O₂ and μSOFCs cannot work at the setting parameters. Therefore, serious anode CP by electrolyte gas leak diffusion is induced when the reaction of H₂ and O₂ occurs, especially for μSOFCs. Advanced techniques should be developed to fabricate μSOFCs with well controlled thickness and quality. Dense electrolyte films with dramatically decreased defects, cracks or pinholes are required as the film thickness decreases.

Fig. 3 (a and b) Plots of (a) η^l_a, and (b) the proportion of η^l_a to total anode CP versus . The electrolyte thickness varies from 100 nm to 100 μm. SOFCs work at a output current density of 0.1 A cm⁻² and T = 700 °C. (c and d) The proportion of η^l_a to the total cathode CP versus i when electrolyte thickness is (c) 100 nm and (d) 10 μm. SOFCs operate at 700 °C. It is assumed that the leaked H₂ and O₂ react completely (x = 1).

The cathode CP induced by electrolyte gas leak diffusion is given in Fig. 4. As shown in Fig. 4a, η^l_c increases with increasing and decreasing electrolyte thickness. For instance, η^l_c increases from 0 to 4 mV and 64 mV, respectively for cSOFCs and μSOFCs when increases from 0 to 1%. As shown in Fig. 4b, η^l_c/η^t_c increases from 0 to ∼35% and ∼99%, respectively, for cSOFCs and μSOFCs when increases from 0 to 1%. The result indicates that although η^l_c is smaller than η^l_a under the same operation conditions, η^l_c/η^t_c is typically larger than η^l_a/η^t_a. Electrolyte gas leak diffusion has a larger impact on cathode CP compared to anode CP. Fig. 4c and d show the variation of η^l_c/η^t_c versus i for μSOFCs and cSOFCs, respectively. η^l_c/η^t_c decreases with increasing i for both types of SOFCs since η^l_c changes little but η^nl_c increases significantly when i increases. At i = 1 A cm⁻², η^l_c/η^t_c is ∼30% for μSOFCs but is only 0.5% for cSOFCs when is 0.01%. When increases to 1%, η^l_c/η^t_c is ∼90% and ∼30% for μSOFCs and cSOFCs, respectively. The result illustrates that cathode CP arising from the electrolyte gas leak diffusion is dominant in the total cathode CP for μSOFCs even when the output current density is large.

Fig. 4 (a and b) Plots of (a) η^l_c, and (b) the proportion of η^l_c to total CP polarization versus . The electrolyte thickness varies from 100 nm to 100 μm. SOFCs work at a output current density of 0.1 A cm⁻² and T = 700 °C. (c and d) The proportion of η^l_c to the total cathode CP versus i when electrolyte thickness is (c) 100 nm and (d) 10 μm. SOFCs operate at 700 °C. It is assumed that the leaked H₂ and O₂ react completely (x = 1).

3.3 CP of SOFCs with partial reaction between leaked H₂ and O₂ (0 < x < 1)

Partial reaction between leaked H₂ and O₂ is also investigated when x varies between 0 and 1, as shown in Fig. 5. η^l_a, η^l_c, η^l_a/η^t_a and η^l_c/η^t_c all increase with increasing x, and such trends are more obvious when the electrolyte thickness decreases. For instance, at L_m = 10 μm, η^l_a increases from 0 to 0.18 mV, η^l_a/η^t_a increases from 0 to ∼0.4%, η^l_c holds at ∼0.004 mV, and η^l_c/η^t_c holds at ∼0.6% when x increases from 0 to 1. At L_m = 100 nm, η^l_a increases from 0 to ∼15 mV, η^l_a/η^t_a increases from 0 to ∼24%, η^l_c increases from 0.3 to 0.4 mV, and η^l_c/η^t_c increases from 31% to 38% when x increases from 0 to 1. The result indicates that larger CPs for both anode and cathode are induced by electrolyte gas leak diffusion with increasing reaction ratio between the leaked H₂ and O₂, especially for anode CP of μSOFCs. Therefore, gas leak diffusion through electrolyte must be avoided for μSOFCs with catalytically active anodes and cathodes.

Fig. 5 Plots of electrolyte gas leak diffusion induced (a and b) anode and (c and d) cathode concentration polarization versus x. Both and are kept at 0.001%. SOFCs operate at 700 °C.

The most promising advantage of μSOFCs is that it can reduce largely the ohmic loss when SOFCs operate at a low temperature below 500 °C.⁴¹ To evaluate the influence of the electrolyte gas leak diffusion on SOFC performance under different temperatures, the impact of operating temperature on both anode and cathode CP induced by electrolyte gas leak diffusion is also investigated, as shown in Fig. S3–S5.† η^l_c and η^l_c/η^t_c decrease with T at x = 0, but both anode and cathode CPs induced by electrolyte gas leak diffusion vary little at x = 1. Considering that x decreases with decreasing T since the activity of catalyst at electrode/electrolyte interface decreases with lowering T, the electrolyte gas leak diffusion induced CP decreases with T, indicating that operation of μSOFCs at lower temperatures reduces the adverse impact of gas leak diffusion of the electrolyte.

Through the analysis, the key factors in designing μSOFCs are proposed, including the thickness of electrodes/electrolyte as well as the maximum tolerance of electrolyte gas leak and cell operating temperatures. It is noted that the fuel gas of H₂ is used in the model. Other fuel gases such as CH₄ or CO can be considered to experience similar electrolyte gas leak diffusion induced polarization. The model developed here is typically applicable to ionic electrolytes with low electronic conductivity, like YSZ. Ionic electrolytes with notable electronic conductivities, such as CeO₂ and Bi₂O₃ based electrolytes, also suffer unignorable electronic current leakage when the electrolyte thickness falls into a nanoscale regime. The difference in the gas leak diffusivities for different types of gases is also ignored since it depends on the undefined electrolyte crack structure.

4. Conclusion

In this work, electrolyte gas leak diffusion induced polarization for both μSOFCs and cSOFCs is studied analytically. The effects of cell parameters including electrode diffusivity, electrolyte leak diffusivity, electrolyte thickness, and cell operating parameters including current density output and working temperature are discussed. Our results show that if leaked H₂ and O₂ do not react directly, electrolyte gas leak diffusion contributes little to anode concentration polarization due to proportional variation of H₂ and H₂O, but results in an increased cathode concentration polarization. Both significant anode and cathode concentration polarizations by electrolyte gas leak diffusion are induced when direct reaction between leaked H₂ and O₂ occurs. μSOFCs experience several orders of magnitude larger concentration polarization arising from electrolyte gas diffusion as compared with cSOFCs, suggesting that high gas-tight electrolyte sub-micro/nano films are required for μSOFCs. The electrolyte gas leak induced concentration polarization decreases with lowering cell operation temperature. In addition, other factors must be considered in the evaluation of electrode polarization in thin-film based fuel cells, including the non-trivial decrease of ohmic loss while employing sub-micro/nano-electrolyte films.

Nomenclature

	The effective diffusivity of leaked O2 and N2 in the anode
	The effective diffusivity of leaked H2 and H2O in the cathode (m^2 s^−1)
D^effA–B	The effective binary diffusivity of gas species A and B (m^2 s^−1)
Dm	The effective diffusivity of gas in the electrolyte (m^2 s^−1)
F	Faraday's constant (C mol^−1)
i	Current density (A m^−2)
ias	Limiting current density of anode (A m^−2)
ics	Limiting current density of cathode (A m^−2)
JH2	Gas flux of H2 at the anode (mol m^−2 s^−1)
JH2O	Gas flux of H2O at the anode (mol m^−2 s^−1)
JO2	Gas flux of O2 at the cathode (mol m^−2 s^−1)
	Gas flux of H2 at the anode/electrolyte interface due to cracked electrolyte (mol m^−2 s^−1)
	Gas flux of H2O at the anode/electrolyte interface due to cracked electrolyte (mol m^−2 s^−1)
	Gas flux of O2 at the cathode/electrolyte interface due to cracked electrolyte (mol m^−2 s^−1)
La	The thickness of the anode (m)
Lc	The thickness of the cathode (m)
Lm	The thickness of the electrolyte membrane (m)
MA	The molecule weight of the components A (kg mol^−1)
	The partial pressures of H2 at the anode/electrolyte interface (Pa)
	The partial pressures of H2O at the anode/electrolyte interface (Pa)
	The partial pressures of O2 at the anode/electrolyte interface (Pa)
	The partial pressures of O2 in the anode-side flow field (Pa)
	The partial pressures of N2 in the anode-side flow field (Pa)
	The partial pressures of H2 at the cathode/electrolyte interface (Pa)
	The partial pressures of H2O at the cathode/electrolyte interface (Pa)
	The partial pressures of O2 at the cathode/electrolyte interface (Pa)
	The pressures of H2 in the cathode-side flow field (Pa)
	The pressures of H2O in the cathode-side flow field (Pa)
	The partial pressures of H2 in the anode-side flow field (Pa)
	The partial pressures of H2O in the anode-side flow field (Pa)
	The partial pressures of O2 in the cathode-side flow field (Pa)
pt	Total pressure (Pa)
R	Ideal gas constant (J mol^−1 K^−1)
T	Temperature (K)
x	The ratio of O2 flux reacting with H2 at the anode/electrolyte interface to the total

Greek letters

η	Concentration polarization (V)
η^la	The contribution of the electrolyte gas leak diffusion to the anode CP (V)
η^lc	The contribution of the electrolyte gas leak diffusion to the cathode CP (V)
η^nla	The anode concentration without gas leak diffusion in the electrolyte (V)
η^nlc	The cathode concentration without gas leak diffusion in the electrolyte (V)
η^ta	The total concentration polarization of the anode (V)
η^tc	The total concentration polarization of the cathode (V)
σH2–H2O	Average collision diameter of H2/H2O (m)
ϕ	The porosity of porous electrode
τ	The tortuosity of porous electrode
Ω	The collision integral (Å)

Superscripts & subscripts

A	Gas species of A
B	Gas species of B
a	Anode
c	Cathode
m	Electrolyte membrane
t	Total
nl	SOFCs with no gas leak diffusion in electrolyte
l	SOFCs with gas leak diffusion in electrolyte
ai	At the interface of anode/electrolyte
ci	At the interface of cathode/electrolyte
af	In the anode-side flow field
cf	In the cathode-side flow field

Acknowledgements

The work is supported by UESTC new faculty startup fund, the National Natural Science Foundation of China (grant no. 21403031 and 51501030) and the Fundamental Research Funds for the Chinese Central Universities (grant no. ZYGX2014J088 and grant no. ZYGX2015Z003).

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra10071f

‡ These authors contributed equally to this work.

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