Determination of the effective conductivities of solid oxide fuel cell electrodes using the first‐order homogenization method

Approximately 80% of the world's primary energy supply consists of fossil fuels. In order to reduce the CO2 consumption in the world, fuel cells will be indispensable for the environmentally friendly generation of electrical energy. Many experimental and numerical studies show that the composition and the microstructure morphology of porous electrodes have a great influence on the durability and conversion efficiency of fuel cells. In order to establish a relationship between the specific microstructure of the electrodes and the performance of the overall fuel cell, the macroscopic physical quantities need to be determined. In this work, the first‐order computational homogenization method capable of capturing anisotropic effects is applied in order to estimate different types of homogenized conductivities. The effective conductivities can be used to conduct numerical simulations on the fuel cell level in order to obtain correlations between the microstructure characteristics (e.g., volume fraction of pores, morphology) and the performance.


INTRODUCTION
In a world where climate change has become one of the biggest challenges, we need more energy-efficient and sustainable power generation.The emissions of greenhouse gases are much lower for solid oxide fuel cells (SOFCs) compared to other conventional power systems [1].In comparison to other types of fuel cells, SOFCs operate at high temperatures and thus do not need expensive catalyst materials for the electrochemical reactions.Furthermore, they have high efficiency compared to other types of fuel cells.They also have a higher tolerance with respect to impurities and are therefore suitable for different fuels [2,3].Due to their advantages and their promising fuel cell technology, optimizing the microstructure of the porous electrodes (see Figure 1) is a great opportunity that could lead to a higher electric efficiency and a longer durability.For instance, in order to enable the fuel and the air to flow to the cell active sites in which the electrochemical reactions take place, the microstructure must exhibit both sufficient porosity and high pore connectivity.Due to time and cost constraints, both a large amount of (i) experiments with various physical or geometrical properties and of (ii) numerical simulations with a fully resolved microstructure are nearly impossible.Therefore, the first-order homogenization method [4] is applied in the present paper to determine the effective thermal, electric and ionic conductivities for lanthanum strontium cobalt ferrite (LSCF) often used as the cathode material.In Section 2, the equations that need to be solved and the principles of the homogenization method are presented.In Section 3, the generation of the representative volume element () is described that is used as an example for investigating the material symmetry and for the effective conductivities in Section 4. A brief conclusion and outlook follow in Section 5.

GOVERNING EQUATIONS
The equations required to determine the effective conductivities are described in Sections 2.1 and 2.2.In addition, the principles of the first-order homogenization are briefly explained in Section 2.3.

Thermal field problem
For the estimation of the effective thermal conductivity, the thermal field problem expressed by the energy balance needs to be solved: where  indicates the heat capacity,  the temperature,  the heat flux and  th thermal sources.Furthermore,  is the nabla operator and ( ) the partial time derivative.As the constitutive law, the well-known Fourier law is used with the temperature-dependent second-order thermal conductivity tensor .

Electrical field problem
The continuity equation describes the charge conservation and has to be used to calculate the effective electric/ionic conductivity.Here,  is the charge density,  the current density,  el the source of electrons or ions and  = {ions, electrons} the species.A linear relationship between the electric field  and the flux of ions or electrons is assumed.This is expressed by Ohm's law: The electric and ionic conductivities   depend on the temperature, resulting in a one-way coupling between the thermal field problem and the electric field problem.The electric field is defined as  = − with  being the scalar electric potential.

First-order homogenization
The main assumption of the first-order homogenization is the separation of scales [4,5], that is, must be satisfied.Therefore, the field equations can be used in their steady-state form at the microscale [6].In addition, the volume density terms must be set to zero to satisfy the energy equivalence between the microscale and the macroscale.The primary variable has to be split into a macroscopic part, a prescribed constant gradient and the fluctuations (⋅) at the microscale [7].Thus, the split of the primary variables (i.e., the temperature  and the electric potential ) reads The coordinates are represented by .Using all these facts and inserting the Fourier law (2) into Equation ( 1) yields to the partial differential equation at the microscale: The Hill-Mandel lemma states the equivalence of the dissipation power density between the separated scales, that is, the thermal dissipation power density in Equation ( 8) 1 [8,9] and the ponderomotive power density in Equation ( 8) 2 [10].The averaging operator over the volume  of the domain  can be introduced as Therefore, three different types of boundary conditions can be used to satisfy condition (8) 1 [11]: (i) the uniform temperature gradient as Dirichlet boundary conditions, (ii) periodic boundary conditions with periodic temperature fluctuations and anti-periodic heat fluxes and (iii) the uniform heat flux as Neumann boundary conditions.These types of boundary conditions can also be transferred to the electrical field problem to fulfill condition (8) 2 .However, in this work, only boundary conditions of type (i) with are used, respectively, as the domain of the  is not periodic, see Section 3.

F I G U R E 2
Microstructure generated by the material twin concept.

REPRESENTATIVE VOLUME ELEMENT
The algorithm for generating the used  is explained in detail in Shao et al. [12].A material twin concept that relies on probability functions and the statistical continuum theory is developed to reconstruct 3D porous microstructures with desired physical and anisotropic properties.
The microstructure is characterized by microstructural descriptors, the -point correlation functions.Here, points or vectors are randomly placed into the microstructure and the probability of finding  points within the same corresponding phase is calculated.For instance, the one-point correlation function describes the volume fractions of the material and the pore and the two-point correlation function expresses the morphology of the microstructure.Higher order correlation functions can additionally predict the shapes of each phase and the spatial distribution [10,13].
The effective thermal conductivity of isotropic two-phase materials can be determined analytically by the statistical strong-contrast formulation that uses the previously defined -point correlation functions [13,14].For a fast prediction of  ef f , only the one-point and two-point correlation functions are used [12].
Within the algorithm of Shao et al., first an isotropic microstructure with corresponding isotropic effective properties is generated.Based on the calculated effective property, the voxels can be rearranged to achieve a desired (anisotropic) effective behavior.The rearrangement and the generation of the microstructure is done, if the convergence criterium for the effective property is satisfied.This procedure leads to a microstructure as depicted in Figure 2.
In order to determine various effective properties and not be limited by the statistical strong-contrast formulation, the microstructure needs to be meshed for the finite element method.However, with 150 voxels in each direction, the entire microstructure is too large to be meshed with finite elements.Due to this reason, 216 sections with 25 3 voxels are cut from the whole microstructure and meshed with CGAL [15], see Figure 1 (left).Thus, the simulations with different loads (e.g., macroscopic temperature gradient) are executed for this number of these sections and the microstructural response is averaged with Please note that there may be an influence of the section size on the effective property which has to be investigated in future works.

RESULTS
In the following, the results of the material symmetry in Section 4.1 and of the effective conductivities in Sections 4.2-4.4 are shown for the respective microstructure.For the constitutive behavior of the perovskite structure LSCF and the material parameters for the thermal, electric and ionic conductivities used in this study, please refer to refs.[16,17] and the references therein.
F I G U R E 3 Representation of the slightly transversely isotropic behavior of the microstructure.The isotropic plane is represented by the -plane (left), the -direction can be considered as the preferred direction (middle) and the transverse isotropy is indicated by the ellipsoid (right).

Class of anisotropy
The requirement of material symmetry is that the energetic state does not change, if the material element is rotated by a rotational tensor  that belongs to the symmetry group  ⊆ (3) [18]: The orthogonal group ( 3) is defined as For  ∈ (3) the material is isotropic.Otherwise it is anisotropic [19].As it can be seen in Figure 3, the material cannot be considered isotropic, because although the -plane (Figure 3 (left)) is isotropic,  can be identified as the preferred direction (Figure 3 (middle)).However, having a closer look at the ellipsoid (Figure 3 (right)) and the values of the norm of ⟨⟩, it can be seen that the transverse isotropy is only slightly expressed.

Thermal conductivity
For the determination of the effective thermal conductivity, the averaged quantities ⟨⟩ and ⟨⟩ need to be used.As the relationship between the heat flux and the temperature gradient is linear (see Equation ( 2)), we can simply calculate  ef f for different initial or operating temperatures.Here,  ef f for 873 K and 1273 K are shown: The isotropic -plane can be recognized by comparing the values   and   .The slightly expressed transverse isotropy with  as the preferred direction can be seen through   .However, the almost independence of the effective thermal conductivity from temperature can be realized comparing  ef f , and the graphs in Figure 4(a).

Electric conductivity
In order to get the effective electric conductivity, the averaged electric field ⟨⟩ and the flow of electrons ⟨ el ⟩ must be obtained.Due to the linear relationship between ⟨ el ⟩ and ⟨⟩ at both scales, we can use Ohm's law (4) to establish  el,ef f F I G U R 4 Courses of the averaging flows versus the averaged loads.Here, ℎ and  are scalars for the temperature gradient or the electric fields applied as a load in a particular direction ( 1 ,  2 ,  3 ), respectively.
for certain temperatures: One can see a slightly decrease of the effective electric conductivity for increasing initial temperatures in Figure 4(b).Furthermore, the transversely isotropic behavior can also be detected.

Ionic conductivity
The determination of the effective ionic conductivity is similar to the previous section, as the averaged electric field ⟨⟩ and flow of ions ⟨ ion ⟩ can be inserted into Ohm's law (4) to calculate  ion,ef f for various temperatures: The slight transverse isotropy can be observed in Figure 4(c) and on the basis of the values for two different temperatures.In addition, a large dependence of the temperature on the effective ionic conductivity can be noticed, as  ion strongly increases with increasing initial temperature  ini .Although the ionic conductivity is for large temperatures compared to low temperatures,  ion still remains the limiting factor for the electrochemical reactions that take place within the electrodes, since the ionic conductivity is much lower than the electric conductivity.

CONCLUSION
In this work, the effective conductivities of solid oxide fuel cell cathodes made of lanthanum strontium cobalt ferrite are determined by the first-order computational homogenization method.This procedure enables the incorporation of effective conductivities at the macroscale to find correlations between microstructure and electrical performance.It can be analogously used for other material compositions of the electrodes, even for systems with two different materials that conduct only ions or electrons, respectively.However, a large number of sections of the whole microstructure are required to obtain reasonable results for the effective properties.In particular, the generation of finite element meshes is timeconsuming.In addition, the sections must be not too small, otherwise a dependence on the section size occurs.For these reasons, the use of Fast Fourier Transform (FFT) based methods is planned.Furthermore, other important quantities at the fuel cell level should be estimated, such as the effective permeability or the effective Young's modulus.Moreover, the coupling of different fields will be expressed in terms of the generalized standard materials in future works.

A C K N O W L E D G M E N T S
This work is conducted within the M-ERA.Net project MEDIATE.MEDIATE project is co-financed with tax funds on the basis of the budget passed by the Saxon State Parliament, Germany.The authors would like to acknowledge the support of the Fond National de la Recherche (FNR), Luxembourg.
Open access funding enabled and organized by Projekt DEAL.

F I G U R E 1
Meshed section of the microstructure of the porous fuel cell cathode (left) and structure of a solid oxide fuel cell (right).