# The design optimization of nanostructured hierarchical electrodes for solid oxide cells by artificial impregnation

Davide Cademartori, Maxime Hubert, Peter Cloetens, M. Paola Carpanese,

Jérôme Laurencin

► To cite this version:

Davide Cademartori, Maxime Hubert, Peter Cloetens, M. Paola Carpanese, Jérôme Laurencin. The design optimization of nanostructured hierarchical electrodes for solid oxide cells by artificial impregnation. Materials & Design, 2024, 238, pp.112663. 10.1016/j.matdes.2024.112663. hal-05095824

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Submitted on 3 Jun 2025

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<figure><img src="image_1.png" /></figure>
# The design optimization of nanostructured hierarchical electrodes for solid oxide cells by artificial impregnation

Davide Cademartori<sup>a, *</sup>, Maxime Hubert<sup>b</sup>, Peter Cloetens<sup>c</sup>, M. Paola Carpanese<sup>a</sup>, Jérôme Laurencin<sup>b</sup>

<figure><img src="image_2.png" /></figure>

## ABSTRACT

Microstructural correlations of impregnated freeze tape cast scaffolds for solid oxide cells are studied by coupling experimental and modelling approaches. The functional and supporting layers of the hierarchical porous backbones are initially reconstructed by synchrotron X-ray micro- and nano-holotomography, respectively. A particle-based model was then built to numerically infiltrate the scaffold walls with hemispherical nanoparticles. The electrode microstructural properties are evaluated on the artificially impregnated electrodes as function of the catalyst loading. A parametric investigation on the effect of the nanoparticle size is performed to analyse the evolution of the electrode characteristics. It has been shown that the volume fraction of the infiltrated phase necessary to reach the percolating threshold is increased while increasing the nanoparticle size. The density of active sites presents a maximum as function of the catalyst loading that depends on the particle size. The volume fraction of infiltrated phase required to reach the percolating threshold in the diffusion layer is one order of magnitude lower than in the functional layer (~1 vol% compared to 3–8 vol%, respectively). The analyses contained in this paper aim at guiding the manufacturing process to the shaping of innovative electrodes microstructures combining both high activation and mass transfer properties.

## 1. Introduction

The development of renewable energy sources is nowadays becoming a priority in response to climate change. However, because of the intermittent nature of these natural power sources, new technologies able to balance the energy fluctuations between production and demand are required [1–3]. In this context, Solid Oxide Cells (SOCs) are gaining interest due to their high operating efficiency, fuel flexibility and operational reversibility. In particular, the latter aspect offers the opportunity of combining a fuel cell (SOFC) and an electrolyser (SOEC) in a single energy system being able to work alternatively between the two operating modes [4,5]. Therefore, SOCs are expected to play a key role in an energy scenario where the surplus of renewable electricity is firstly converted and stored as carbon-free fuels (P2X-SOEC) to be turned back into electrical power (X2P-SOFC) when demand exceeds production [6,7].

The architecture of the state-of-the-art SOCs includes a thin dense electrolyte sandwiched between a thick supporting fuel electrode and a thin air electrode. The conventional electrolyte is made of Yttria Stabilized Zirconia (YSZ) [8] while the fuel electrode is usually a cermet of Ni and YSZ [9,10]. A Mixed Ionic and Electronic Conductor (MIEC), such as

Lanthanum Strontium Manganite (LSM) [11,12] or Lanthanum Strontium Cobalt Ferrite (LSCF), mixed with an ionic conductor like Gadolinium-Doped Ceria (GDC) [13,14], is classically used for the air electrode.

SOCs operate at high temperatures (700–800 °C) to enable the ionic conduction in the electrolyte and the electro-catalytic activity within the electrodes. The charge transfer reactions take place at the electrochemical active sites of the porous electrodes where gas, electron-conducting and ionic-conducting phases meet. Therefore, the diffusion of reacting species as well as the electrochemical reactions are strongly affected by the morphological characteristics of the electrode microstructure [15–21]. Moreover, its structural stability over time is also dependent on the microstructural properties [22,23]. For instance, the amount and distribution of pores in the thick fuel electrode play a key role in effective gas diffusion, affecting the mass transfer and hence reactions rate in the functional layer [24–27]. In traditional SOCs electrodes manufacturing techniques (e.g. tape casting, screen printing, extrusion etc. [28–30]), organic pore-forming agents, which burn out during the high temperature sintering step, are usually added into the initial ceramic slurries to generate a network of pores [28,31,32]. Even though the pores are randomly distributed throughout the electrodes,

<footnote><sup>a</sup> Department of Civil, Chemical and Environmental Engineering, University of Genoa (UNIGE-DICCA), Via all'Opera Pia 15, 16145 Genoa, Italy</footnote>

<footnote><sup>b</sup> Univ. Grenoble Alpes, CEA, LITEN, DTCH 17 rue des Martyrs, 38054 Grenoble, France</footnote>

<footnote><sup>c</sup> European Synchrotron Radiation Facility (ESRF), 38000 Grenoble, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: davide.cademartori@edu.unige.it (D. Cademartori).</footnote>
the microstructure can include unconnected pores which do not promote the gas diffusion [15,33]. Performance limitations addressed to an insufficient amount of porosity, which hinders gas diffusion and contributes to the increase of the degradation rate during operation, are well documented in literature [12,34,35]. In particular, the sponge-like standard electrode architecture was found to facilitate the gas pressure build-up at the electrode-electrolyte interface, especially in SOEC operating mode [34,35]. In the air electrode, an insufficient gas diffusion rate can even lead to the delamination of the electrode from the electrolyte. At the fuel electrode the high steam partial pressure can cause a local re-oxidation of Ni into NiO, that can induce mechanical damage [35-37]. The enhancement of gas diffusion within the electrodes is thus essential to improve the SOCs performance. To this end, alternative manufacturing routes such as freeze tape casting (i.e. a materials processing technique finding applications in different research fields [38-40]) have recently been explored to produce graded electrodes with a preferred orientation of porosity [25,41,42]. Microstructural and electrochemical characterizations carried out on graded porous SOCs electrodes fabricated by freeze tape casting highlighted the lower tortuosity of their gas diffusion channels and enhanced mass transfer properties [25,43,44]. Wu et al. manufactured a graded porous oxygen electrode combining a low tortuosity factor (≈1) and a high volume fraction (≈75 %) for the pore phase with a high mechanical strength [45]. The electrode consisted of a hierarchical porous yttria-stabilized zirconia (YSZ) scaffold and a nanoscale $La_{0.6}Sr_{0.4}CoO_{3-δ}$ (LSC) electrocatalytic coating. The proposed architecture showed a very low polarization resistance of 0.0094 Ω·cm² at 800 °C compared to the traditional composite electrode. In recent works Cademartori et al. highlighted also the operational advantages of the freeze tape casting technique (i.e. use of water as solvent, manufacturing of the functional and support layer in a single tape, faster manufacturing cycles) and the very low gas diffusion resistance measured on the manufactured graded porous architectures [41,46]. The high electrochemical performance of freeze tape cast electrodes, in terms of low polarization resistance and high-power output, was also reported in previous studies [25,47-49] which outlined the operational benefits provided by hierarchical porosity.

In addition to gas diffusion, the SOCs performance is also critically dependent on the electrocatalytic activity of the electrodes. The efficiency of electrodes is controlled both by the material intrinsic properties and microstructural parameters such as the density of Triple Phase Boundary (TPB) length ($λ_{TPB}$) [16,50]. With regard to the latter aspect, it has been shown in literature that the density of electrochemically active sites is directly related to the particle size [51-53]. It is also worth mentioning that the increase of the specific surface area, which also promotes the electrocatalytic activity, is obtained when reducing the particles size of the powders [50,54,55]. The classical electrodes are commonly manufactured starting from ceramic suspensions where the raw powders hold a particle size distribution d50 which lies in the range of 0.5-10 µm [56-58]. The subsequent sintering step, which is carried out at high temperature to obtain densified ceramic structures, enlarges the grain size thus reducing the final $λ_{TPB}$. To reduce the particle size, different fabrication methods have been developed to design nano-structured electrodes aiming at boosting the effective number of active reaction sites and specific surface area [59-61]. The infiltration method is a well-known technique, which enables the addition of nanoparticles into a single or bi-component scaffold by its impregnation throughout a precursor solution, metal salt precipitation, molten salts, etc. [62-65]. Electrodes infiltration is usually repeated several times to achieve a sufficient loading of the electrocatalyst and impart the desired properties to the backbone. Another shaping route to enhance the electrochemical performance consists in the exsolution of nanoparticles which produces a uniformly dispersed electrocatalyst phase onto the scaffold walls [59,66,67]. The infiltration technique has been extensively applied using classical isotropic backbones produced, for example, by tape casting or screen-printing [41,60,68-70]. Recently, graded porous

backbone manufactured by freeze tape casting have also been infiltrated to obtain promising nano-structured electrodes holding tailored microstructures [45,47,48].

To date, only few models have been developed to investigate the microstructural properties of infiltrated electrodes. Most of them have been focused on the prediction of the electrode effective characteristics making use of percolation theory concepts and numerical methods [71-74]. All the investigations have been carried out on single or bi-component synthetically generated backbones holding the traditional sponge-like porosity. However, to the best of the authors knowledge, the microstructural correlations for infiltrated electrodes using graded backbones have not been studied yet.

This work investigates the morphological properties of SOCs fuel electrodes composed by a graded porous YSZ backbone, experimentally manufactured by freeze tape casting. The hierarchical porous backbone was reconstructed by X-ray nano-holotomography and X-ray microscopy to characterize the 3D morphological properties of the functional and supporting layer, respectively. Subsequently, the segmented 3D images of the ionic conducting scaffold were modified using an in-house infiltration algorithm for the generation of synthetic nanoparticles. The impact of the catalyst loading ($V_f$) on the tortuosity factor ($τ$), percolating fraction ($φ$) and $λ_{TPB}$, was investigated by progressively increasing the surface coverage fraction of catalyst ($f$) on the freeze tape cast backbone. The study was conducted considering different nanoparticles size distributions. The percolating threshold of the impregnated catalyst was also analysed in the supporting layer, with the purpose of identifying the required $V_f$ and $f$ to provide sufficient electronic conductivity to the overall YSZ scaffold. It must be pointed out that all the performed simulations were intended to provide general microstructural correlations, which do not depend on the specific nanoparticles materials. However, in this study the synthetic nanoparticles will be assigned to the Ni phase in order to facilitate the understanding of the performed analyses by referring to a potential real scenario (i.e. a Ni-infiltrated YSZ fuel electrode).

## 2. Materials and methods

### 2.1. Manufacturing of freeze tape cast YSZ scaffolds

The fabrication of freeze tape cast backbones has followed the same manufacturing procedure reported in [41] for the production of hierarchical supports of GDC and YSZ. A water based-suspension was initially prepared using as received 8YSZ (TZ-8Y, Tosoh Corporation) and adding an acrylic latex emulsion as water insoluble binder to prevent any interference between the additives and the water crystallization process. The volumetric composition of the 8YSZ-based starting suspension is reported in Table 1.

The green freeze tape cast scaffolds were subsequently employed to manufacture three-layered porous-dense-porous YSZ architectures by the addition of a thin dense electrolyte sandwiched between two hierarchical backbones. The YSZ electrolyte is obtained by screen printing an organic based ink, featuring terpineol (Sigma Aldrich) as solvent and polyvinyl butyral (PVB, Sigma Aldrich) as binder, on the surface of the tape which was in direct contact with the cold source. A second YSZ freeze cast backbone is then added by die pressing over the electrolyte to

<caption>**Table 1**</caption>

Composition of the YSZ-based starting suspension for the manufacturing of freeze tape cast scaffolds.

<table><thead><tr><td>Components</td><td>Volumetric percentage %</td></tr></thead><tbody><tr><td>Water</td><td>55 %</td></tr><tr><td>Dispersant</td><td>2.8 %</td></tr><tr><td>8YSZ</td><td>16 %</td></tr><tr><td>Surfynol SE-F</td><td>0.15 %</td></tr><tr><td>Acrylic Latex Emulsion</td><td>25.75 %</td></tr><tr><td>Arabic Gum</td><td>0.3 %</td></tr></tbody></table>
form a porous-dense-porous architecture. Finally, after a heating ramp of 1 °C/min with two dwelling steps of 1 h at 550 °C and 1000 °C, the three-layered structure is co-sintered at 1450 °C under air for 5 h.

## 2.2. Morphological Characterization: SEM analysis and 3D reconstruction

Fragments of the produced samples were embedded under vacuum in epoxy resin and then polished using diamond paste down to 0.25 µm. The polished cross sections were then observed with a Scanning Electron Microscope (SEM, Phenom ProX, Thermo Fisher Scientific). The YSZ porous scaffold was reconstructed by X-ray micro-tomography using a Zeiss Xradia 510 Versa Microscope. The experiment was performed on a 2 mm diameter pillar extracted from the three-layered structure. The data were acquired following the procedure previously detailed in [41] with a voxel size of 1.9 × 1.9 × 1.9 µm³. To analyse the fine micro-structure of the functional layer, a complementary experiment was performed at the Nano Imaging beamline ID16A of the European - Synchrotron Radiation Facility (ESRF) [75]. For this purpose, a micro-pillar with a diameter of ≈ 40 µm containing the functional layer of the YSZ freeze tape cast scaffold and a part of the dense electrolyte was prepared using a plasma-focused ion beam (pFIB) Vion (FEI™). The details about the extraction procedure has been reported in a previous article [76]. For the holotomographic experiment, the data acquisition procedure and the algorithm for the reconstruction have been previously adapted to the SOCs materials [77]. With this technique, a representative volume of ≈ 50 × 30² × π µm³ is typically obtained with a voxel size of 25 nm and a spatial resolution of ≈ 50 nm [76].

## 2.3. Model description

A particle-based model was developed and implemented on the Matlab® software to emulate the infiltration of nanoparticles on the freeze tape cast scaffold. The model was built with specific constraints to control the morphology and distribution of the infiltrated phase.

### 2.3.1. Generation of Ni nanoparticles

The synthetic generation of Ni nanoparticles is based on a multi-steps algorithm schematically illustrated in Fig. 1. First, the segmented volume to be artificially impregnated is uploaded and the original voxel size of the 3D reconstruction can be refined to accommodate the addition of smaller nanoparticles. The voxel size is decreased using the Matlab® function *interpn* which returns the interpolated values of the original grid on a refined one formed by halving *n* times the existing intervals of the starting 3D matrix in each dimension. The returned grid eventually contains $2^{n-1}$ interpolated points between the original sample values. The reduction in voxel size is thus achieved by adjusting the value of *n* to obtain the desired refinement of the original reconstructions. For the functional layer, the pre-processing of the 3D scaffolds was carried out setting *n* = 1 to allow the generation of small nanoparticles with mean radius ranging from 25 nm to 120 nm. On the other hand, for the supporting layer, the voxel size was decreased to a minimum value of 60 nm (corresponding to *n* = 5). Indeed, both the large RVE of the supporting layer (i.e. 100 × 100 × 100 µm³, Section 3.2.1.4.) combined with the initial voxel size (1.9³ µm³) prevent a finer discretization of the computed domain that would result in too high computational effort for the simulations. In other words, a compromise between size and resolution is necessary to avoid excessively long simulating times.

The second step performed by the algorithm is the identification of the voxels at the interface between the solid and the gas phases (Fig. 1b). The edge labelling is obtained using a function which searches the local gradient maxima in the binary volume. A fraction $\chi$ of the voxels identified at the interface is then selected by the infiltration algorithm to host the nanoparticles. The selected voxels are randomly distributed on the scaffold surface and used as seeds where the artificial nanoparticles are positioned. In other words, the algorithm reassigns a desired fraction $\chi$ of pore voxels at the solid-pore interface to the seeds for the subsequent nanoparticle growth. It must be pointed out that no seed overlap is allowed onto the ionic conducting phase scaffold, thus avoiding the multiple selection of the same seed position for the nanoparticles

<figure><img src="image_3.png" /><figcaption>Fig. 1. a) segmentation of the ysz backbone (grey: backbone, black: porosity); b) identification of the interface scaffold-porosity; c) nanoparticles deposition over the backbone surface; d) restoration of the scaffold which shape the nanoparticles as hemispheres; e) simulation of the sintering effect using morphological operations.</figcaption></figure>
growth. Therefore, if a nanoparticle is rejected, the same nanoparticle is tested again by randomly selecting a new voxel belonging to the backbone surface. When a position is accepted, the whole procedure is repeated to identify a new seed until the targeted fraction $\chi$ is reached.

The seed position is then converted into an electrocatalyst nanoparticle by the algorithm (Fig. 1c). For this purpose, a sphere centred on each seed position is added in the microstructure. Its dimension follows a normal probability distribution for which the mean ($\mu$) and standard deviation ($\sigma$) of the population are input model parameters (Equation (1)):

$$f(r) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{r-\sigma}{\sigma}\right)^2} \quad (1)$$

where $r$ is the sphere radius. The normal probability distribution is well representative of the different size of the impregnated nanoparticles which can be obtained experimentally. It is worth noting that the original scaffold is restored after the nanoparticle growth (Fig. 1d). As a result, the microstructure is impregnated with hemispheres of radius $r$ on the backbone surface. Only a fraction of the original porosity is thus converted into electrocatalyst without any modification to the initial backbone. The iterative procedure is stopped when all the seeds have been expanded allowing a single layer of nanoparticles to be deposited over the scaffold surface, thus avoiding a multi-layered impregnation. It must be pointed out that clusters of nanoparticles can be experimentally obtained when infiltrating SOCs electrodes, especially at high catalyst loadings. However, the formation of these clusters is closely linked to the characteristics of each backbone (shape and constrictivity of pores) and the operating parameters of each impregnation process, such as the concentration of the starting solution or the number of infiltration repetitions. Therefore, the artificial simulation of multiple nanoparticle layers should be adapted to any specific experimental case, thus being out of the scope of this study. Furthermore, the manufacturing processes which shape nanostructured electrodes are usually adjusted to avoid the formation of these clusters because they decrease the electrode performance (e.g. due to their lower specific surface area or by blocking gas diffusion). For these reasons, with the purpose of using the particle-based model to shape reference synthetic architectures holding optimal microstructural properties, only monolayers of nanoparticles were considered as also previously reported in [73].

In the model, the sintering between the nanoparticles, which leads to the creation of a neck curvature between the infiltrated phase, is also taken into account. A morphological operation is applied to the electrocatalyst phase. In particular, the nanoparticles are first diluted and then eroded using a spherical structuring element (Fig. 1e). After this step, the electrocatalyst volume fraction is slightly increased.

### 2.3.2. Input and output parameters for simulations

The proposed modelling framework has been conceived to be flexible, easily adapted to different scaffold microstructures and able to simulate the electrodes morphologies obtained starting from different impregnation procedures. The algorithm for the generation of the impregnated phase requires as input parameters (i) the 3D reconstruction of the backbone to be decorated, (ii) the desired coverage fraction $\chi$ of the scaffold surface, (iii) the values of $\mu$ and $\sigma$ of the normal distribution for the nanoparticle size and (iv) the size of the spherical structuring element for the morphological operation.

In this study, real reconstructions of the backbone were used as starting point for the infiltration. The effect of the electrocatalyst loading has been studied by a sensitivity analysis, setting different $\chi$ values, to investigate the evolution of the microstructural parameters (such as $\lambda_{TPB}$) around the percolating threshold. Since different particle size of electrocatalyst can be obtained depending on the impregnation method (i.e. adoption of a precursor solution for the in-situ generation of the nanoparticles or direct impregnation of raw powders dispersed in a suspension), its impact on the microstructural properties has been also

studied varying the values of $\mu$ and $\sigma$. Therefore, the mean particle radius was changed between 25 nm and 120 nm to sweep a large range of manufacturing conditions. Finally, the size of the structural element for the morphological opening was chosen between one-sixth to one-fifth of the mean particle diameter to reproduce the typical necking arising during the sintering of the electrocatalyst particles [78]. It should also be pointed out that the sensitivity analysis on the nanoparticles size can reproduce the agglomeration of the infiltrated catalyst at a fixed volume fraction (e.g. Ni particles coarsening during operation).

After the numerical infiltration, several properties are computed on the synthetic microstructure. In particular, the volume fraction of each phase is extracted from the 3D reconstruction as well as the specific surface area, the tortuosity factor and the density of triple phase boundary length ($\lambda_{TPB}$). The procedures to compute all these microstructural parameters have been already described in detail in previous works [16,76,79,80]. The particle size distribution and mean diameter of the nanoparticles are also computed since the final geometry of the electrocatalyst is not anymore spherical.

## 3. Results and discussion

### 3.1. Microstructure of the freeze tape cast YSZ scaffold

#### 3.1.1. SEM micrographs

Fig. 2 shows the cross sections of the manufactured YSZ tape observed after sintering and the pillar extracted from the functional layer of the three-layered porous-dense-porous YSZ architecture. The global view of the scaffold reported in Fig. 2a highlights the presence of anisotropic gas diffusion channels, holding hierarchical well-ordered porosity growing directly from the surface exposed to the cold source up (bottom part of the image) to the top of the tape (top part of the image). On the other hand, Fig. 2b and c reveal that the microstructure of the functional layer recalls a highly porous sponge-like electrode. Indeed, during the lyophilization step of the manufacturing process, the tape bottom surface, which is in direct contact with the freezing plate, is subjected to the coldest temperature while the top surface is exposed to air. The high porosity observed in the active zone stems from the higher rate of ice nucleation and the low temperature of the freezing plate while the pore size is defined by the ice growth rate [81,82]. A detailed investigation on the effects of the freeze tape casting manufacturing parameters on the resulting morphology of the YSZ scaffolds has been already detailed in [41].

#### 3.1.2. 3D reconstructions of the functional and diffusion layers

The 2D slices of the freeze tape cast active layer before and after segmentation and the 3D cubic segmented sub-volume obtained from the X-ray nano-holotomography are displayed in Fig. 3a, b and e, respectively. The solid and porous phases can be identified by their corresponding grey-levels in the micrograph (dark grey: pores, bright grey: 8YSZ). For the image segmentation, the two phases in the electrode have been labelled univocally using the threshold, equal to 92, identified with the Otsu's approach [83]. This comparison highlights the correct segmentation since all the details of the 8YSZ microstructure are preserved without introducing artefacts.

The same procedure described for the functional layer was also applied to the segmentation of the diffusion layer dataset (Fig. 3c, d and f). However, the reconstruction obtained using laboratory X-ray microtomography has a higher signal-to-noise ratio that makes the grey level signal slightly more convoluted. Nevertheless, the correct thresholding between the phases was possible as illustrated in Fig. 3c and d. The segmented volume of the diffusion layer is shown in Fig. 3f.

#### 3.1.3. Microstructural properties of the backbone

The evolution of porosity along the electrode thickness (referred as z axis) has been plotted for the functional and supporting layers (Fig. 4).
<figure><img src="image_4.png" /><figcaption>Fig. 2. a) SEM micrograph showing the overview of the manufactured 8YSZ scaffold highlighting the presence of anisotropic gas diffusion channels and b) zoom on the functional layer which reveals porosity highly porous microstructure in the first 25–30 µm of the backbone; c) Pillar extracted from the from the functional layer of the three-layered porous-dense-porous YSZ architecture.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. a)–b): grey scale and corresponding segmented 2d slice extracted from the 3d reconstruction of the functional layer (top view); c)-d): grey scale and corresponding segmented 2d slice extracted from the 3d reconstruction of the diffusion layer (top view); e): 3d image taken from the whole reconstruction of the functional layer; f): 3d image taken from the whole reconstruction of the supporting layer.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. Evolution of the porosity as a function of the electrode thickness for (a) functional layer and (b) support layer.</figcaption></figure>

As it can be observed in Fig. 4a for the functional layer, the sharp increase of $V_{f,pores}$ in the first four micrometres of the sample highlights the transition from the thin dense electrolyte towards the porous layer. The transition zone is followed by a region spreading over ≈20 µm in the electrode where the pore volume fraction is almost constant to around 60 %-65 %. In the second transition zone towards the substrate, the porosity decreases down to ≈44 %. The evolution of porosity in the support layer is shown in Fig. 4b which highlights a rather stable $V_{f,pores}$ value around 44 % in the region close to the functional layer followed by a progressive decrease up to the free surface.

In the functional layer, the corresponding low percentage of YSZ phase is likely to stem from the combination of the relatively low solid loading in the initial water-based suspension and the freezing temperature selected for the lyophilisation step. In freeze tape casting the ice growth rate is controlled by the temperature gradient which is established during the freezing process [25,42,82,84]. The suspension at the interface with the cold source is subjected to the coldest temperature, being in direct contact with the freezing plate while the top surface is exposed to air. The large amount of porosity present in the functional layer can be attributed to the high rate of ice nucleation while the pore size is defined by the ice growth rate. Low freezing temperatures translate into faster crystallization rate and larger nucleation which limit the ice crystal dimensions [25,84]. The temperature gradient, which is established along the tape thickness, leads to slower nucleation and ice growth rates which are responsible for the creation of the anisotropic microstructure found in the supporting layer. On the other hand, the freeze tape casting of aqueous suspensions for solid oxide electrodes usually requires solids loading which range between 10 and 35 vol% [42,85]. In this work, a relatively low solid loading has been employed for the fabrication of graded porous scaffolds, with the purpose of maximizing the mass transfer properties of the manufactured sintered backbones. Furthermore, the selected 16 % solid loading has been found to optimize the rheological properties of the starting suspension viscosity, thus facilitating its initial deposition by tape casting. These manufacturing conditions were specifically selected to enhance the pore volume fraction even though they eventually lead to the formation of closed microporosities in the ceramic bulk [41,46]. However, the high amount of porosity close to the electrolyte-electrode interface might decrease the mechanical robustness of the cell, thus neutralizing the potential high advantages provided by a hierarchical architecture. Suspensions containing higher solid loading and lower freezing temperatures will thus be necessary to strengthen the ionic-conducting scaffold robustness.

In order to measure the average microstructural properties of the scaffolds 3D reconstructions, it is necessary to analyse a volume sufficiently large to be statistically representative of the investigated characteristics. Such a volume can be referred to as the Representative

Volume Element and its identification, both for the functional and diffusion layers, was performed by varying the edge size ($l$) of segmented cubic subdomains randomly cut out from the original 3D reconstructions and measuring the corresponding $V_{f,pores}$. Considering cubic domains characterized by the same $l$, a sufficient large number of realizations allows the statistical determination of the average quantity of porosity and its standard deviation. The $l$ value ranged from 0.3 µm to 21.8 µm for the functional layer and from 5 µm to 190 µm for the supporting one. It must also be outlined that the RVE of a graded electrode corresponds to the whole analysed domain if a long-distance homogeneity across the microstructure is not present. In our case, the investigation of the RVE is performed in the region where $V_{f,pores}$ is almost constant both for the functional and support layers (cf. circled areas in Fig. 4a and b). The mean, maximum and minimum values of the $V_{f,pores}$ as well as the standard deviation are reported as function of $l$ in Fig. 5.

As it can be observed in Fig. 5a and b, $V_{f,pores}$ and its associated standard deviation tend to an acceptable solution when $l$ is around 12 µm. This sub-volume can thus be considered sufficiently large to be statistically representative and is taken as the minimum threshold for the RVE of the functional layer. Consequently, in this study, a volume size of $15 \times 15 \times 15 \ \mu m^3$ was used as RVE of the functional layer freeze tape cast scaffold for the subsequent artificial infiltration. With regard to the supporting layer, a cubic volume of $100 \times 100 \times 100 \ \mu m^3$ was found to be sufficiently large to measure a standard deviation equal to 5 % for $V_{f,pores}$ (Fig. 5c and 5d). The larger RVE dimension obtained for this layer can be explained considering the freezing conditions of the manufacturing process. The higher characteristic size of both porosity and solid phase stems from the evolution of the solidification front speed within the ceramic matrix during the lyophilization step: the temperature gradient across the tape thickness promotes the increase of the ice crystals size while growing perpendicularly from the cold source [84,85] and thus requires a larger RVE.

The microstructural characteristics of both layers have been calculated on the RVEs and the obtained values are reported in Table 2. The $V_f$, the interfacial specific surface areas between the phases (Spa,b) and the mean size ($d_m$) of porosity and YSZ have been measured using the methods reported in [86]. Furthermore, the apparent tortuosity factors ($\tau$) of the two phases have been computed by homogenization following the procedure reported in [79].

Porosity and YSZ, both in the functional and diffusion layers, are found to be fully connected thus promoting the uniform oxygen diffusion in the gas and solid phases. The smaller crystals generated in the scaffold layer, close to the cold surface, give rise to an isotropic-like electrode morphology which increases the tortuosity factor of the gas diffusion channels compared to the supporting layer. A quasi-isotropic microstructure of the functional layer can be beneficial to the electrochemical activity of the electrode, due to the higher specific surface area
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 5.** a)–c) mean, max and minimum values of the pore volume fraction calculated in sub-volumes of different size, cut out from the segmented 3d reconstruction of the functional and supporting layers, respectively; b-d) evolution of the standard deviation from the mean porosity value while increasing the domain size of the functional and supporting layers subdomain, respectively.</figcaption></figure>

<table><caption>**Table 2**

Microstructural properties of the functional and supporting layers.</caption><thead><tr><th>Calculated microstructural properties</th><th>Functional Layer</th><th>Supporting Layer</th><th></th></tr></thead><tbody><tr><td>Pore bulk volume fraction (V<sub>f,pores</sub>)</td><td>61.5</td><td>42.5</td><td>(%)</td></tr><tr><td>8YSZ volume fraction (V<sub>f,YSZ</sub>)</td><td>38.50</td><td>57.50</td><td>(%)</td></tr><tr><td>Specific surface area between 8YSZ and pores (Sp<sub>YSZ-pores</sub>)</td><td>1.81</td><td>0.07</td><td>(µm<sup>-1</sup>)</td></tr><tr><td>Mean pore diameter (d<sub>pores</sub>)</td><td>1.13</td><td>6.41</td><td>(µm)</td></tr><tr><td>Mean 8YSZ particle diameter (d<sub>YSZ</sub>)</td><td>0.47</td><td>9.51</td><td>(µm)</td></tr><tr><td>Pore tortuosity factor on the z axis (τ<sub>z,pores</sub>)</td><td>1.83</td><td>1.32</td><td>(-)</td></tr><tr><td>Average pore tortuosity factor (τ<sub>pores</sub>)</td><td>2.32</td><td>5.86</td><td>(-)</td></tr><tr><td>8YSZ tortuosity factor on the z axis (τ<sub>z,YSZ</sub>)</td><td>1.85</td><td>1.89</td><td>(-)</td></tr><tr><td>Average 8YSZ tortuosity factor (τ<sub>YSZ</sub>)</td><td>2.33</td><td>2.24</td><td>(-)</td></tr></tbody></table>

between pores and YSZ, expected to accommodate a larger $\lambda_{TPB}$ after the impregnation of the electrocatalyst. On the other hand, the measured large mean pore diameter and high value of $V_{f,pores}$ are expected to enhance the gas diffusion since the Knudsen coefficient ($D_i^{knudsen}$) is directly proportional to $d_{pores}$, Equation (2) [87,88], and the effective gas diffusion coefficient scales directly with $V_{f,pores}$, Equation (3):

$$D_i^{knudsen} = d_{pores} \frac{1}{3} \left( \frac{8RT}{\pi M_i} \right)^{1/2} \quad (2)$$

$$D_{gas}^{eff} = \frac{V_{f,pores}}{\tau_{pore}} D_{gas} \quad (3)$$

where $M_i$ is the molar mass for the gas species $i$ and $D_{gas}$ is the diffusion coefficient calculated with the Bosanquet equation [88]. Further details on the expected advantages of a hierarchical microstructure towards the optimization of mass transfer can be found in [41].

The main morphological difference between the functional and supporting layers lies in the higher anisotropy of the latter. Indeed, the evolution of the temperature gradient over the functional layer leads to the growth of orderly distributed crystals. Therefore, compared to the active layer, the support shows lower $\tau_{z,pores}$ and $Sp_{YSZ-pores}$. These evolutions must be analysed in the frame of the physical processes that are to be promoted within the electrode substrate. The decrease of $\tau_{z, pores}$ is expected to facilitate the gas diffusion and limit the overpotential [41]. On the other hand, a lower $Sp_{YSZ-pores}$, caused by the combination of diminished $V_{f,pores}$ and $\tau_{z,pores}$, is not expected to impact the electrode performances since the fuel redox reactions take place closer to the interface with the dense electrolyte (i.e. in the functional layer). Previous studies on the experimental and modelling investigation of the electrochemical active thickness of SOCs electrodes highlighted an activity range between 5 and 15 µm [89,90]. Therefore, an enhanced design of the supporting layer morphology, including a hierarchical porous structure, is expected to minimize the concentration losses without affecting the electrochemical activity of the electrode.

## 3.2. Simulation of infiltrated freeze tape cast backbones

### 3.2.1. Functional layer

The random iterative algorithm was applied to the functional layer of the YSZ freeze tape cast backbone to study the main electrode microstructural properties of the infiltrated scaffold. Fig. 6 displays the image renderings of several impregnated microstructures. In particular, Fig. 6a–c highlight the effect of a different electrocatalyst mean diameter
<figure><img src="image_11.png" /><figcaption>Fig. 6. Volume rendering of the functional layer impregnated using $\phi_{Ni} = 0.5$ and $\sigma = 0.5$ with a) $\mu = 25$ nm, b) $\mu = 60$ nm and c) $\mu = 120$ nm, respectively. Volume rendering of the functional layer impregnated using $\mu = 60$ nm and $\sigma = 0.5$ with d) $\phi_{Ni} = 0$, e) $\phi_{Ni} = 0.5$ and f) $\phi_{Ni} = 0.95$, respectively.</figcaption></figure>

$\mu$ for constant percolating fraction ($\phi_{Ni} = 0.5$) and $\sigma = 0.5$, while Fig. 6d-f show the impregnated volumes at different $\phi_{Ni}$ values for $\mu = 60$ nm and $\sigma = 0.5$. The insets of Fig. 6d-f reveal the homogenous distribution and the necking of the catalyst particles, whose disposition on the backbone surface recalls the microstructures typically observed in classical infiltrated electrodes [61,69,91,92].

The main properties of interest for an infiltrated-pure ionic-conducting backbone, are the connected density of TPBs ($\lambda_{TPB,conn}$), the percolating fraction ($\phi_{Ni}$) and apparent tortuosity factor of the catalyst phase ($\tau_{z,Ni}$). A large database of synthetic microstructures has been synthetically generated to study the evolution of these parameters as function of $V_{f,Ni}$ and the YSZ specific surface coverage $f$, which has been expressed as, Equation (4):

$$f = \frac{Sp_{YSZ-Ni}}{Sp_{YSZ-Ni} + Sp_{YSZ-Pores}} \quad (4)$$

where $Sp_{YSZ-Ni} + Sp_{YSZ-Pores}$ is equal to the backbone specific surface area without infiltration as reported in Table 2. Fig. 7 shows the evolution of $f$ as a function of $V_{f,Ni}$ for different $\mu$ and $\sigma$ values. As expected, the surface coverage evolves almost linearly with the catalyst loading. Besides, it can be observed that a smaller nanoparticle size requires a lower $V_{f,Ni}$ to achieve a high value of $f$. Indeed, the increase of $\mu$ enlarges the volumetric fraction of the single nanoparticles which is not in contact with the scaffold walls and results in a lower surface coverage for a given volume fraction. For instance, at $\mu=25$ nm, the scaffold is almost entirely covered by the catalysts for a Ni volume fraction of only 2.8 %, whereas a loading of 5.8 % is required to reach the same coverage for $\mu=60$

<figure><img src="image_12.png" /><figcaption>Fig. 7. YSZ specific surface coverage as function of $V_{f,Ni}$ for different $\mu$ and $\sigma$ values.</figcaption></figure>

(Fig. 7). It can also be highlighted that, at a fixed catalyst volume fraction, the YSZ coverage decreases as the particle size distribution widens.

### 3.2.1.1. Percolating properties.
For an electrode made of a pure ionic
<figure><img src="image_13.png" /><figcaption>Fig. 8. a)–b) variation of $\phi_{Ni}$ as function of $V_{f,Ni}$ and $f$ for different $\mu$ and $\sigma$ values, respectively.</figcaption></figure>

conductor (such as YSZ), the electrochemical performance of a Ni-infiltrated scaffold is highly dependent on the percolating volume fraction of the electronic conductor phase ($\phi_{Ni}$). In Fig. 8, the percolating ratio has been plotted as function of $V_{f,Ni}$ and $f$ for different particle size distributions (i.e. changing $\mu$ and $\sigma$).

For both plots, the percentage of percolating phase increases steeply before reaching a plateau. The initial transition points (i.e. the first value at which $\phi_{Ni} \neq 0$) correspond to the percolating thresholds, which represent the minimum $V_{f,Ni}$ and $f$ values at which a percolating Ni-infiltrated cluster allows the electrocatalyst phase to be connected from the top to the bottom surface of the investigated electrode domain. Fig. 8a outlines that the $V_{f,Ni}$ percolating threshold ($V_{f,Ni}$ perc) decreases drastically with the mean particle size. For instance, the full connectivity of Ni catalyst is achieved for a loading of only 3.2 % with $\mu=25$, thus being considerably lower than the typical $V_{f,Ni}$ values of sintered composite electrodes [93–95]. On the other hand, a higher standard deviation for the particle distribution yields a higher percolating threshold. All these trends are consistent with the results shown in Fig. 7 on the relationship between $V_{f,Ni}$ and $f$.

With the purpose of minimizing the number of infiltration repetitions and reducing the manufacturing complexity, a low electrocatalyst loading able to effectively cover the scaffold walls and quickly reach the percolating threshold can be regarded as an experimental target. Therefore, in the analysed scenario, it can be recommended to preserve a small particle size with a narrow distribution of the electrocatalyst. On the other hand, a larger nanoparticle size, which is usually obtained either when directly impregnating raw powders synthesized ex-situ or using higher sintering temperatures, requires a higher Ni volume fraction to reach the critical loading. A $V_{f,Ni}$ perc shifted to higher values can also involve a larger number of infiltration repetitions which make the manufacturing process more complex and slower.

Distribution of nanoparticles with higher dispersion from the observed expected value can also be found in scaffolds which show a heterogeneous distribution of the infiltrated phase [96,97]. As already mentioned, the increase of $\sigma$ shifts the threshold of the percolating phase towards higher $V_{f,Ni}$ values (Fig. 8a). Larger $\sigma$ values involve the addition of a higher number of Ni particles holding radii which deviate from $\mu$, thus including larger particles that provide a higher impact on the resulting $V_{f,Ni}$ perc. Therefore, a population of nanoparticles characterized by higher standard deviations increases $V_{f,Ni}$ perc. From this point of view, the morphological characterization of the real particle size distribution generated by the manufacturing process appears to be essential as it can have a non-negligible impact on the percolating threshold.

<figure><img src="image_14.png" /><figcaption>b)</figcaption></figure>

Fig. 8b highlights the presence of a corresponding critical percolation value for the surface coverage, $f_{perc}$. In this case, the percolating threshold for the Ni catalyst is reached when $f$ ranges between 0.4 and 0.5 for $\mu > 25$ nm. This threshold is lowered to 0.38 when $\mu = 25$ nm. However, the amount of percolated Ni phase, increases more slowly when $\mu = 25$ nm, thus showing that smaller nanoparticles entail a larger surface coverage to obtain high percolation. This result also highlights the key role of the catalyst particle size on percolation.

#### 3.2.1.2. TPB length.
In addition to the analysis of the percolating properties of the synthetically infiltrated scaffold, the evolution of $\lambda_{TPB}$ as function of $V_{f,Ni}$ and $f$ has been also carried out. It is worth reminding that the connected TPB lines are the active sites in the microstructure where the electrochemical reactions take place. Therefore, the design of the electrodes microstructure should target their maximization to optimize the cells electrochemical performance. As it can be observed in Fig. 9, the density of total and connected TPB length ($\lambda_{TPB,tot}$ and $\lambda_{TPB,conn}$ respectively) reach a maximum and then decrease when additional electrocatalyst particles are added.

The characteristic “bell” shape of $\lambda_{TPB}$ as function of the catalyst loading can be explained considering the evolution of the catalyst coverage over the backbone walls. During the initial infiltration stages, there is a high probability that the randomly deposited particles are isolated from each other. In this region, the addition of new particles progressively increases the number of TPB lines in the microstructure. However, as the catalyst loading is increased, the probability of contact and overlap between two adjacent particles becomes higher and higher, thus leading $\lambda_{TPB}$ to reach a maximum. After this maximum, the density of electrochemically active sites decreases to zero that corresponds to a fully covered backbone by Ni. As for the investigated infiltrated freeze tape cast microstructure, the $\lambda_{TPB,conn}$ maximum can already be reached for a catalyst loading of 2.2 % when $\mu=25$ nm and $\sigma = 0.5$ (Fig. 9b). Even if the shift is rather limited, it can be noticed that the peak of $\lambda_{TPB,conn}$ is measured at higher $V_{f,Ni}$ than for the $\lambda_{TPB,tot}$ one. Indeed, the $\lambda_{TPB,conn}$ maximum is obtained before the Ni phase is fully connected. For example, for a catalyst loading of 2.2 % (for $\mu=25$ nm and $\sigma = 0.5$), the volume fraction of unconnected Ni phase is still 7 %.

The strong dependence of $\lambda_{TPB}$ from the nanoparticle size is highlighted in Fig. 9a and b. As it can be observed, a larger mean radius lowers the maximum $\lambda_{TPB}$ achievable, thus potentially limiting the global electrochemical performance. However, it must also be pointed out that, for every investigated particle size, the $\lambda_{TPB}$ values obtained within the range of sufficient Ni percolation are one or two order of
<figure><img src="image_15.png" /><figcaption>c)</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>b)</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>d)</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 9. a)–b): $\lambda_{TPB,tot}$ and $\lambda_{TPB,conn}$ as function of $V_{f,Ni}$ for different $\mu$ and $\sigma$ values, respectively; c)-d) $\lambda_{TPB,tot}$ and $\lambda_{TPB,conn}$ as function of $f$ for different $\mu$ and $\sigma$ values, respectively.</figcaption></figure>

magnitude higher than the $\lambda_{TPB}$ usually measured for conventional Ni-YSZ cermets [15,80,98]. Therefore, Ni-infiltrated YSZ electrodes are expected to exhibit higher electrochemical activity than traditional Ni-YSZ composites even when the mean Ni nanoparticle diameter ranges between 100 and 250 nm.

### 3.2.1.3. Tortuosity factor.
A freeze tape cast YSZ scaffold requires the

infiltrated Ni phase to provide the electronic path for the electrons involved in the charge transfer reactions. The effective conductivity in the x, y or z direction ($\sigma_{eff}^{el,j=x,yorz}$) of the electronic phase affects the ohmic contribution of the electrode polarization resistance and can be expressed according to Equation (5) [80]:

<figure><img src="image_19.png" /><figcaption>b)</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 10. a)–b): Evolution of $\tau_{z,Ni}$ as function of $V_{f,Ni}$ and $f$, respectively.</figcaption></figure>
$$ \sigma_{el,j}^{eff} = \frac{(V_{f,el} \Phi_{Ni})}{\tau_{el,j}} \sigma_{el} \qquad (5) $$

where $\sigma_{el}$ is the bulk electronic conductivity of Ni (S·m⁻¹) and $V_{f,el}$ and $\tau_{el,j}$ are the volume fraction and tortuosity factor of the electron conducting phase. Due to the strong anisotropy of the freeze tape cast microstructure, only the results for the out of plane direction (i.e. z component of $\tau$) are discussed hereafter. Indeed, a low $\tau_{z,Ni}$ is targeted to reduce the path length for the current collection. Fig. 10 shows the evolution of $\tau_{z,Ni}$ as function of $V_{f,Ni}$ and f.

As it can be seen in Fig. 10a, $\tau_{z,Ni}$ decreases sharply when the Ni loading is increased whatever the particle size. This drop can be clearly ascribed to the percolating threshold previously discussed (cf. Fig. 8). Nevertheless, even if the percolating threshold is exceeded, the tortuosity factor still continues to decrease slowly until reaching a lower bound when the YSZ surface is fully covered by Ni (Fig. 10b). Fig. 10a outlines that $\tau_{z,Ni} < 15$ is obtained at lower $V_{f,Ni}$ when infiltrating smaller nanoparticles. On the other hand, an higher electrocatalyst size facilitates the coverage of the scaffold walls thus requiring a lower surface coverage fraction to achieve acceptable $\tau_{z,Ni}$ values (Fig. 10b). Indeed, the full coverage of the surface is expected to generate a continuous thin layer of electrocatalyst minimizing the tortuosity. However, the complete coverage of the scaffold walls would also drastically reduce $\lambda_{TPB}$ and increase the activation overpotential, as suggested by the bell-shape of Fig. 9. From this point of view, a trade-off between the maximization of $\lambda_{TPB}$ and the minimization of the tortuosity factor should be identified.

The evaluation of the impact of $\tau_{z,Ni}$ on the effective electrical conductivity was then carried out on a specific set of artificially infiltrated electrodes. The synthetic architectures holding the highest $\lambda_{TPB}$ values for the three investigated $\mu$ size show different $\tau_{z,Ni}$ values (i.e. 83, 62.1 and 9.6 for $\mu$ equal to 25, 60 and 120 nm, respectively). As shown in Equation (5), the effective electrical conductivity $\sigma_{el}^{eff}$ depends both on $V_{f,el}$ and $\tau_{el}$. In the investigated hierarchically porous scaffolds, the addition of larger nanoparticles allows to obtain an increased Ni volume fraction along with a lower tortuosity factor. Therefore, the effective electrical conductivity is expected to evolve with the nanoparticles size (i.e. it decreases when decreasing the particle diameter). The effective electrical conductivity of four generated nanostructured electrodes, decorated with different loadings of nanoparticles holding $\mu = 25$ nm, was investigated to assess the impact of $\tau_{z,Ni}$ when using smaller nanoparticles. The characteristics of the analyzed electrodes have been taken from the microstructural correlations shown in Figs. 8-10 and their values are reported in Table 3.

The evolution of $\text{Log}_{10}(\sigma_{el}^{eff})$ for the investigated nanostructured electrodes is reported in Fig. 11 as function of the electrode thickness (Z) and compared to the $\sigma_{io}^{eff}$ of the YSZ scaffold. Specifically, the analysis was carried out considering a plausible operating temperature of 700 °C and bulk conductivities, $\sigma_{Ni}$ and $\sigma_{BYSZ}$, equal to $80 \cdot 10^3$ S·m⁻¹ and 1.713 S·m⁻¹, respectively. Fig. 11 was plotted by extracting the microstructural properties of the synthetic architectures as function of Z using the methods previously reported in [23].

It can clearly be observed that the gap between $\sigma_{el}^{eff}$ and $\sigma_{io}^{eff}$ is progressively reduced for higher $\tau_{z,Ni}$ values until, when $\tau_{z,Ni} = 8412$, the

<table><caption>**Table 3**
Microstructural correlations of electrodes infiltrated with different amount of Ni for $\mu = 25$ nm.</caption><thead><tr><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Input Ni loading ($\mu$ = 25 nm)</td><td></td><td>V<sub>f,Ni</sub></td><td>0.013</td><td>0.017</td><td>0.022</td><td>0.032</td><td>(-)</td></tr><tr><td rowspan="4">Computed microstructural properties</td><td>τ<sub>z,Ni</sub></td><td></td><td>8412</td><td>541</td><td>83</td><td>15</td><td>(-)</td></tr><tr><td>λ<sub>TPB</sub></td><td></td><td>15</td><td>22</td><td>29</td><td>21</td><td>(µm<sup>-2</sup>)</td></tr><tr><td>conn</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Φ<sub>Ni</sub></td><td></td><td>0.5</td><td>0.82</td><td>0.93</td><td>0.98</td><td>(-)</td></tr></tbody></table>

effective electronic conductivity becomes lower than the ionic one. Therefore, the evolution of $\sigma_{Ni}^{eff}$ with $\tau_{z,Ni}$ shows that an insufficient addition of small nanoparticles over a pure ion-conducting scaffold could lead to a significant charge transport co-limitation, that will affect the electrode performance. From this analysis, it appears that a tradeoff needs to be found to maximize the density of TPBs by keeping a sufficient electronic conductivity for the catalyst. This optimum is expected to be strongly dependent on the material properties and could be identified by the adoption of electrochemical models [52,99].

### 3.2.1.4. Supporting layer.
As the porosity is not significantly changed by the addition of nanoparticles, it can be easily assumed that the gas diffusion properties remain unchanged after infiltration (cf. Table 2). However, along with gas diffusion, the electronic conduction of Ni particles must be optimised within the supporting layer to ensure effective current collection [19,79,100]. Indeed, as described in section 3.2.1.3, the microstructural properties of the electrocatalyst phase can drastically limit its effective electronic conductivity, especially when considering a pure ionic conductor as scaffold material. In this frame, special attention has been paid to investigate the properties of the Ni particles infiltrated in the YSZ freeze tape cast backbone used as support layer.

The combination of the required RVE dimension and grid refinement from the initial 3D reconstruction of the supporting layer, lead to select a voxel size of 60 × 60 × 60 nm³ for the microstructural computations. The study of the percolating properties was then conducted using a single value for the mean particle diameter ($\mu = 120$ nm) due to the high computational effort required. The obtained results can be compared with those calculated for $\mu = 120$ nm in the functional layer, with the objective of identifying the required $V_{f,Ni}$ to effectively impregnate the YSZ walls in the overall layer. For the purposes of experimental infiltration techniques, the scale factor $\gamma$, defined as:

$$ \gamma = \frac{V_{f,NiFL}}{V_{f,NiSL}} \qquad (6) $$

where FL stands for functional layer and SL for supporting layer, can be calculated at $\mu = 120$ nm. This factor can then be used to evaluate, as a rule of thumb, the required $V_{f,NiSL}$ for $\mu = 25$ nm and $\mu = 60$ nm.

In Fig. 12, the volume renderings of the supporting layer are displayed for the synthetically infiltrated microstructure at percolating ratio $\phi_{Ni} = 0$, $\phi_{Ni} = 0.5$ and $\phi_{Ni} = 0.95$, respectively.

The comparison between the functional and supporting layers for the evolution of the surface coverage f as a function of $V_{f,Ni}$ is reported in Fig. 13. It is found that the supporting layer can be more easily covered by the impregnation method compared to the functional one. Indeed, 95 % of the YSZ surface is covered when $V_{f,Ni}$ is just equal to 0.0093 while $V_{f,Ni} = 0.108$ was required to obtain the same surface coverage in the functional layer. This large difference can be explained considering that the SpYSZ-pores of the supporting layer is two orders of magnitude lower than the one of the functional one (cf. Table 2). For this reason, a lower amount of Ni is found sufficient to cover the scaffold surface on the coarse and dense microstructure of the support.

### 3.2.1.5. Percolating properties.
Fig. 14 shows the parametric curves which reports $\phi_{Ni}$ as a function of $V_{f,Ni}$ and f (with $\mu = 120$ nm and $\sigma = 0.5$). It can be noticed that the percolating threshold for the Ni phase is reduced of one order of magnitude compared to the functional layer, being now lower than 1 % while the threshold is around 5 % in the functional layer with $\mu=120$ nm (cf. Fig. 8). As for the coverage, this behaviour is addressed to the denser and coarser microstructure of the supporting layer compared to the functional layer one.

From a practical point of view, the obtained results show that the main challenge in an impregnation process which generates a uniform nanoparticles coverage of the freeze tape cast scaffold is the effective decoration of the functional layer. Indeed, achieving the optimum $V_{f,Ni}$
<figure><img src="image_21.png" /><figcaption>Fig. 11. Evolution of Log₁₀(σ<sub>el</sub><sup>eff</sup>) as function of the electrode thickness for different τ<sub>Ni</sub> values.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 12. a)–c): Volume renderings of the supporting layer RVE impregnated using μ = 120 nm and σ = 0.5 with φ<sub>Ni</sub> = 0, φ<sub>Ni</sub> = 0.5 and φ<sub>Ni</sub> = 0.95, respectively.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 13. Comparison between the 8YSZ specific surface coverage of the functional and supporting layer as function of V<sub>f,Ni</sub> for μ = 120 nm and σ = 0.5.</figcaption></figure>

for the functional layer guarantees the full percolation of the electronic
conducting phase in the supporting layer. The associated decrease of
λ<sub>TPB</sub> while increasing *f* is not expected to affect the electrochemical
performance of the electrode, knowing that the active thickness is

limited to 5–15 µm [89,90]. On the other hand, nanostructured elec-
trodes featuring φ<sub>Ni</sub> = 1 and hierarchical porosity would ensure both the
minimization of the ohmic and concentration losses, respectively.

3.2.1.6. Tortuosity factor. The comparison between the evolution of τ<sub>z,Ni</sub> for μ = 120 nm and σ = 0.5, in both the functional and supporting layer, as function of V<sub>f,Ni</sub> and *f*, is reported in Fig. 15.

In the supporting layer, the evolution of τ<sub>z,Ni</sub> with both V<sub>f,Ni</sub> and *f*, is characterized by a sharp decrease as observed in the functional layer. Fig. 15a shows that the requested V<sub>f,Ni</sub> to reach τ<sub>z,Ni</sub> < 15 is one order of magnitude lower than the value measured for the functional layer, in accordance with the surface coverage values reported in Fig. 13. Therefore, the minimization of τ<sub>z,Ni</sub> in the supporting layer of a freeze tape cast electrode is obtained when the functional layer is effectively impregnated. In other words, the obtained results show that the optimization of the microstructural characteristics for the infiltrated functional layer also ensures the effective electronic conduction in the supporting layer.

**4. Conclusions**

In this work, the microstructure of an 8YSZ freeze tape backbone was
reconstructed by X-ray holotomography and X-ray micro-tomography. A
random-sequential-addition algorithm was developed to synthetically
infiltrate the scaffold.

The investigation of the freeze tape cast backbone revealed two
different morphologies for the supporting and functional layers. The
functional layer holds d<sub>pores</sub> = 2.1 µm and Sp<sub>YSZ-pores</sub> = 1.8 µm<sup>-1</sup> with a
τ<sub>z,pores</sub> = 1.83 while the supporting layer features d<sub>pores</sub> = 6.4 µm and
Sp<sub>YSZ-pores</sub> = 0.066 µm<sup>-1</sup> with a τ<sub>z,pores</sub> = 1.3. The functional layer
presents an isotropic-like microstructure which recalls the one of
traditional electrodes holding sponge-like porosity. The porosity
anisotropy is more evident within the supporting layer where ordered
gas diffusion channels are homogeneously distributed throughout the
YSZ bulk.

The artificial impregnation of the functional and supporting layer
pointed out that the microstructural properties of the electrodes are
highly dependent on the size and distribution of the infiltrated
nanoparticles.

Regarding the functional layer, the percolation of the Ni electro-
catalyst with a mean particle size of μ = 25 nm is reached when the
surface coverage fractions is equal to f<sub>Ni perc</sub>≈0.382. This threshold is in
<figure><img src="image_24.png" /><figcaption>Fig. 14. a)–b): $\phi_{Ni}$ as function of $V_{f,Ni}$ and $f$ at $\mu = 120$ nm and $\sigma = 0.5$, respectively.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Fig. 15. a)–b): Comparison of the evolution of $\tau_{z,Ni}$ in the functional and supporting layers, for $\mu = 120$ nm and $\sigma = 0.5$, as function of $V_{f,Ni}$ and $f$, respectively.</figcaption></figure>

the range $f_{Ni_{perc}} = 0.4–0.5$ when $\mu \ge 60$ nm. Moreover, at the percolating threshold, the catalyst loading $V_{f,Ni}$ increases from 1.2 % to 4.5 % by increasing the particle size from $\mu=25$ nm to $\mu=120$ nm. The peak in density of total and active TPB length in the functional layer were found in the range $\lambda_{TPB, TOT} = 37–20 \ \mu m^{-2}$ and $\lambda_{TPB, conn} = 30–18 \ \mu m^{-2}$, respectively, thus being one-two orders of magnitude higher than those measured in traditional composite electrodes. The tortuosity factor of the Ni phase has been also computed as function of the electro-catalyst loading. The conditions leading to charge transport colimitation in a Ni-infiltrated 8YSZ freeze tape cast fuel electrode have been also discussed.

For what concerns the supporting layer, the percolation is reached for a much lower volume fraction of Ni compared to the functional layer. This difference has been attributed to the larger $Sp_{YSZ\_pores}$ of the functional layer. In this condition, the microstructural optimisation for the functional layer infiltration can be carried without penalizing the supporting layer.

Even if the study was referred to a Ni-YSZ cermet, it must be stressed that the developed impregnation framework aims at providing general microstructural correlations, independent from the electrode materials, and valid for hierarchically porous nanostructured electrodes. Therefore, the integration of the microstructural correlations reported in this work with the specific electric and thermal properties of common SOCs materials is expected to guide the shaping of highly performing electrodes. Specifically, the use of validated electrochemical models

grounded on the microstructural findings reported in this work can drive the identification of the most suitable fuel and air electrode materials. The potential of the presented modelling framework in predicting the microstructural characteristics of nano-structured electrodes makes it a powerful tool to design high performing SOCs and guide the research community towards the optimization of their electrochemical performance.

### CRediT authorship contribution statement

**Davide Cademartori:** Investigation, Writing – original draft, Methodology. **Maxime Hubert:** Conceptualization, Writing – review & editing, Supervision. **Peter Cloetens:** Investigation. **M. Paola Carpanese:** Writing – review & editing, Supervision. **Jérôme Laurencin:** Conceptualization, Writing – review & editing, Supervision.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
## Data availability

Data will be made available on request.

## Acknowledgements

This work was carried out with the scientific support of the ID16A beamline of the European Synchrotron Radiation Facility (proposal MA-5514). The authors also acknowledge the contribution to the micro-X-Ray CT given by Dr. Michael Schulz, Dr. Alex Gustschin and Dr. Yiyong Han from the Technical University of Munich and Prof. Massimo Viviani and Dr. Juan Felipe Basbus from the Italian National Research Council (CNR-ICMATE).

Eventually, the authors thank the support of the Italian project - PROMETH2eus (ID: RSH2B 000039-CUP: F37G22000080006) funded by the Next Generation EU (Piano Nazionale di Ripresa e Resilienza-PNRR, Missione 2, Componente 2) and coordinated by the department DICCA of the University of Genoa.

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