Validation of a physically-based solid oxide fuel cell anode model combining 3D tomography and impedance spectroscopy

This study presents a physically-based model for the simulation of impedance spectra in solid oxide fuel cell (SOFC) composite anodes. The model takes into account the charge transport and the charge-transfer reaction at the three-phase boundary distributed along the anode thickness, as well as the phenomena at the electrode/electrolyte interface and the multicomponent gas diffusion in the test rig. The model is calibrated with experimental impedance spectra of cermet anodes made of nickel and scandia-stabilized zirconia and satisfactorily validated in electrodes with different microstructural properties, quantiﬁed through focused ion beam SEM tomography. Besides providing the material-speciﬁc kinetic parameters of the electrochemical hydrogen oxidation, this study shows that the correlation between electrode microstructure and electrochemical performance can be successfully addressed by combining physically-based modelling, impedance spectroscopy and 3D tomography. This approach overcomes the limits of phenomenological equivalent circuits and is suitable for the interpretation of experimental data and for the optimisation of the electrode microstructure.


Introduction
Solid oxide fuel cells (SOFCs) are electrochemical systems which convert the chemical energy of a fuel, such as hydrogen, directly into electric energy and heat, allowing for high efficiency of power generation [1,2], low emission of pollutants [3] and fuel flexibility [4]. The performance and durability of the cell are strongly dependent on the electrodes [5e7], which are porous layers, typically made of a composite of ceramics and/or metallic particles, wherein electrochemical reactions occur. There is significant interest in relating the electrode microstructure to electrochemical performance [8e12]. Understanding how the microstructure affects the electrochemical response would allow researchers to better design the electrodes to increase power density and extend lifetime [13e17].
This task can be accomplished by using physically-based models. Physically-based electrochemical models solve conservation equations for charged and chemical species to provide a mechanistic description of the transport and reaction processes occurring within the electrodes [18e21]. The microstructural characteristics of the electrodes are included in the model with different levels of complexity. In some recent studies, model equations are solved within the three-dimensional microstructure of the electrode [22e24], which is reconstructed with tomographic techniques [25e28]. However, the computational cost of this type of simulations is currently too high to allow for a systematic use of this method. Typically, the continuum approach is adopted in developing electrochemical models [19,29e31]. In this approach the particle-level details are represented through effective microstructural properties, which can be obtained from the application of empirical correlations [32], percolation models [33e35], packing algorithms [36,37] or 3D tomography [38]. Several modelling studies have been presented to predict the microstructural contribution to electrochemical performance and enhance the power density via an optimisation of the microstructural properties [19,37,39e43].
However, in most of the studies published so far the validation of the model is lacking [32,44,45]. In order to quantify the correlation between microstructure and electrochemical performance, models must be properly calibrated with highquality electrochemical experimental data and then verified in different electrode microstructures. Typically electrochemical models are calibrated by tuning microstructural or electrochemical parameters on a single set of polarization curves [30,46,47] without assessing their validity in different samples. Furthermore, electrochemical impedance spectroscopy (EIS) data contain more information than polarization curves and allow for the decoupling of different processes according to their characteristic timescale, thus representing a more comprehensive benchmark for the validation of electrochemical models [48e53].
This study describes the validation of a physically-based electrochemical model by using impedance spectroscopy data for different operating conditions and different electrode microstructures. The study focuses on SOFC composite anodes made of the same materials, nickel and scandiastabilized zirconia (ScSZ), prepared with different volume fractions and fabrication techniques, including infiltrated electrodes. Focused ion beam SEM (FIB-SEM) tomography is used to obtain the microstructural parameters required by the model, while EIS data at different operating conditions are used for its calibration and validation. The results of the study comprise the material-specific kinetic parameters of Ni:ScSZ electrodes, which can be used along with the model to optimise the microstructural design.
The paper is organized as follows: in Section 2 the electrochemical model is presented. Section 3 reports the results of the study, comprising the fitting and deconvolution of impedance spectra, the discussion of model parameters and the cross verification of the model in different microstructures. The general conclusions of the study are reported in Section 4. Fig. 1a schematically represents the main phenomena occurring within the anodes modelled and tested in this study. The anode consists of a porous composite layer of Ni and ScSZ particles randomly dispersed and sintered on the top of an yttria-stabilized zirconia (YSZ) electrolyte [54]. Hydrogen is supplied to the anode and diffuses throughout the diffusive stagnant layer, which is expected to be present in electrodes tested in button cell configuration [55e57]. Hydrogen is electrochemically oxidized at the three-phase boundaries (TPBs) among Ni, ScSZ and pores spread throughout the thickness of the anode. Water counter-diffuses back while electrons migrate along the percolating electron-conducting networks created by Ni particles. Oxygen ions, coming from the electrolyte, are transferred to the ScSZ phase at the anode/electrolyte interface and then reach the TPBs to take part in hydrogen electro-oxidation.

Modelling
The electrochemical model, developed in the next section, mathematically describes these transport and reaction phenomena. The following set of simplifying assumptions are considered in order to keep the model as simple as possible and reduce the number of unknown parameters: Fig. 1 e a) Schematic representation of the charge-transfer and charge transport phenomena occurring within the anode (not to scale). On the top, the three contributions to electrode impedance are reported. b) Tomographic reconstruction of the anode Ni40 (green: Ni, grey: ScSZ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 1 ( 2 0 1 6 ) 2 2 3 8 1 e2 2 3 9 3 no mixed ionic-electronic conduction, i.e., Ni is regarded as a pure electronic conductor while ScSZ as a pure ionic conductor; charge-transfer reactions are described according to Butler-Volmer kinetic expressions [34,58]; the gas transport within the pores of the anode is neglected due to the small thickness of the electrode. This assumption is further verified in Section 3.2 (see Fig. 5); isothermal conditions; only the transport and reaction phenomena occurring along the thickness of the electrode are modelled, resulting in a 1D distributed model. For the button cell configuration used here, this 1D approximation has been shown to be sufficient to extract accurate results [51,59,60]; the electrode microstructure is modelled as a continuum: the effective microstructural properties, such as the TPB length per unit volume and the effective conductivities, which are evaluated from the tomographic reconstruction of the electrodes (see Fig. 1b), are assumed homogeneous throughout the electrode thickness.
Although the model is applied to Ni:ScSZ anodes, it is generally applicable to any type of composite SOFC electrodes that satisfy the assumptions listed above.

Model equations
According to the description provided in the previous section, the impedance of the electrode consists of three processes occurring in series (see Fig. 1a): oxygen ion transfer at the electrode/electrolyte interface; distributed charge-transfer at the TPB spread throughout the electrode thickness, coupled with electron and oxygen ion migration; gas diffusion in the stagnant layer.
In this section the impedance of the electrode is derived according to physically-based sub-models describing these processes. For the sake of conciseness, model equations are reported in the frequency domain only. The symbol Dz is used to represent the phasor of the a.c. perturbation of a generic variable z' around its steady-state value z, which is the opencircuit condition in this case: where j is the imaginary unit and u the angular frequency, with u ¼ 2p f, where f is the frequency. For a comprehensive description of the Fourier transformation of model equations from time domain to frequency domain the reader is referred to Lasia [61] and Bertei et al. [52]. The impedance of the oxygen ion transfer Z Ot between YSZ and ScSZ at the electrode/electrolyte interface is represented through an oxygen transfer resistance r ScSZ/YSZ in parallel to a capacitance c ScSZ/YSZ , as follows: where A ScSZ/YSZ is the fraction of interfacial area between ScSZ and YSZ at the electrode/electrolyte interface. r ScSZ/YSZ is the specific charge-transfer resistance associated to the Faradaic contribution of the oxygen ion transfer at ScSZ/YSZ interface, while c ScSZ/YSZ is a lumped parameter taking into account the double-layer capacitance and the other capacitive contributions associated to the oxygen ion transfer (oxygen ion transfer at the electrode/electrolyte interface is taken into account in several well-established physically-based models [62e64] and has been identified in the analysis of impedance spectra [52,65]). The impedance related to the distributed charge-transfer at the TPB within the electrode is obtained from the conservation equations of electronic and ionic current along the thickness of the anode: where x ¼ 0 refers to the electrode/current collector interface and x ¼ L an to the electrode/electrolyte interface (i.e., from right to left in Fig. 1a). Note that the current perturbation DI is totally in electronic form at x ¼ 0 while it is in ionic form at x ¼ L an [34,42].  [20,37]. In Eq. (4), DV Ni and DV ScSZ represent the phasors of the electronic and ionic potential in Ni and ScSZ, respectively.
The source terms per unit volume in the right-hand side of Eq. (3) refer to the Faradaic current due to the charge-transfer reaction at the TPB Di v TPB and the current associated to the electric double-layer between Ni and ScSZ Di v Ni=ScSZ . According to the Butler-Volmer kinetics [58] and assuming ideal capacitive behaviour of the Ni/ScSZ interface [18], Di v TPB and Di v Ni=ScSZ result as follows: where i 0TPB is the exchange current density per unit of TPB length, c Ni/ScSZ the specific capacitance of the Ni/ScSZ interface while L v TPB and A v Ni=ScSZ are microstructural properties corresponding to the TPB length and the Ni/ScSZ contact area per unit volume, respectively. F, R and T represent the Faraday constant, the gas constant and the absolute temperature, respectively.
By defining the activation overpotential Dh as [34]: i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 1 ( 2 0 1 6 ) 2 2 3 8 1 e2 2 3 9 3 and introducing the dimensionless coordinatex ¼ x=L an , the substitution of Eqs. (4)e(6) into Eq. (3) provides the following system: where the dimensionless parameter G, which depends on frequency, is defined as: Eqs.
(3)e(6) represent the extension to frequency domain of the steady-state model of porous composite SOFC electrodes proposed by Costamagna et al. [34], who provide the solution of Eq. (7) in order to compute the total overpotential of the electrode Dh tot as follows: with: Finally, the impedance of the electrode Z ed due to the distributed charge-transfer reaction coupled with electron and ion migration is equal to: It is noteworthy that Eq. (11) was also derived, with different notation, by Paasch et al. [66] and by Lasia [61], and is present in ZView (element DX-1) [67] to represent the impedance of an electrode with finite thickness and finite resistivity of the conducting phases. A modified version of Eq. (11) was also applied by Sonn et al. [68] and Ramos et al. [60] in Ni:YSZ and Ni:ScSZ cermet anodes, respectively. When the conductivity of a phase is much larger than that of the other one, U approaches 0 and Eq. (11) equals the impedance of a finite-length Gerischer element [61,69].
The multicomponent diffusion in the stagnant layer is described according to the MaxwelleStefan approach [70]. In the following, a ternary mixture of H 2 , H 2 O and inert gas (e.g., N 2 ) is considered, where the species are labelled as 1, 2 and 3, respectively. The derivation that follows represents the extension to the frequency domain around OCV of the steadystate diffusion model proposed by Aravind et al. [57].
Under the assumption of uniform pressure P, the conservation equations for H 2 and H 2 O within the stagnant layer read as follows: where, in this case, x ¼ 0 represents the interface between electrode and stagnant layer while x ¼ L st represents the thickness of the stagnant layer (i.e., from left to right in Fig. 1a). Dy i,0 is the phasor of molar fraction, which is the a.c. perturbation generated by the dynamics of the electrochemical processes occurring within the electrode, which are assumed to be faster than the diffusion in the stagnant layer. The boundary condition at x ¼ L st sets a constant molar fraction of H 2 and H 2 O in the supply of gases outside the stagnant layer where the perturbations vanish, thus resulting in Dy i ¼ 0.
The phasor of the molar flux DN i is computed according to the MaxwelleStefan approach [70]: where D ik is the binary diffusivity between species i and k, calculated according to the Fuller et al. equation [71].
Considering that, at steady-state, the reaction stoichiometry sets N 2 ¼ eN 1 and N 3 ¼ 0, the flux expression can be recast in a Fick-type form as [57]: where the a.c. perturbation Dy i in D i has been neglected. In particular, D i represents the average diffusivity of species i within the stagnant layer in open-circuit conditions. By substituting Eq. (14) into Eq. (12), two independent second-order ordinary differential equations are obtained, whose solution is: By substituting Eq. (15) into Eq. (14), the phasor of the flux is obtained: which is related to the a.c. current density DI experienced by the electrode according to the hydrogen oxidation reaction stoichiometry: Eqs. (16) and (17) link the phasors of the molar fractions Dy i,0 to the a.c. current density DI.
Following the Taylor expansion of the Nernst equation of hydrogen oxidation reaction, the a.c. perturbation of the i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 1 ( 2 0 1 6 ) 2 2 3 8 1 e2 2 3 9 3 concentration overpotential at the interface between electrode and stagnant layer results as [72]: By substituting Eqs. (16) and (17) into Eq. (18), the impedance contribution due to the diffusion within the stagnant layer Z st results as follows: It is worth noting that Eq. (19) represents the sum of two finite-length Warburg elements [69], which is consistent with the mathematical description of H 2 and H 2 O diffusion within a finite stagnant layer.
Finally, the total impedance of the system Z is calculated by summing up all the contributions: where the factor 2 takes into account that there are two anodes in the symmetric setup considered in this study, R ey is the ohmic resistance of the YSZ electrolyte and L cables takes into account the inductance of the cables connecting the cell to the EIS analyser. The total impedance is schematically represented by the equivalent circuit reported in Fig. 2. It is worth reminding that the equivalent circuit represents the analytical solution of the physically-based model derived in this section, thus it is not a phenomenological circuit build a posteriori for the empirical interpretation of EIS data as commonly done in the literature.

Samples, tests and measured parameters
The model described in Section 2 is used to interpret the impedance spectra of symmetric anodes prepared and tested by Somalu et al. [54,73]. The samples refer to cermet anodes made of Ni and ScSZ (more specifically, 10Sc1CeSZ), sintered at 1350 C, with different solid volume fractions of Ni. Only the samples Ni30 and Ni40, corresponding to 30 and 40% vol. of Ni respectively, are considered: Ni30 is used as a benchmark for model calibration (Section 3.2) because it shows the largest impedance, thus allowing for a more accurate deconvolution, while Ni40 is used for verification purposes (Section 3.4). An additional sample, referred as 10x, is considered in Section 3.4. The sample 10x consists of a scaffold of ScSZ infiltrated 10 times with Ni(NO 3 ) 2 solution, following a fabrication procedure similar to that reported by Lomberg et al. [74]. Thus, in all the samples analysed in this study the materials are the same, only the electrode microstructure differs.
The impedance of the anodes was recorded over the frequency range 10 À2 e10 5 Hz in an in-house built apparatus [73] using a symmetric setup at open-circuit over the temperature range 600e800 C under 97% H 2 e3% H 2 O and 48.5% H 2 e48.5% N 2 e3% H 2 O, referred in the following as 100% H 2 and 50% H 2 .
The microstructural parameters required by the model are obtained from the analysis of the FIB-SEM reconstruction of the anodes after the tests (see Fig. 1b), as described by Tariq et al. [75,76]. Table 1 summarizes the microstructural parameters evaluated for Ni30 and used in Section 3.2. The effective conductivity of Ni in the samples was determined according to the van der Pauw technique with a 4-point probe [54,77]. The bulk ionic conductivity of ScSZ is taken from the literature [78] and approximated with the correlation: Calibration and deconvolution The model is calibrated by comparing model simulations with experimental EIS data of Ni30 at different temperatures and hydrogen partial pressures. The calibration allows for the fitting of the unknown material-specific parameters related to the charge-transfer reactions at the electrode/electrolyte interface, r ScSZ/YSZ and c ScSZ/YSZ , and at the distributed TPB, i 0TPB and c Ni/ScSZ . The thickness of the stagnant layer L st and the water molar fraction y H2O are regarded as open fitting parameters of the diffusion model Z st . Note that, although the water molar fraction is a known operating parameter, its value is allowed to vary in the simulations to take into account the complex three-dimensional convection and diffusion phenomena in the test rig which cannot be captured by a 1D diffusion model (Eq. (19)), as further discussed in Section 3.3. Except for L st , which is assumed constant in all the experimental conditions, the other parameters are expected to depend on temperature while being independent of hydrogen partial pressure. In addition, the material-specific parameters r ScSZ/YSZ , c ScSZ/YSZ , i 0TPB and c Ni    constant in different electrode microstructures. A complex non-linear least square fitting is used to minimize the sum of squared residuals [79].
The calibration of the model for different operating temperatures is reported in Fig. 3. In the whole range of temperatures investigated, the spectra show two features in the Nyquist plot: a low-frequency contribution, which is almost unaffected by temperature, and a depressed feature at medium/high frequency, which shrinks as temperature increases. The model accurately reproduces the shape and frequency-dependence of experimental data, showing minor inaccuracies for frequencies <0.1 Hz. Fig. 4 shows the comparison between model simulations and experimental data for different hydrogen partial pressures for the anode Ni30 at OCV and 700 C. The Figure shows that increasing the hydrogen partial pressure affects only the low-frequency contribution, whose resistance is almost halved, while the feature at medium/high frequency remains unvaried. Also in this case, the model provides a satisfactory fitting of the experimental data, especially in the medium/ high frequency range.
Besides reproducing the experimental data, the model allows for the physically-based deconvolution of the impedance spectra into three elementary contributions, as reported in Fig. 5a. The low-frequency (LF) contribution is due to the diffusion of gases in the stagnant layer, reproduced with Z st according to Eq. (19). This interpretation is supported by the weak temperature-dependence of this feature (Fig. 3), the decrease in resistance as the hydrogen partial pressure increases (Fig. 4), and a characteristic frequency in the order of 1e5 Hz (see Fig. 3b) compatible with gas transport processes taking place in the test rig [55e57,72]. The large resistance associated to the LF is due to the arrangement of the test setup, as discussed in Section 3.3, and can be reduced with a proper design of the test rig.
The depressed feature at medium/high frequency is decoupled into two contributions: a medium-frequency feature (MF) and a smaller high-frequency feature (HF). The MF feature is associated to the electrochemical processes occurring along the anode thickness, described according to Z ed . The frequency range is compatible with the distributed charge-transfer reaction as reported in the literature for similar electrodes [52,60,80]. This contribution is the one mostly affected by the microstructural properties of the electrode, since its impedance strongly depends on parameters such as L v TPB , A v Ni=ScSZ and s eff ScSZ according to Eq. (11). As anticipated in Section 2, gas diffusion across the anode thickness is not considered in Z ed . This simplifying assumption is verified in the inset of Fig. 5a: the impedance of the MF feature is not affected when porous gas diffusion is included in the model according to the approach presented by Bertei et al. [81], thus porous gas diffusion can be neglected as done in Eq. (11). This is also corroborated by the findings of Ramos et al. [60], who report a negligible porous gas diffusion contribution in electrolyte-supported Ni:ScSZ functional layers.
Finally, the HF contribution is ascribed to the impedance Z Ot of the oxygen transfer at the electrode/electrolyte interface (Eq. (2)), which is expected to be present in the high frequency range according to the literature [52,62,63,65]. The presence of three contributions in the experimental spectra is corroborated by the distribution of relaxation times (DRT), reported in Fig. 5b. The DRT method allows for the deconvolution of the spectra without any a priori assumptions, providing the resistance and characteristic frequency of each process. More details on the DRT method can be found elsewhere [82e84]. In this study the DRT is used as a semi-quantitative tool to corroborate the findings of the physically-based model. In particular, both the characteristic frequency and the resistance of each process identified with the DRT are compatible  with the independent deconvolution provided by the physically-based model in Fig. 5a. This is a good indication about the validity of the model proposed in this study, which is further discussed in the next section.

Discussion of fitted parameters
The calibration reported in Section 3.2 showed that the model satisfactorily reproduces the experimental impedance spectra in a wide range of operating conditions. However, before accepting the calibration as successful, the fitted parameters must be analysed and compared with the literature in order to assess their soundness. The parameters fitted in Section 3.2 are summarized in Table 2 and reported in Fig. 6 as a function of temperature. Fig. 6 shows that all the material-specific parameters exhibit clear linear trends in the Arrhenius plot, which allow for the calculation of their activation energies, reported in Table 2.
The activation energy of i 0TPB , equal to 115 kJ mol À1 , is similar to what reported in the literature for Ni:ScSZ anode functional layers, i.e., 120 kJ mol À1 [85] and 111e122 kJ mol À1 [60]. In addition, this value compares reasonably well with the activation energy of the hydrogen charge-transfer in a very similar system, such as Ni:YSZ. For Ni:YSZ electrodes, de Boer [86] and Bieberle et al. [87] report an activation energy of 152 and 85 kJ mol À1 , respectively: in this study, the fitted E act ¼ 115 kJ mol À1 lies between these two extreme values. At 700 C the i 0TPB fitted in this study for Ni:ScSZ is equal to 46.0$10 À6 A m À1 , while in Ni:YSZ anodes the i 0TPB is 27.5$10 À6 or 3.8$10 À6 A m À1 according to Bieberle et al. [87] or de Boer [86], respectively, for 97% H 2 e3% H 2 O. The ratio between the exchange current densities in Ni:ScSZ and Ni:YSZ lies between 1.67 (considering Bieberle et al. [87]) and 12.1 (considering de Boer [86]): this is in qualitative agreement with Shi et al. [85], who report an i 0TPB in Ni:ScSZ about 5 times larger than in Ni:YSZ.
The resistance of the oxygen transfer r ScSZ/YSZ decreases sharply as the temperature increases (see Fig. 6). On the other hand, both the capacitances c Ni/ScSZ and c ScSZ/YSZ show a weak dependence on temperature as expected [18] and lie within the typical range 0.1e30 F m À2 reported in the literature [59,80,85,88]. Note that the fitted material-specific parameters do not vary with the hydrogen partial pressure within the range investigated, which is in fair agreement with the weak p H2 -dependence of the hydrogen electrochemical oxidation kinetics reported in the literature [86e89]. The effect of water partial pressure on the hydrogen electro-oxidation kinetic parameters, which can be more significant than p H2 according to Ramos et al. [60], will be subject of future analysis.
The thickness of the stagnant layer L st remains constant with temperature and hydrogen partial pressure and is in the order of centimetres (see Table 2). This high value is attributed to the arrangement of the test rig and it is consistent with Aravind et al. [57], who used a similar experimental setup. However, the transport of gases in the test rig, ruled by a complex three-dimensional flow field, possibly influenced by pressure-driven flows [56,57], is expected to be more complex than the 1D diffusion considered in this study (see Eq. (19)). This supports the inaccuracies in the fitting of the LF feature in Section 3.2. Furthermore, this explains why the water molar fraction y H2O had to be left as an open fitting parameter during the calibration, as anticipated in Section 3.2. In particular, Table 2 shows that, in order to capture the LF resistance, y H2O must be reduced by about an order of magnitude if compared with the water molar fraction estimated from the operating conditions, equal to ca. 3%. Similar inaccuracies in the simulation of diffusion in the stagnant layer are acknowledged by Ramos et al. [60] too. Hence, the model of gas diffusion in the stagnant layer Z st cannot be considered satisfactorily validated due to the complex geometry of the test rig. Nevertheless, it can be used to fit the LF feature and decouple its contribution from the total impedance in order to focus on the medium/high frequency range. On the other hand, the models of the HF and MF processes (Z Ot and Z ed , respectively) and the corresponding parameters are robust and consistent with the literature. Since the microstructural contributions to impedance, which are the main target of this study, lie in the medium and high frequency range, the two models Z Ot and Z ed are verified in different electrode microstructures as discussed in the next section.

Model verification with different electrode microstructures
The quantification of the correlation between electrode microstructure and electrochemical performance requires the model to be tested in samples made with the same conducting materials but different microstructures. In this section the model presented in Section 2 and calibrated in Sections 3.2 and 3.3 is tested against the experimental EIS data of two different samples, namely Ni40 and 10x, referring to a cermet anode with 40% vol. of Ni and a ScSZ scaffold infiltrated 10 times with Ni, respectively, as presented in Section 3.1. As discussed in the previous sections, the contribution of the anode microstructure to impedance lies in the medium/high frequency range, thus only this region of the spectra is investigated in this section. In order to corroborate the models Z Ot and Z ed in samples Ni40 and 10x, the following approach is used: the same material-specific parameters r ScSZ/YSZ , c ScSZ/YSZ , i 0TPB and c Ni/ScSZ reported in Table 2, obtained during the calibration of Ni30 (Section 3.2) and discussed in Section 3.3, are kept. Then, the microstructural parameters A ScSZ/YSZ , L v TPB , A v Ni=ScSZ and k eff ScSZ are fitted to reproduce the experimental impedance spectra of Ni40 and 10x. If the models Z Ot and Z ed and corresponding material-specific parameters are sound, the fitted microstructural properties will compare well with FIB-SEM results and will not vary with the operating temperature. Since the validation of the diffusion model of the stagnant layer Z st is not the main concern here, the parameters L st and y H2O are treated as open fitting parameters to reproduce the LF feature and are not subject to further discussion.
The comparison between model simulations and experimental data of samples Ni40 and 10x is reported in Figs. 7 and 8, respectively. As expected, the experimental impedance spectra in different samples differ in the medium/high frequency range, while the LF feature, ascribed to gas transport processes in the test rig, remains almost unaffected, which is consistent with the literature [60]. The Figures show that the model accurately reproduces the impedance in the medium/ high frequency range for both samples Ni40 and 10x in different operating conditions. Table 3 reports the microstructural parameters used to reproduce the spectra of Ni40 and 10x, in comparison with the microstructural parameters obtained through their FIB-SEM  reconstruction. The analysis of Table 3 reveals some interesting insights. For sample Ni40, the fitted value of L v TPB remains almost constant with temperature and is essentially coincident with the TPB density evaluated by FIB-SEM tomography. Similarly, the fitted normalized conductivity factor k eff ScSZ does not vary significantly with temperature and is only slightly larger than that obtained from the tomographic reconstruction of Ni40. Both these results are positive indications regarding the validity of the model. The fitted contact area A v Ni=ScSZ shows some variability with temperature and is larger than the value of 1.74$10 6 m À1 obtained by FIB-SEM reconstruction. Since A v Ni=ScSZ is coupled to c Ni/ScSZ , which affects the characteristic frequency of charge-transfer reaction at the TPB, this variability reveals only the difficulty to reproduce the frequencydependency of the MF feature, without undermining the general validity of the Z ed model. Similarly, the interfacial area fraction A ScSZ/YSZ , which is constant and smaller than 1, is larger than expected, since it should be equal to 0.361 according to the tomographic results reported in Table 3. This inaccuracy of the model is attributed to the difficulty of decoupling the oxygen transfer process, which represents a small contribution at high frequency, from the strongly overlapping feature associated with test rig inductance, as evident from Fig. 5a. Furthermore, this may indicate that the contact between electrode and the electrolyte is not ideal, as assumed by the model. However, it is noteworthy that the fitted interfacial area fraction A ScSZ/YSZ is smaller than 1, which is the physical upper limit for the area fraction.
The analysis of the microstructural parameters obtained in anode 10x provides additional positive indications regarding the applicability of the model to electrodes with different microstructures. The interfacial area fraction A ScSZ/YSZ , equal to 0.60, matches well with the estimated porosity f of the ScSZ scaffold, which is ca. 40% [90], since ideally A ScSZ/YSZ z 1ef for a scaffold. In addition, the normalized conductivity factor of ScSZ k eff ScSZ is larger than in cermet electrodes as expected [91] and lies below the Hashin-Shtrikman upper bound of conductivity [92], confirming that it is realistic. The contact area A v Ni=ScSZ , equal to 0.9$10 6 m À1 , is similar to that measured in cermet electrodes Ni30 and Ni40. This result is credible because A v Ni=ScSZ is proportional to the surface area per unit volume of the ScSZ scaffold, which is similar to that of ScSZ in cermet electrodes with the same volume fraction of scandiastabilized zirconia as reported by Kishimoto et al. [90]. Finally, the TPB density L v TPB estimated in the infiltrated anode 10x is roughly one order of magnitude larger than the TPB length in cermet anodes, which is in good agreement with both experimental observations with FIB-SEM [90] and theoretical considerations [91,93].
Based on the good matching of EIS data in Figs. 7 and 8 as well as the considerations regarding the microstructural parameters, the validation of the model in samples Ni40 and 10x, prepared with different fabrication techniques and showing remarkably different microstructures, can be considered satisfactory.
As anticipated in Section 2, the model is not limited to Ni:ScSZ electrodes but can be used, in principle, for any anode made of a mixture of pure ion-conducting and pure electronconducting materials. In the following, the model is applied to the cermet Ni:YSZ anode sintered at 1400 C reported by Miyawaki et al. [94]. According to their experimental setup, a negligible oxygen transfer impedance Z Ot is expected because they used the same ion-conducting material as electrolyte and within the electrode. In addition, the LF contribution, associated to the gas transport in the test rig, is minimal in their EIS spectra. Therefore, the Miyawaki et al. [94] data represent a useful benchmark to verify the electrode model Z ed , which is the contribution mostly affected by microstructural properties as mentioned before.
According to the FIB-SEM reconstruction reported by Miyawaki et al. [94], the microstructural parameters of the cermet Ni:YSZ electrode are L v TPB ¼ 1.5$10 12 m À2 , k eff YSZ ¼ 0.093 and L an ¼ 8 mm. In the operating conditions reported by the authors, i.e., 1000 C and 97% H 2 e3% H 2 O, the bulk ionic conductivity of YSZ is 10.23 S m À1 [20,95] and the exchange current density i 0TPB is equal to 3.2$10 À4 A m À1 [86,87]. By applying Eq. (11) for u ¼ 0 Hz, the calculated polarization resistance R ed ¼ Z ed (u ¼ 0) is equal to 0.32 U cm 2 . According to Miyawaki et al. [94], the experimental polarization resistance of the anode at 50 mA cm À2 (i.e., close to OCV) is about 0.34 U cm 2 . Therefore, the model Z ed presented in this paper provides a good prediction of the electrode polarization resistance also in Ni:YSZ anodes without adjusting or fitting any parameters.
Concluding, besides the reasonable trends of fitted parameters discussed in Section 3.3, the satisfactory verification with different microstructures and the positive application to Ni:YSZ  i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 1 ( 2 0 1 6 ) 2 2 3 8 1 e2 2 3 9 3 anodes without any adjusted parameters confirm the reliability and good predictability of the model proposed in this study.

Conclusions
In this paper a physically-based model for the simulation of impedance spectra in SOFC anodes was presented, taking into account the main electrochemical and transport phenomena and their coupling with the electrode microstructure, reconstructed with FIB-SEM tomography. The model was calibrated to reproduce the EIS data of cermet Ni:ScSZ anodes in different operating conditions. The material-specific parameters obtained in the fitting, comprising the capacitance, the exchange current density and the activation energy of the hydrogen electrochemical oxidation, were in good agreement with values reported in the literature. The model was then validated in anodes with different microstructure, including an infiltrated electrode, and the fair prediction of the microstructural properties was confirmed by FIB-SEM tomography.
The study shows that the correlation between electrode microstructure and electrochemical performance can be addressed by using physically-based models, assisted by 3D tomography to provide the meaningful microstructural properties, and verified on impedance spectra for different microstructures. This approach turns out to be the most suitable to develop and validate interpretative and predictive modelling tools, capable to decouple the different contributions that build up the electrode resistance as well as to accurately infer kinetic and microstructural properties from impedance data, going beyond the capabilities of phenomenological equivalent circuits.
The model presented in this study and the corresponding material-specific parameters can be used to provide design indications to optimise the electrode microstructure or as a diagnostic tool to quantify the microstructural evolution upon application on EIS data at different stages of degradation. Further research will be dedicated to the extension of the model to simulate anode impedance under d.c. current and the corresponding effect of water partial pressure.

Nomenclature
A v Ni=ScSZ contact area of Ni/ScSZ interface per unit volume (m À1 ) A ScSZ/YSZ area of ScSZ/YSZ interface per unit of electrode area (À) c Ni/ScSZ specific capacitance of the double-layer at Ni/ScSZ interface (F m À2 ) c ScSZ/YSZ specific capacitance associated the oxygen ion transfer at the ScSZ/YSZ interface (F m À2 ) D i average diffusivity of species i within the stagnant layer, with i ¼ 1, 2 ¼ H 2 , H 2 O (m 2 s À1 ) D ik binary diffusivity between gas species i and k (m 2 s À1 ) E act activation energy (J mol À1 ) f frequency (Hz)