H Moussaoui

R K Sharma

J Debayle

Y Gavet

G Delette

J Laurencin
email: jerome.laurencin@cea.fr

Microstructural correlations for specific surface area and triple phase boundary length for composite electrodes of solid oxide cells

Keywords: SOFC, SOEC, triple phase boundary lengths, specific surface area, microstructural correlations

Microstructural correlations for specific surface area and triple phase boundary lengths.

• Calibration of the semi-analytical relationships on a large dataset of synthetic microstructures.

• Validation of the semi-analytical correlations on 3D reconstructions.

Introduction

Solid Oxide Cells (SOCs) are electrochemical devices operating at high-temperature (700-900°C). This technology is nowadays considered as one of the most promising solutions in order to match the electricity consumption with the fluctuation of the production by the renewable energy sources. Indeed, the SOCs present a very good reversibility between fuel cells and electrolysis modes associated to a high efficiency and an excellent fuel flexibility [START_REF] Godula-Jopek | Hydrogen Production by Electrolysis[END_REF][START_REF] Graves | [END_REF][START_REF] Singhal | High Temperature Solide Oxide Fuel cells, Fundamental, Design and Applications[END_REF].

However, before considering the industrial deployment of this technology, improvements in SOCs performance at lower operating temperature are still required in order to reduce the cost and to enhance the long-term durability. In this objective, several attempts have been paid to improve SOCs performance at intermediate temperature with the development of new materials and the optimization of the electrode microstructure.

Nowadays, porous materials based on Mixed Ionic Electronic Conductors (MIECs) such as Lanthanum Strontium Cobalt (LSC) or Lanthanum Strontium Cobalt Ferrite (LSCF) are commonly used as O2 electrode in SOCs. Moreover, the addition of highly ionic conducting materials like Ceria doped Gadolinium Oxide (CGO) has recently attracted attention for improved O2 electrode. Indeed, compared to the single solid-phase materials, the LSCF-CGO or LSC-CGO composite exhibits higher performances at intermediate temperatures and also presents a better mechanical compatibility with the classical electrolyte in Yttria Stabilized Zirconia (YSZ) [START_REF] Dusastre | Optimisation of composite cathodes for intermediate temperature SOFC applications[END_REF][START_REF] Kim | Evaluation of La0.6Sr0.4Co0.2Fe0.8O3-Gd0[END_REF][START_REF] Kim | Investigation of La0.6Sr0.4CoO3-δ-Gd0.1Ce0.9O1.95 composite cathode with different volume ratios by three dimensional reconstruction[END_REF][START_REF] Leng | Development of LSCF-GDC composite cathodes for lowtemperature solid oxide fuel cells with thin film GDC electrolyte[END_REF]. At the present time, the porous cermet made of Nickel and YSZ is considered as the standard composite for the H2 electrode in SOC technology. Indeed, the Ni-YSZ cermet combines a high electronic conductivity due to the Nickel with a high ionic conductivity thanks to the YSZ. It also provides a very high electrochemical activity for fuel oxidation or steam reduction as well as a good mechanical compatibility with the dense YSZ electrolyte [START_REF] Hanna | Fundamentals of electro-and thermochemistry in the anode of solid-oxide fuel cells with hydrocarbon and syngas fuels[END_REF][START_REF] Prakash | Properties and development of Ni/YSZ as an anode material in solid oxide fuel cell: A review[END_REF].

The SOCs performances are highly sensitive to the complex 3D microstructure of both the H2 and O2 electrodes [START_REF] Kim | Evaluation of La0.6Sr0.4Co0.2Fe0.8O3-Gd0[END_REF][START_REF] Abe | Microstructural control of Ni-YSZ cermet anode for planer thin-film solid oxide fuel cells[END_REF][START_REF] Ostergard | Manganite-zirconia composite cathodes for SOFC: influence of structure and composition[END_REF][START_REF] Suzuki | Impact of Anode Microstructure on Solid Oxide Fuel Cells[END_REF][START_REF] Virkar | the role of electrode microstructure on activation and concentration polarizations in solid oxide fuel cells[END_REF][START_REF] Usseglio-Viretta | Quantitative microstructure characterization of a Ni-YSZ bi-layer coupled with simulated cathode polarisation[END_REF]. One of the key microstructural parameters is the active Triple Phase Boundaries lengths (TPBl) defined by the lines where the percolated ionic, electronic and gas phases meet. For example, the efficiency of the H2 electrode is directly controlled by the density of active TPBl since the electrochemical reaction in the Ni-YSZ cermet takes place at these electroactive sites [START_REF] Hanna | Fundamentals of electro-and thermochemistry in the anode of solid-oxide fuel cells with hydrocarbon and syngas fuels[END_REF]. For the O2 electrode, the higher performances of the LSCF-CGO composite with respect to the pure LSCF can be explained by the extension of TPBl in the bulk of the electrode [START_REF] Dusastre | Optimisation of composite cathodes for intermediate temperature SOFC applications[END_REF][START_REF] Kim | Evaluation of La0.6Sr0.4Co0.2Fe0.8O3-Gd0[END_REF]. Indeed, it has been suggested that the oxygen oxidation/reduction at TPBl according to the so-called "surface path" could control the global electrochemical reaction for the LSCF-CGO electrode [START_REF] Hubert | Role of microstructure on electrode operating mechanisms for mixed ionic electronic conductors: From synchrotron-based 3D reconstruction to electrochemical modelling[END_REF][START_REF] Laurencin | Reactive Mechanisms of LSCF Single-Phase and LSCF-CGO Composite Electrodes Operated in Anodic and Cathodic Polarisations[END_REF]. Aside from the TPBl, the interfacial specific surface areas Si/j between the electrode phases can also contribute to the global reaction mechanism (with i/j=electronic/pore, ionic/pore or electronic/ionic interfaces). For instance, it is suspected that the charge transfer at TPBl in the Ni-YSZ cermet can be co-limited by a gas interaction step with the YSZ or Ni surface [START_REF] Hanna | Fundamentals of electro-and thermochemistry in the anode of solid-oxide fuel cells with hydrocarbon and syngas fuels[END_REF]. For the O2 electrode, the reaction of oxygen reduction under SOFC mode is assumed to occur on the whole LSCF/pore interfacial surface area according to the so-called "bulk path" [START_REF] Adler | Electrode Kinetics of Porous Mixed-Conducting Oxygen Electrodes[END_REF][START_REF] Monaco | Experimental validation of a La0.6Sr0.4Co0.2Fe0.8O3-δ electrode model operated in electrolysis mode: Understanding the reaction pathway under anodic polarization[END_REF]. In this reaction mechanism, the ionic charge transfer between LSCF and CGO is proportional to the contact surface area between the two materials.

It is also widely recognized that the electrode microstructure plays a key role on the cell durability [START_REF] John | Evolution of the electrochemical interface in high-temperature fuel cells and electrolysers[END_REF][START_REF] Lay-Grindler | Micromodelling of Solid Oxide Electrolysis Cell: from performance to durability[END_REF][START_REF] Moçoteguy | A review and comprehensive analysis of degradation mechanisms of solid oxide electrolysis cells[END_REF]. For instance, the degradation mechanism in the Ni-YSZ electrode is related to the corresponding changes in the TPBl density and specific surface areas due to a microstructural evolution over time [START_REF] Faes | Nickel-zirconia anode degradation and triple phase boundary quantification from microstructural analysis[END_REF][START_REF] Jiao | Quantitative Characterization of SOFC Nickel-YSZ Anode Microstructure Degradation Based on Focused-Ion-Beam 3D-Reconstruction Technique[END_REF][START_REF] Lay-Grindler | Degradation study by 3D reconstruction of a nickel-yttria stabilized zirconia cathode after high temperature steam electrolysis operation[END_REF][START_REF] Hubert | Impact of Nickel agglomeration on solid oxide cell operated in fuel cell and electrolysis modes[END_REF]. As the material deterioration in operation is associated to thermally activated processes, the decrease of the operating temperature should allow mitigating the degradation in performances. In this case, an electrode microstructural optimization is still necessary in order to maintain the high performances at these 'intermediate' operating temperatures. An important strategy to enhance the catalytic activity in the electrodes at lower temperatures is to increase the number of active sites via an increase in both the density of TPBl, LTPB, and the specific surface areas, Si/j, [START_REF] Kim | Evaluation of La0.6Sr0.4Co0.2Fe0.8O3-Gd0[END_REF][START_REF] Ostergard | Manganite-zirconia composite cathodes for SOFC: influence of structure and composition[END_REF][START_REF] John | Evolution of the electrochemical interface in high-temperature fuel cells and electrolysers[END_REF]. For this microstructural optimization, it is essential to link these properties to the "basic" morphological electrode parameters that can be more easily controlled by the manufacturer. These "basic" parameters correspond to (i) the porosity (θp), (ii) the relative phase volume fraction of solid phases (α= θio/θel) and (iii) the parameters describing the shape and the distribution of the particles (ionic and electronic conductors).

In order to establish these electrode microstructural correlations, a first strategy is based on the manufacturing of several electrodes by changing both the composition and the Particle Size Distribution (PSD) of the initial powder. The microstructural properties (LTPB and Si/j) can be measured on 3D reconstructions obtained by X-ray Computed Tomography (XCT) [START_REF] Cronin | Three-dimensional reconstruction and analysis of an entire solid oxide fuel cell by full-field transmission X-ray tomography[END_REF][START_REF] Grew | Nondestructive nanoscale 3D elemental mapping and analysis of a solid oxide fuel cell anode[END_REF][START_REF] Hubert | Efficient correction of wavefront inhomogeneities in X-ray holographic nanotomography by random sample displacement[END_REF][START_REF] Shearing | Exploring microstructural changes associated with oxidation in Ni-YSZ SOFC electrodes using high resolution X-ray computed tomography[END_REF][START_REF] Villanova | 3D phase mapping of SOFC YSZ/Ni cermet at the nanoscale by X-Ray holotomography[END_REF] or by Focused Ion Beam Scanning Electron Microscopy (FIB-SEM) [START_REF] Holzer | Quantitative relationships between composition, particle size, triple phase boundary length and surface area in nickelcermet anodes for solid oxide fuel cells[END_REF][START_REF] Vivet | 3D Microstructural characterization of a solid oxide fuel cell anode reconstructed by focused ion beam tomography[END_REF][START_REF] Wislon | Three -dimensional reconstruction of solid-oxide fuel-cell anode[END_REF]. This experimental approach requires a large number of samples for parametric studies. However, the number of experiments is generally very limited since this method is time consuming. Therefore, the dataset is usually insufficient to fully validate the complex microstructural relationships [START_REF] Chen | Percolation theory in solid oxide fuel cell composite electrodes with a mixed electronic and ionic conductor[END_REF][START_REF] Gokhale | A stochastic geometry model for total triple phase boundary length in composite cathodes for solid oxide fuel cells[END_REF].

An alternative approach consists in theoretical developments based on geometrical considerations. In this frame, numerical approaches can be used to derive some pure phenomenological microstructural relationships. They are based on the generation of synthetic 3D microstructures obtained by the random packing of initial spheres [START_REF] Ali | Geometrical modeling of microstructure of solid oxide fuel cell composite electrodes[END_REF][START_REF] Bertei | Percolating behavior of sintered random packings of spheres[END_REF][START_REF] Cai | Modelling the 3D microstructure and performance of solid oxide fuel cell electrodes: computational parameters[END_REF][START_REF] Kenney | Computation of TPB length, surface area and pore size from numerical reconstruction of composite solid oxide fuel cell electrodes[END_REF][START_REF] Metcalfe | Connected three-phase boundary length evaluation in modeled sintered composite solid oxide fuel cell electrodes[END_REF] or by geostatistical simulations [START_REF] Moussaoui | Stochastic Geometrical Modeling of Solid Oxide Cells Electrodes Validated on 3D Reconstructions[END_REF][START_REF] Neumann | Stochastic 3D modeling of complex three-phase microstructure in SOFC electrodes with completely connected phases[END_REF][START_REF] Suzue | Micro modeling of solid oxide fuel cell anode based on stochastic reconstruction[END_REF].

In the frame of the percolation theory, analytical models are also proposed to express LTPB and the specific surface area [START_REF] Chen | Percolation theory in solid oxide fuel cell composite electrodes with a mixed electronic and ionic conductor[END_REF][START_REF] Bertei | Percolation theory in SOFC composite electrodes: effects of porosity and particle size distribution on effective properties[END_REF][START_REF] Chan | Cathode micromodel of solid oxide fuel cell[END_REF][START_REF] Chen | Percolation micro-model to predict the effective properties of the composite electrode with poly-dispersed particle sizes[END_REF][START_REF] Costamagna | Micro-modelling of solid oxide fuel cell electrodes[END_REF][START_REF] Janardhanan | Three-phase boundary length in solidoxide fuel cells: a mathematical model[END_REF][START_REF] Sunde | Simulations of composite electrodes in fuel cells[END_REF][START_REF] Sunde | Monte Carlo simulations of polarization resistance of composite electrodes for solid oxide fuel cells[END_REF][START_REF] Zhang | Random-packing model for solid oxide fuel cell electrodes with particles size distributions[END_REF]. Most of these approaches are limited to mono-sized spheres with approximations on the estimation of the mean coordination number and the contact angles between the particles. Nevertheless, the microstructures of real composite electrodes are strongly disordered and the particles tend to have a complex and non-spherical shape. To overcome these issues, Gokhale et al. [START_REF] Gokhale | A stochastic geometry model for total triple phase boundary length in composite cathodes for solid oxide fuel cells[END_REF] have proposed to use the concept of "extended" microstructure and developed an analytical model for the estimation of LTPB in composite electrodes taking into account the actual PSD of spheres. In this case, from our best knowledge, no correlation has been proposed for the interfacial specific surface areas.

In this work, semi-analytical expressions linking the density of TPBl and the interfacial specific surface areas to the "basic" parameters of the microstructure are studied and calibrated on two sets of synthetic 3D microstructures. These two databases have been generated by using inhouse models based on the sphere packing and the truncated Gaussian random field methods.

Finally, the reliability of the microstructural relationships has been checked with real 3D electrode reconstructions. The article is organized as follows: the theoretical background and the development of semi-analytical correlations are detailed in Section 2. The fitting of the parametric equations are presented and discussed in Section 3. For each property (LTPB and Si/i), the correlations deduced from simulated microstructure datasets have been compared to the data extracted from the real reconstructions in Section 4.

Theoretical background, development and methodology

Theoretical background

Microstructural engineering of composite electrode materials is a multivariate complex problem. Indeed, the density of TPBl and the interfacial specific surface areas, which are the key properties controlling the electrochemical response, depend on several "basic" microstructural parameters which can be linked to the powder before sintering [START_REF] Zhang | Random-packing model for solid oxide fuel cell electrodes with particles size distributions[END_REF]. By considering the medium described by a random packing of convex particles, these "basic" parameters of the microstructure can be ascribed to the features of the particles collection which can be defined as follows [START_REF] Gokhale | A stochastic geometry model for total triple phase boundary length in composite cathodes for solid oxide fuel cells[END_REF][START_REF] Zhang | Quantitative characterization and modelling of the microstructure of solid oxide fuel cell composite electrodes[END_REF]:

(a) The phase volume fractions are expressed through the porosity (θp) and the ratio of the ionic to the electronic phase volume fractions (α= θio/θel) with:

𝜃 𝑖𝑜 + 𝜃 𝑒𝑙 + 𝜃 𝑝 = 1 (1) (b)
The morphology of the electrode particles can be described for each phase i by two shape factors K1i and K2i defined by:

𝐾 1𝑖 = 〈𝑆 𝑖 〉 〈𝐷 𝑖 2 〉 𝑎𝑛𝑑 𝐾 2𝑖 = 〈𝑉 𝑖 〉 〈𝐷 𝑖 3 〉 𝑤𝑖𝑡ℎ i = ionic or electronic particles (2)
where 𝐷 𝑖 , 𝑆 𝑖 and 𝑉 𝑖 denote the mean diameter, the surface area and the volume for the ionic or electronic particle, respectively (note that 〈𝑋 𝑖 〉 represents the arithmetic mean of 𝑋 𝑖 :

〈𝑋 𝑖 〉 = ∫ 𝑋 𝑖 𝑝(𝑋 𝑖 ) 𝑑𝑋 𝑖 ∞ 0
with p(Xi) is the probability density function) . For the ideal case of spherical particles, the ratio βi=K1i/K2i is equal to 6 [START_REF] Gokhale | A stochastic geometry model for total triple phase boundary length in composite cathodes for solid oxide fuel cells[END_REF].

(c) The Particle Size Distribution (PSDi) of each solid phase i is fully defined by the mean value 〈𝐷 𝑖 〉, the standard deviation σ 𝑖 2 , the Coefficient of Variation (CVi) and the skewness γi:

𝜎 𝑖 2 = ∫ [𝐷 𝑖 -〈𝐷 𝑖 〉] 2 𝑝(𝐷 𝑖 )𝑑𝐷 𝑖 ∞ 0 ; 𝐶𝑉 𝑖 = 𝜎 𝑖 〈𝐷 𝑖 〉 ; 𝛾 𝑖 = 1 𝜎 𝑖 3 ∫ [𝐷 𝑖 -〈𝐷 𝑖 〉] 3 𝑝(𝐷 𝑖 )𝑑𝐷 𝑖 ∞ 0 (3)
Therefore, the density of TPBl and the interfacial specific surface areas have to be expressed as a function of θp, α, βi, 〈𝐷 𝑖 〉, CVi and 𝛾 𝑖 .

As mentioned in the introduction, phenomenological equations can be derived from the generation of synthetic microstructures obtained by geometrical stochastic simulations (based on the random packing of particles or geostatistical modelling). As a general matter, these methods require intensive computational means for the generation of a large dataset of 3D microstructures and must also involve complex optimization algorithms of data mining [START_REF] Stenzel | Big data for microstructureproperty relationships: a case study of predicting effective conductivities[END_REF].

Aside from these numerical computations, some analytical developments based on the percolation theory have been proposed to establish the microstructural correlations. Most of them are focused on the density of TPBl [START_REF] Chen | Percolation theory in solid oxide fuel cell composite electrodes with a mixed electronic and ionic conductor[END_REF][START_REF] Bertei | Percolation theory in SOFC composite electrodes: effects of porosity and particle size distribution on effective properties[END_REF][START_REF] Chan | Cathode micromodel of solid oxide fuel cell[END_REF][START_REF] Chen | Percolation micro-model to predict the effective properties of the composite electrode with poly-dispersed particle sizes[END_REF][START_REF] Janardhanan | Three-phase boundary length in solidoxide fuel cells: a mathematical model[END_REF][START_REF] Sunde | Simulations of composite electrodes in fuel cells[END_REF][START_REF] Sunde | Monte Carlo simulations of polarization resistance of composite electrodes for solid oxide fuel cells[END_REF][START_REF] Zhang | Random-packing model for solid oxide fuel cell electrodes with particles size distributions[END_REF], while Chen et al. [START_REF] Chen | Percolation theory in solid oxide fuel cell composite electrodes with a mixed electronic and ionic conductor[END_REF] and Costamagna et al. [START_REF] Costamagna | Micro-modelling of solid oxide fuel cell electrodes[END_REF] have dealt with the interfacial specific surface area between the electronic/pore and ionic/electronic phases, respectively. All these mathematical models, which are based on pure geometrical considerations, are free from the computational cost of 3D simulations but require some simplifications. All these models consider spherical particles for the electronic and ionic conductors which are loosely packed. The particles of each constituent have the same coordination number while the distance between the particle centers (or the contact angle) depicting the spheres intersection is constant. In the first studies, the spheres of each phase were assumed to have the same size [START_REF] Chan | Cathode micromodel of solid oxide fuel cell[END_REF][START_REF] Costamagna | Micro-modelling of solid oxide fuel cell electrodes[END_REF][START_REF] Sunde | Simulations of composite electrodes in fuel cells[END_REF]. This simplification has been recently improved by extending the modelling to poly-dispersed particles [START_REF] Chen | Percolation theory in solid oxide fuel cell composite electrodes with a mixed electronic and ionic conductor[END_REF][START_REF] Bertei | Percolation theory in SOFC composite electrodes: effects of porosity and particle size distribution on effective properties[END_REF][START_REF] Chen | Percolation micro-model to predict the effective properties of the composite electrode with poly-dispersed particle sizes[END_REF][START_REF] Janardhanan | Three-phase boundary length in solidoxide fuel cells: a mathematical model[END_REF][START_REF] Zhang | Random-packing model for solid oxide fuel cell electrodes with particles size distributions[END_REF].

Nevertheless, they still do not capture the effects of particle shapes/morphologies and the distribution on the coordination number. However, a distribution of the coordination number is expected in real microstructures exhibiting a scattering on the particle size. Indeed, there is a clear correlation between the particle size and the number of bonds in the lattice. Moreover, a distribution on the contact angles related to the particles intersection could also affect the distribution on the coordination number. Finally, it is worth noting that all these models rely on the knowledge of the particles intersection and/or the coordination number. Nevertheless, the definition and the computation of these properties for real partially sintered microstructures is not straightforward and requires further simplifying assumptions [START_REF] Gokhale | A stochastic geometry model for total triple phase boundary length in composite cathodes for solid oxide fuel cells[END_REF].

To overcome these issues, Gokhale et al. [START_REF] Gokhale | A stochastic geometry model for total triple phase boundary length in composite cathodes for solid oxide fuel cells[END_REF] have proposed a stochastic geometry based analytical expression for the total LTPB related to the overall phases (i.e. including the percolated phases and the disconnected clusters). The model is applicable for particles of any convex shapes and size distribution. For this purpose, they have used the notion of "extended" microstructure, widely employed in modeling phase transformations [START_REF] Avrami | Kinetics of Phase Change. I: General Theory[END_REF][START_REF] Avrami | Kinetics of Phase Change. II: Transformation-Time Relations for Random Distribution of Nuclei[END_REF][START_REF] Avrami | Kinetics of Phase Change. III: Granulation, Phase Change, and Microstructure[END_REF][START_REF] Fanfoni | The Johnson-Mehl-Avrami-Kohnogorov model: A brief review[END_REF][START_REF] Gokhale | Application of microstructure modelling to the kinetics of recrystallization[END_REF][START_REF] Gokhale | A test for randomness of the spatial distribution of particles during phase transformation[END_REF][START_REF] Jones | Kinetics of the simultaneous decomposition of austenite into several transformation products[END_REF]. With such an approach, the effects of the particle shape and size distribution can be taken into account:

𝐿 𝑇𝑃𝐵 = 𝜋 4 𝛼 (1 + 𝛼)² 𝜃 𝑝 (𝑙𝑛𝜃 𝑝 )²𝛽 𝑖𝑜 𝛽 𝑒𝑙 1 〈𝐷 𝑖𝑜 〉〈𝐷 𝑒𝑙 〉 (1 + 𝐶𝑉 𝑖𝑜 2 ) (1 + 3𝐶𝑉 𝑖𝑜 2 + 𝛾 𝑖𝑜 𝐶𝑉 𝑖𝑜 3 ) (1 + 𝐶𝑉 𝑒𝑙 2 ) (1 + 3𝐶𝑉 𝑒𝑙 2 + 𝛾 𝑒𝑙 𝐶𝑉 𝑒𝑙 3 ) (4)
As seen in the previous equation, the density of total TPBl is then explicitly expressed as a function of the 'basic' parameters of the microstructure. The consistency of this theoretical relation has been checked by Zhang et al. [START_REF] Zhang | Computer simulations of topological connectivity of the triple phase boundaries in solid oxide fuel cell composite cathodes[END_REF] thanks to 3D simulated volumes generated by an algorithm developed to produce the same microstructure as defined in the analytical model. As expected, they have found a very good agreement between the simulated and predicted TPBl.

They have also checked that eq. ( 4) stands to accurately predict the active density of TPBl as long as the electrode phases are well connected (i.e. solid phase volume fraction higher than 30 % and porosity higher than 10% in their case).

Theoretical development: analytical expression for Si/j and Si

In this work, we have derived the analytical expressions for the interfacial specific surface areas Si/j as a function of the microstructural parameters of the composite electrode. The electrode specific surface areas Sp can be obviously deduced from the former relations with 𝑆 𝑝 = 𝑆 𝑖𝑜/𝑝 + 𝑆 𝑒𝑙/𝑝 . The same procedure as the one proposed by Gokhale et al. [START_REF] Gokhale | A stochastic geometry model for total triple phase boundary length in composite cathodes for solid oxide fuel cells[END_REF] for the density of TPBl has been followed. The main steps of this study are detailed hereafter.

Concept of "extended" microstructure

The approach is based on the concept of a virtual domain called the "extended" microstructure [START_REF] Avrami | Kinetics of Phase Change. I: General Theory[END_REF][START_REF] Avrami | Kinetics of Phase Change. II: Transformation-Time Relations for Random Distribution of Nuclei[END_REF][START_REF] Avrami | Kinetics of Phase Change. III: Granulation, Phase Change, and Microstructure[END_REF]. In this domain, the centers of convex particles are randomly distributed by a homogeneous Poisson point process. The ionic and electronic particles in the "extended" microstructure are free to interpenetrate while the remaining empty space corresponds to the porosity (Fig. 1). With such a definition, the volumes of the ionic and electronic phases are overestimated in the "extended" domain since the particles cannot overlap in the "real" microstructure. Nevertheless, the properties of the "actual" microstructure can be deduced from the virtual domain by taking into account the particles overlap. Indeed, only a fraction of any incremental volume change in the "extended" microstructure 𝑑𝑉 𝑖 𝑒𝑥 contributes to a change in the real microstructure 𝑑𝑉 𝑖 (the subscript i denotes the ionic or electronic phase). Assuming that the particles are randomly distributed, this fraction of 𝑑𝑉 𝑖 𝑒𝑥 corresponds to the volume in the extended microstructure which is formed in the complementary phase of the solid particles (i.e. the porosity); whereas the volume that drops within the overlap must be ignored (Fig. 1). On average, any increment of volume in the real medium is thus proportional to the volume increase in the extended microstructure as follows [START_REF] Jones | Kinetics of the simultaneous decomposition of austenite into several transformation products[END_REF]:

𝑑𝑉 𝑒𝑙 = [1 - 𝑉 𝑒𝑙 + 𝑉 𝑖𝑜 𝑉 𝑡𝑜𝑡 ] 𝑑𝑉 𝑒𝑙 𝑒𝑥 𝑎𝑛𝑑 𝑑𝑉 𝑖𝑜 = [1 - 𝑉 𝑒𝑙 + 𝑉 𝑖𝑜 𝑉 𝑡𝑜𝑡 ] 𝑑𝑉 𝑖𝑜 𝑒𝑥 (5)
where 𝑉 𝑡𝑜𝑡 denotes the volume of the considered domain. The terms 𝑉 𝑖 and 𝑉 𝑖 𝑒𝑥 are related to the volumes of the phase i for the extended and real microstructures, respectively. Knowing that ). Jones and Bhadeshia [START_REF] Jones | Kinetics of the simultaneous decomposition of austenite into several transformation products[END_REF] have integrated the increment of volumes to find the relations between the extended and real volumes for two phases:

𝑉 𝑒𝑙 𝑒𝑥 𝑉 𝑡𝑜𝑡 = -1 1 + 𝛼 𝑙𝑛 [1 -(1 + 𝛼) 𝑉 𝑒𝑙 𝑉 𝑡𝑜𝑡 ] 𝑎𝑛𝑑 𝑉 𝑖𝑜 𝑒𝑥 𝑉 𝑡𝑜𝑡 = -𝛼 1 + 𝛼 𝑙𝑛 [1 -( 1 + 𝛼 𝛼 ) 𝑉 𝑖𝑜 𝑉 𝑡𝑜𝑡 ] (6)
It can be noticed that the two previous relations in eq. ( 6) imply that:

𝜃 𝑖𝑜 𝑒𝑥 𝜃 𝑒𝑙 𝑒𝑥 = 𝜃 𝑖𝑜 𝜃 𝑒𝑙 (7)
The combination of eq. ( 6) and eq. ( 1) allows the volume fraction of the ionic and electronic conductors in the extended microstructure to be expressed as a function of the porosity of the real medium:

𝜃 𝑒𝑙 𝑒𝑥 = -1 1 + 𝛼 𝑙𝑛[𝜃 𝑝 ] 𝑎𝑛𝑑 𝜃 𝑖𝑜 𝑒𝑥 = -𝛼 1 + 𝛼 𝑙𝑛[𝜃 𝑝 ] (8)
These last equations will be used to express the interfacial specific surface areas.

Fig. 1: Schematic representation of two overlapping particles in the "extended" microstructure (given here for two spherical particles in ionic and electronic conducting phases).

2.2.2. Specific surface area in the "real" and "extended" microstructures for the ionic/pore and electronic/pore interfaces In the following, S 𝑖𝑜 𝑒𝑥 and S 𝑒𝑙 𝑒𝑥 denote the specific surface areas for the ionic and electronic phases in the extended microstructure, whereas 𝑆 𝑖𝑜/𝑝 and 𝑆 𝑒𝑙/𝑝 represent the interfacial surface areas per unit volume of sample between the ionic/porosity and electronic/porosity phases in the corresponding real microstructure. As previously discussed for the volumes, only a part of the

Ionic Electronic Porosity R io dR io R el  dV io
surface area in the extended microstructure can contribute to the surface area in the real microstructure. By analogy with the volume increments in eq. ( 5), the interfacial surface area of the ionic or electronic particles in contact with porosity in the real microstructure is only equal to the fraction of the surface area that falls within the porosity in the extended microstructure. As a consequence, it can be shown starting from eq. ( 5) [START_REF] Gokhale | Application of microstructure modelling to the kinetics of recrystallization[END_REF][START_REF] Gokhale | A test for randomness of the spatial distribution of particles during phase transformation[END_REF] the following relations:

𝑆 𝑒𝑙/𝑝 = 𝜃 𝑝 𝑆 𝑒𝑙 𝑒𝑥 𝑎𝑛𝑑 𝑆 𝑖𝑜/𝑝 = 𝜃 𝑝 𝑆 𝑖𝑜 𝑒𝑥 (9)
These last equations will be used to express the interfacial specific surface areas as a function of the geometrical characteristics of the electrode phases.

2.2.3 Expression of the ionic/pore and electronic/pore interfacial specific surface areas versus the "basic" electrode microstructural parameters

The specific surface area for each solid phase in the extended microstructure, 𝑆 𝑒𝑙 𝑒𝑥 or 𝑆 𝑖𝑜 𝑒𝑥 , is directly given by the mean surface area of the electronic or ionic particles, 〈𝑆 𝑒𝑙 〉 or 〈𝑆 𝑖𝑜 〉, multiplied by the number of particles per unit volume, 𝑁 𝑒𝑙 or 𝑁 𝑖𝑜 :

𝑆 𝑒𝑙 𝑒𝑥 = 𝑁 𝑒𝑙 〈S el 〉 𝑎𝑛𝑑 𝑆 𝑖𝑜 𝑒𝑥 = 𝑁 𝑖𝑜 〈S io 〉 (10)
The same relations can be also written for the volume fractions in the extended microstructure:

𝜃 𝑒𝑙 𝑒𝑥 = 𝑁 𝑒𝑙 〈𝑉 𝑒𝑙 〉 𝑎𝑛𝑑 𝜃 𝑖𝑜 𝑒𝑥 = 𝑁 𝑖𝑜 〈𝑉 𝑖𝑜 〉 (11)
Where 〈𝑉 𝑒𝑙 〉 and 〈𝑉 𝑖𝑜 〉 denote respectively the mean value for the volume of the electronic and ionic particles. For each solid phase, the combination of eq. ( 9) to (11) yields the following expressions:

𝑆 𝑒𝑙/𝑝 = 𝜃 𝑝 〈𝑆 𝑒𝑙 〉 〈𝑉 𝑒𝑙 〉 𝜃 𝑒𝑙 𝑒𝑥 𝑎𝑛𝑑 𝑆 𝑖𝑜/𝑝 = 𝜃 𝑝 〈𝑆 𝑖𝑜 〉 〈𝑉 𝑖𝑜 〉 𝜃 𝑖𝑜 𝑒𝑥 (12)
The relations given in eq. ( 8) can be introduced in eq. ( 12) in order to express the interfacial specific surface area with the parameters of the "real" microstructure only:

𝑆 𝑒𝑙/𝑝 = -1 1 + 𝛼 𝜃 𝑝 𝑙𝑛[𝜃 𝑝 ] 〈𝑆 𝑒𝑙 〉 〈𝑉 𝑒𝑙 〉 𝑎𝑛𝑑 𝑆 𝑖𝑜/𝑝 = -𝛼 1 + 𝛼 𝜃 𝑝 𝑙𝑛[𝜃 𝑝 ] 〈𝑆 𝑖𝑜 〉 〈𝑉 𝑖𝑜 〉 (13)
The first terms in eq. ( 13) describe the variation of the interfacial specific surface areas with the volume fractions of the three ionic, electronic, and pore phases. The ratio,

〈𝑆 𝑒𝑙 〉 〈𝑉 𝑒𝑙 〉 or 〈𝑆 𝑖𝑜 〉 〈𝑉 𝑖𝑜 〉
, is related to the morphology of the electronic or ionic particles. Indeed, by using eq. ( 2), the ratio can be replaced by the mean diameter and the shape factor of particles as follows:

𝑆 𝑒𝑙/𝑝 = -1 1 + 𝛼 𝜃 𝑝 𝑙𝑛[𝜃 𝑝 ]𝛽 𝑒𝑙 〈𝐷 𝑒𝑙 2 〉 〈𝐷 𝑒𝑙 3 〉 𝑎𝑛𝑑 𝑆 𝑖𝑜/𝑝 = -𝛼 1 + 𝛼 𝜃 𝑝 𝑙𝑛[𝜃 𝑝 ]𝛽 𝑖𝑜 〈𝐷 𝑖𝑜 2 〉 〈𝐷 𝑖𝑜 3 〉 (14)
The ratio, βel=K1el/K2el or βio=K1io/K2io, depends only on the electronic or ionic particle shape whereas the last term,

〈𝐷 𝑒𝑙 2 〉 〈𝐷 𝑒𝑙 3 〉 or 〈𝐷 𝑖𝑜 2 〉 〈𝐷 𝑖𝑜 3 〉
, contains the characteristics describing the particle size distribution. Indeed, with the definitions of the standard deviation σ 𝑖 2 , the coefficient of variation CVi and the skewness γi given in eq. ( 3), it can be shown that:

〈𝐷 𝑖 2 〉 = 〈𝐷 𝑖 〉 2 [1 + 𝐶𝑉 𝑖 2 ] 𝑎𝑛𝑑 〈𝐷 𝑖 3 〉 = 〈𝐷 𝑖 〉 3 [1 + 3𝐶𝑉 𝑖 2 + 𝛾 𝑖 𝐶𝑉 𝑖 3 ] (15)
Finally, the combination of eqs. ( 15) and ( 14) allows the surface areas versus the "basic" microstructural parameters of the electrode to be expressed for both the electronic/porosity and ionic/porosity interfaces:

𝑆 𝑒𝑙/𝑝 = -1 1 + 𝛼 𝜃 𝑝 𝑙𝑛[𝜃 𝑝 ]𝛽 𝑒𝑙 1 〈𝐷 𝑒𝑙 〉 [1 + 𝐶𝑉 𝑒𝑙 2 ] [1 + 3𝐶𝑉 𝑒𝑙 2 + 𝛾 𝑒𝑙 𝐶𝑉 𝑒𝑙 3 ] (16a)
𝑆 𝑖𝑜/𝑝 = -𝛼 1 + 𝛼 𝜃 𝑝 𝑙𝑛[𝜃 𝑝 ]𝛽 𝑖𝑜 1 〈𝐷 𝑖𝑜 〉 [1 + 𝐶𝑉 𝑖𝑜 2 ] [1 + 3𝐶𝑉 𝑖𝑜 2 + 𝛾 𝑖𝑜 𝐶𝑉 𝑖𝑜 3 ] (16b)
It is worth reminding that the "extended" microstructure is composed of ionic and electronical particles while the porosity is taken as the complementary phase (Fig. 1). With such a configuration, the contact surface between two overlapping particles is not well defined.

Therefore, the same development cannot be applied to derive the formula expressing the interfacial specific surface area between the electronic and ionic particles 𝑆 𝑖𝑜/𝑒𝑙 . From that point of view, further assumptions are required.

Expression of the interfacial specific surface area between the ionic and electronic phases

With simple geometrical considerations, it can be shown that the surface area 𝑆 𝑖𝑜/𝑒𝑙 is linked to the density of TPBl. Indeed, the interfacial surface area between two convex ionic and electronic particles is proportional to the contact perimeter.

Assuming spherical particles, the density of TPBl is simply given by the number of ionic and electronic contacts per unit volume, Nc, multiplied by the mean circumference of the intersections:

𝐿 𝑇𝑃𝐵 = 𝑁 𝑐 𝜋〈𝑑 𝑐 〉 (17)
where 〈𝑑 𝑐 〉 is the mean value of the contact diameter between the ionic and electronic particles.

Accordingly, the interfacial specific surface area can be expressed as

𝑆 𝑒𝑙/𝑖𝑜 = 𝑎 1 × 𝑁 𝑐 𝜋〈𝑑 𝑐 〉 2 /4
where the term 𝑎 1 is a factor of proportionality taking into account a potential roughness of the solid/solid interface created during the sintering. The interfacial specific surface area between the electronic and ionic particles is then given by:

𝑆 𝑒𝑙/𝑖𝑜 = 𝑎 1 × 𝐿 𝑇𝑃𝐵 〈𝑑 𝑐 〉/4 (18)
Considering the geometry of a Hertzian contact force model for spheres, Amitai et al. [START_REF] Amitai | Theory-based design od sintered granular composite triples three-phase boundary in fuel cells[END_REF] have shown that 〈𝑑 𝑐 〉 can be related to the mean diameters of the ionic and electronic particles:

〈𝑑 𝑐 〉 2 4 ≈ 〈𝐷 𝑒𝑙 〉〈𝐷 𝑖𝑜 〉 〈𝐷 𝑒𝑙 〉 + 〈𝐷 𝑖𝑜 〉 ℓ ( 19
)
where ℓ denotes the penetration length of the particles intersection. Unfortunately, this parameter depends on the local geometry of the intersections (contact angle and sphere radius).

However, it varies over a limited range for "classical" electrode microstructures with a good estimation given by ℓ = 𝑎 2 × min {〈𝐷 𝑒𝑙 〉, 〈𝐷 𝑖𝑜 〉}, where 𝑎 2 is a constant of proportionality depending on the mean contact angle 〈𝛩〉. This approximation is rigorously fulfilled when the ionic and the electronic particles have the same size. Indeed, the penetration length is then simply given by ℓ = 𝑎 2 × 〈𝐷 𝑖=𝑖𝑜 𝑜𝑟 𝑒𝑙 〉 (with 𝑎 2 = (1 -𝑐𝑜𝑠〈𝛩 𝑖=𝑖𝑜 𝑜𝑟 𝑒𝑙 〉)). However, this relation is also valid when one of the mean phase diameters is significantly higher than the other (in this case, ℓ = 𝑎 2 × min {〈𝐷 𝑒𝑙 〉, 〈𝐷 𝑖𝑜 〉}. Therefore, by combining ( 4), ( 18) and ( 19), the interfacial specific surface area between the ionic and electronic phases can be given as a function of the geometrical attributes of the microstructure:

𝑆 𝑒𝑙/𝑖𝑜 = 𝜋 8 𝛼 (1 + 𝛼)² 𝜃 𝑝 (𝑙𝑛𝜃 𝑝 )² 𝛽 𝑖𝑜 𝛽 𝑒𝑙 𝑎 √max {〈𝐷 𝑒𝑙 〉, 〈𝐷 𝑖𝑜 〉} × (〈𝐷 𝑖𝑜 〉 + 〈𝐷 𝑒𝑙 〉) × (1 + 𝐶𝑉 𝑖𝑜 2 ) (1 + 3𝐶𝑉 𝑖𝑜 2 + 𝛾 𝑖𝑜 𝐶𝑉 𝑖𝑜 3 ) (1 + 𝐶𝑉 𝑒𝑙 2 ) (1 + 3𝐶𝑉 𝑒𝑙 2 + 𝛾 𝑒𝑙 𝐶𝑉 𝑒𝑙 3 ) ( 20
)
where 𝑎 is a constant of proportionality (depending on 𝑎 1 and 𝑎 2 ). As for the TPBl, this correlation is expressed for the whole ionic/electronic surface area including the connected and disconnected phases. However, it must be still valid for the connected phases as long as all the electrode phases are above the percolation threshold.

Methodology for calibrating and validating the microstructural correlations

Calibration of the semi-analytical correlations

The "basic" microstructural parameters used in the correlations are associated to the geometric attributes that describe the morphology of the electrode particles collection. If the sintering temperature is not too high, these parameters must not be too far from the geometrical characteristics of the initial powders [START_REF] Zhang | Random-packing model for solid oxide fuel cell electrodes with particles size distributions[END_REF]. However, the full relevance of this claim is questionable as the solid phases in real electrodes exhibit a continuous shape with no remanence of the initial geometry of the powder particles [START_REF] Prakash | Properties and development of Ni/YSZ as an anode material in solid oxide fuel cell: A review[END_REF][START_REF] Moussaoui | Stochastic Geometrical Modeling of Solid Oxide Cells Electrodes Validated on 3D Reconstructions[END_REF][START_REF] Neumann | Stochastic 3D modeling of complex three-phase microstructure in SOFC electrodes with completely connected phases[END_REF]. Nevertheless, considering the real electrodes, the "basic" microstructural parameters can still be measured by image analysis on 3D reconstructions (or assessed on 2D micrographs with stereological assumptions). For example, the parameters 〈𝐷 𝑖 〉, CVi and 𝛾 𝑖 can be measured using the concept of continuous PSD [START_REF] Serra | Image analysis and mathematical morphology: Theoretical advances[END_REF]. Therefore, only the determination of the ratio 𝛽 𝑒𝑙 , 𝛽 𝑖𝑜 and the parameter 𝑎 is not straightforward.

In the present work, the correlations have been calibrated thanks to the generation of a large database of representative 3D synthetic microstructures. In other words, the unknown parameters 𝛽 𝑒𝑙 , 𝛽 𝑖𝑜 and 𝑎 of the correlations have been fitted on the numerical dataset computed on the 3D volumes. Besides, the consistency of the microstructural correlations have been checked by varying independently (i) the electrode composition, (ii) the porosity and (iii) the mean particle size of the ionic and electronic phases (which constitute the main parameters that can be easily adapted for an electrode microstructural optimization).

Generation of the synthetic microstructures

It is worth noting that a special attention has been paid to generate synthetic microstructures which are representative of the morphology of real typical SOC electrodes. In practice, two distinct 3D stochastic models for the generation of synthetic microstructures have been used.

Both methods and their validation have been detailed elsewhere [START_REF] Moussaoui | Stochastic Geometrical Modeling of Solid Oxide Cells Electrodes Validated on 3D Reconstructions[END_REF][START_REF] Moussaoui | [END_REF]. The first method is based on an original sphere-packing algorithm [START_REF] Moussaoui | [END_REF] shortly described hereafter. The empty volume is firstly divided in sub-domains in which initial spheres are randomly positioned.

Then, in order to fill the space, new spheres are added in an iterative process up to reach the targeted electrode density. In the procedure, both distributions in the particle size and their overlapping are controlled. Finally, the microstructure is smoothed with a specific geometrical operation (morphological opening) to remove the unrealistic spherical shapes of the solid phases. The second method is based on a geostatistical modelling approach where the microstructure is simulated using morphological functions. In our case, the truncated plurigaussian random field method has been adapted to simulate the SOCs electrode microstructures [START_REF] Moussaoui | Stochastic Geometrical Modeling of Solid Oxide Cells Electrodes Validated on 3D Reconstructions[END_REF]. Basically, two Gaussian random noises are independently convoluted with two distinct weight functions that contain the statistical information of the composite electrode to mimic. The resulting correlated bigaussian field is then thresholded to obtain the final microstructure.

The representativeness of the two models [START_REF] Moussaoui | Stochastic Geometrical Modeling of Solid Oxide Cells Electrodes Validated on 3D Reconstructions[END_REF][START_REF] Moussaoui | [END_REF] has been checked on the same 3D reconstruction of a typical Ni-YSZ cermet [START_REF] Hubert | Efficient correction of wavefront inhomogeneities in X-ray holographic nanotomography by random sample displacement[END_REF] used as reference for the database generation (named Electrode_1 in the article). It can be noticed that, being radically different, both approaches lead to produce distinct microstructures even if they have to emulate the same porous medium. Therefore, the two models must provide two approximations of the real microstructures, and hence, two approximations for the correlations calibrated on the two numerical datasets.

Validation of the microstructural correlations

Finally, the validity of the correlations to predict the microstructural properties has been checked thanks to various reconstructions acquired on different electrodes. In this frame, the reconstruction of a second typical cermet published in [START_REF] Moussaoui | Stochastic Geometrical Modeling of Solid Oxide Cells Electrodes Validated on 3D Reconstructions[END_REF] has been used. Besides, a typical O2 electrode made of LSCF and CGO (Electrode_3) has been specifically reconstructed by FIB-SEM according to the protocol detailed in [START_REF] Vivet | 3D Microstructural characterization of a solid oxide fuel cell anode reconstructed by focused ion beam tomography[END_REF]. The ability of the correlation to reproduce the loss of TPBl after Ni agglomeration has also been checked on Electrode_1 thanks to 3D volumes reconstructed after long-term operation [START_REF] Hubert | Impact of Nickel agglomeration on solid oxide cell operated in fuel cell and electrolysis modes[END_REF].

Visualization of the real and synthetic electrodes microstructures

The 3D rendering volumes of the three electrode reconstructions are provided in Fig. 2. The synthetic microstructures (digital twins) generated by the sphere-packing and the random fields methods are also given for Electrode_1 taken as reference for the sensitivity analysis. The simulated volumes obtained by changing the electrode porosity are also provided as an illustration. A volume of 15x15x15 µm 3 with a resolution of 50 nm is taken for the synthetic microstructures. As already discussed in Mousaoui et al. [START_REF] Moussaoui | Stochastic Geometrical Modeling of Solid Oxide Cells Electrodes Validated on 3D Reconstructions[END_REF], this size of the 3D volume is sufficiently large to reach a good representativeness of the heterogeneous electrode microstructure (so that it can be considered as a Representative Volume Element (RVE) of the microstructure for which only one realization is required).

The microstructural properties have been measured on the synthetic microstructures or real reconstructions thanks to a set of in-house programs already detailed in [START_REF] Usseglio-Viretta | Quantitative microstructure characterization of a Ni-YSZ bi-layer coupled with simulated cathode polarisation[END_REF][START_REF] Moussaoui | Stochastic Geometrical Modeling of Solid Oxide Cells Electrodes Validated on 3D Reconstructions[END_REF][START_REF] Laurencin | Characterisation of SOFC Ni-8YSZ substrate by synchrotron X-ray nano-tomography. From 3D reconstruction to microstructure quantification[END_REF]. It can be noticed that, in our case, the continuous PSD is computed according to the procedure given in Gelb et al. [START_REF] Gelb | Pore size distributions in porous glasses: a computer simulation study[END_REF]. It is defined by the probability that a voxel belongs to the largest sphere that does not overlap any complementary phase [START_REF] Usseglio-Viretta | Quantitative microstructure characterization of a Ni-YSZ bi-layer coupled with simulated cathode polarisation[END_REF][START_REF] Gelb | Pore size distributions in porous glasses: a computer simulation study[END_REF]. For the continuous shape of the real electrode phase (Fig. 2), it is reminded that the parameters describing the distribution are thus not related to a collection of discrete particles. As a consequence, the ratios 𝛽 𝑒𝑙 , 𝛽 𝑖𝑜 and the parameter 𝑎 for the sintered SOC electrodes cannot be directly associated to the geometrical shapes of distinct particles. In the present work, the parameters 𝛽 𝑒𝑙 , 𝛽 𝑖𝑜 , 𝑎 are thus fitted on the 3D microstructures, in such a way that the relationships for LTPB and Si/j given by eqs ( 4), ( 16) and ( 20) can be seen as semi-analytical correlations. As the correlations have been established for the specific surface area including the connected and disconnected phases, they have been calibrated on the 3D microstructures taking into account the total phases. However, the analyses have been conducted over a range of porosities and composition for which each phase remains almost fully percolated. In practice, the fitting procedure has been performed while keeping the electrode porosity and the ionic and electronic volume fractions higher than the phase percolation threshold (around 18%). Over this range of electrode compositions, the phase percolation remains higher than 93% so that the correlations should be still valid if considering only the connected phases. The full validity of this last assumption has been a posteriori verified by comparing the predicted surface area with the data obtained on the percolated phases.

Ionic/pores interfacial specific surface area

The ionic/pores interfacial surface area 𝑆 𝑖𝑜/𝑝 calculated with eq. ( 16b) is compared to the numerical values computed on the 3D microstructures by changing the porosity 𝜃 𝑝 , the composition 𝛼 = 𝜃 𝑖𝑜 /𝜃 𝑒𝑙 and the mean phase diameters 〈𝐷 𝑖𝑜 〉 and 〈𝐷 𝑒𝑙 〉 in Fig. 3a, 3b and 3c, respectively. As a general comment, a good agreement is observed between the whole numerical dataset, generated either with the Random Field (RF) model or the Sphere Packing (SP) model, and the semi-analytical correlation (16b) for which only one scaling parameter 𝛽 𝑖𝑜 has been fitted.

As previously mentioned, some specific surface areas have also been calculated on the 3D microstructures by considering only the connected phases. These computations have been performed for the microstructural parameters taken at three bounds in the sensitivity analysis: pores, Ni or YSZ volume fractions equal to 18%. As expected, it can be seen in Fig. 3 that the bias, which is introduced with only the percolated phases, remains negligible (as long as the phase volume fractions are not lower than 18%). Fig. 3: Comparison between the interfacial specific surface area 𝑆 𝑖𝑜/𝑝 calculated with the correlation (16b) and the data measured on the 3D volumes generated by the RF and SP models. The surface area 𝑆 𝑖𝑜/𝑝 is plotted versus the porosity (a), the electrode composition (b) and the mean phase diameters (c). The crosses indicate the values for 𝑆 𝑖𝑜/𝑝 calculated with (16b) considering only the connected phases (for 𝜃 𝑁𝑖 , 𝜃 𝑌𝑆𝑍 or 𝜃 𝑝 = 0.18). The scaling parameter 𝛽 𝑖𝑜 in (16b) has been fitted to 𝛽 𝑖𝑜 𝑅𝐹 = 3.369 and 𝛽 𝑖𝑜 𝑆𝑃 = 3.476 on the RF and SP microstructures, respectively. According to eq. (16b), the ionic/pores interfacial surface area is proportional to 𝑆 𝑖𝑜/𝑝 ∝ -𝜃 𝑝 𝑙𝑛[𝜃 𝑝 ] meaning that the property exhibits a maximum at intermediate porosities. This evolution is well retrieved by both SP and RF models as shown in Fig. 3a. This behavior is easily explained since the surface area tends to zero in the case of a dense or a highly porous medium. Moreover, the surface area 𝑆 𝑖𝑜/𝑝 is a function of the electrode composition with a given in eq. (16b). As discussed for the porosity, this dependence is also rather well reproduced by the numerical data (Fig. 3b). It can also be noticed that the ionic/pore surface area increases with the ratio 𝛼 up to reach a plateau. Indeed, the contact surface area between the ionic phase and the porosity must increase with increasing the ionic content in the electrode composition. Finally, regarding eq. ( 16b), 𝑆 𝑖𝑜/𝑝 must be independent on the mean diameter of the electronic conducting phase whereas it is inversely proportional to the mean diameter of the ionic phase. Both of these dependences are perfectly confirmed by the RF and SP numerical models (Fig. 3c). This behavior is explained since a very fine microstructure for the ionic network must result in a high ionic/pore surface area irrespective of the size of the electronic phase.

Electronic/pores interfacial specific surface area

Following the same procedure as 𝑆 𝑖𝑜/𝑝 , the semi-analytical expression (16a) for the electronic/pores interfacial surface area 𝑆 𝑒𝑙/𝑝𝑜𝑟𝑒 has been calibrated with the same database of numerical microstructures. Similarly to the previous property, a general good consistency has been found with the two datasets obtained with the RF and SP models. The reader is invited to refer to the supplementary for the plots of 𝑆 𝑒𝑙/𝑝𝑜𝑟𝑒 as a function of 𝜃 𝑝 , 𝛼 and 〈𝐷 𝑖 〉. It has been confirmed using the numerical dataset that 𝑆 𝑒𝑙/𝑝𝑜𝑟𝑒 follows the same evolution with the porosity as 𝑆 𝑖𝑜/𝑝𝑜𝑟𝑒 . Besides, the electronic/pore surface area has been found to be inversely proportional to the mean electronic phase diameter as predicted by eq. (16a). Finally, it has also been checked that 𝑆 𝑒𝑙/𝑝𝑜𝑟𝑒 is directly proportional to the relative volume fraction of the electronic phase in the solid (i.e. ).

In order to summarize all the results, the ionic/pores and electronic/pores interfacial specific surface areas measured on the set of synthetic microstructures have been plotted as a function of the semi-analytical estimations in Fig. 4. In this representation, all the points are rather well aligned proving the reliability of the proposed correlation for both RF and SP models.

Correlation for the electronic/ionic interfacial specific surface area

As shown in eq. ( 20), the expression of the electronic/ionic interfacial specific surface area is scaled by the product 𝛽 𝑖𝑜 × 𝛽 𝑒𝑙 × 𝑎 . Knowing that 𝛽 𝑖𝑜 and 𝛽 𝑒𝑙 have already been determined on the previous relationships, only the scaling parameter 𝑎 has been fitted for this correlation. The dependences of 𝑆 𝑒𝑙/𝑖𝑜 with the porosity, the composition and the mean phase diameters are shown in Fig. 5a, 5b and 5c. It can be seen that the proposed equation is able to predict accurately all the evolutions of the studied property measured on the synthetic microstructures for both the RF and SP models. Moreover, in accordance with the previous relations for 𝑆 𝑒𝑙/𝑝𝑜𝑟𝑒 and 𝑆 𝑖𝑜/𝑝𝑜𝑟𝑒 , it has been checked that the semi-analytical relation for 𝑆 𝑒𝑙/𝑖𝑜 is still valid for the percolated phases over the investigated range of microstructures (Fig. 5). Fig. 5: Comparison between the specific interfacial surface area 𝑆 𝑒𝑙/𝑖𝑜 calculated with the correlation [START_REF] Lay-Grindler | Micromodelling of Solid Oxide Electrolysis Cell: from performance to durability[END_REF] and the data measured on the 3D dataset for the RF and SP models. The surface area 𝑆 𝑒𝑙/𝑖𝑜 is plotted versus the porosity (a), the electrode composition (b) and the mean phase diameters (c). The crosses indicate the values for 𝑆 𝑒𝑙/𝑖𝑜 calculated with [START_REF] Lay-Grindler | Micromodelling of Solid Oxide Electrolysis Cell: from performance to durability[END_REF] considering only the connected phases (for 𝜃 𝑁𝑖 , 𝜃 𝑌𝑆𝑍 or 𝜃 𝑝 = 0.18). Correlation calibrated with 𝛽 𝑖𝑜 𝑅𝐹 = 3.369, 𝛽 𝑒𝑙 𝑅𝐹 = 2.476, 𝑎 𝑅𝐹 = 3.661 and 𝛽 𝑖𝑜 𝑆𝑃 = 3.476, 𝛽 𝑒𝑙 𝑆𝑃 = 2.454 and 𝑎 𝑆𝑃 = 3.580 on the RF and SP microstructures, respectively. Unlike the specific surface area at electronic/pore or ionic/pore interfaces, the contact surface area between the two solid phases decreases continuously as a function of the porosity (Fig. 5a). Indeed, the volume fractions of both solid phases, and hence their contact, decreases when the microstructure is more porous. Besides, 𝑆 𝑒𝑙/𝑖𝑜 presents a maximum for 𝛼 = 1 when the amounts of ionic and electronic phases are equal (Fig. 5b). As shown in eq. ( 20), the dependence of 𝑆 𝑒𝑙/𝑖𝑜 with the mean phase diameter is more complex than the one established for the surface area between the solid phases and the porosity. Indeed, the analytical correlation (20) predicts a coupled dependence in the mean diameters for both solid phases: 𝑆 𝑒𝑙/𝑖𝑜 ∝ 1 √max {〈𝐷 𝑒𝑙 〉,〈𝐷 𝑖𝑜 〉}×(〈𝐷 𝑖𝑜 〉+〈𝐷 𝑒𝑙 〉)

. Therefore, the electronic/ionic surface area must decrease when both mean diameters increase. More precisely, two regimes can be distinguished when changing one diameter (while the other remains constant in the sensitivity analysis). For instance, let one consider in Fig. 5c the case of changing 〈𝐷 𝑒𝑙 〉 while 〈𝐷 𝑖𝑜 〉 is kept constant to the reference value 〈𝐷 𝑖𝑜 〉 =0.34 µm. If 〈𝐷 𝑒𝑙 〉 > 〈𝐷 𝑖𝑜 〉, then the specific surface area is given by 𝑆 𝑒𝑙/𝑖𝑜 ∝ 1 √〈𝐷 𝑒𝑙 〉√〈𝐷 𝑒𝑙 〉+0.34

. Inversely, when 〈𝐷 𝑒𝑙 〉 < 〈𝐷 𝑖𝑜 〉, the equation is reduced to

𝑆 𝑒𝑙/𝑖𝑜 ∝ 1 √〈𝐷 𝑒𝑙 〉+0.34
. This behavior predicted by the analytical model involves two slopes in the curve 𝑆 𝑒𝑙/𝑖𝑜 = 𝑓(〈𝐷 𝑒𝑙 〉) plotted in Fig. 5c. This evolution is perfectly reproduced by the data computed on the synthetic microstructures generated by the two models (Fig 5c). This result allows validating the mean diameters dependency depicted in eq. ( 20).

As expected, the error given by the correlation is minimum when the two mean diameters are similar in the sensitivity analysis (i.e. around the reference electrode for 〈𝐷 𝑖𝑜 〉 ≈ 0.34 µm).

Indeed, there is no approximation in the expression of the penetration length when the two diameters are equal as discussed in Section 2.2.4. Nevertheless, the error introduced when the two mean diameters differ from each other remains negligible as it can be seen in Fig. 5c.

Consequently, the expression remains valid over a wide range of fine or coarse microstructures for the ionic or electronic phases.

All the results of the sensitivity analysis are reported in Fig. 6a. The values coming from the analytical correlation are compared to the numerical data measured on the 3D volumes for both SP and RF models. For all the dataset, the analytical predictions match all the values measured on the synthetic microstructures, highlighting the global consistency of the proposed correlations.

Correlation for the density of TPBl

The density of TPBl has been calculated with the correlation (4) by keeping the same scaling parameters 𝛽 𝑖𝑜 and 𝛽 𝑒𝑙 previously determined for the specific surface areas. In other words, no additional fitting parameter has been considered at this stage.

In this way, it has been found that the correlation ( 4) is able to accurately capture the TPBl evolutions as a function of the porosity, the electrode composition and the mean diameters of the electronic and ionic conducting phases measured on the synthetic microstructures (cf. supplementary). All the results are summarized in Fig. 6b where the densities of TPBl calculated with the semi-analytical correlation have been confronted to the data measured on the RF and SP synthetic microstructures. As shown in Fig. 6b, a good agreement is found between the analytical predictions and the numerical dataset, meaning that eq. ( 4) provides an accurate estimation of TPBl density for a large range of representative microstructures. It is worth reminding that Zhang et al. [START_REF] Zhang | Computer simulations of topological connectivity of the triple phase boundaries in solid oxide fuel cell composite cathodes[END_REF] have already checked the validly of eq. ( 4) with simulated microstructures made of spherical powders or needle-shape particles (cf. Section 2.1).

In their case, the electrode was thus composed by a collection of well-defined particles.

Therefore, as predicted by the theory, it was possible to ascribe 𝛽 𝑖𝑜 and 𝛽 𝑒𝑙 to the shape of the constitutive element of the microstructures. In this work, it has been shown that the correlation can still be used for typical sintered SOCs electrodes with phases exhibiting a continuous and smooth shape (Fig. 2). However, in this particular case, it is reminded that the correlation is not fully analytical since 𝛽 𝑖𝑜 and 𝛽 𝑒𝑙 , which can no more be related to the particle shape, have to be considered as simple scaling parameters (without any geometrical meaning).

As done before with the interfacial specific surface areas, the active density of TPBl has been As mentioned for the specific surface area, it can be observed that eq. ( 4) is able to accurately predict the active TPBl as long as the electrode phases are well connected. Indeed, it has been found that the TPBl calculated with eq. ( 4) matches the data measured on the percolated microstructures as long as each volume fraction is higher than 18 %. It can be noticed that the condition to calculate the active density of triple-phase boundary is more restrictive than the one requested for the specific surface area. Indeed, the three electrode phases have to be connected at the same time for the active TPBls whereas only two are necessary for Si/j.

Discussion and validation

The consistency of the calibrated correlations with respect to the RF and SP models are discussed as a first element of validation. The final check-up of the model reliability is then conducted by comparing the predictions with the properties measured on the real electrodes reconstructions.

Model prediction with respect to the RF and SP microstructures

The calibration of the previous correlations for Si/j and TPBl has been carried out on two sets of representative synthetic microstructures generated by two fundamentally different mathematical models. In spite of the large range of investigated cases in the sensitivity analyses, it has been shown that the semi-analytical equations are able to capture all the microstructural evolutions with only three fitting parameters. This preliminary remark constitutes a first evidence for the model validation. Moreover, the scaling parameters are equal to 𝛽 𝑖𝑜 𝑅𝐹 = 3.369, 𝛽 𝑒𝑙 𝑅𝐹 = 2.476 and 𝑎 𝑅𝐹 = 3.661 for the correlations calibrated on the dataset coming from the random field microstructures. By using the sphere packing algorithm, the fitting parameters have been estimated to 𝛽 𝑖𝑜 𝑆𝑃 = 3.476, 𝛽 𝑒𝑙 𝑆𝑃 = 2.454 and 𝑎 𝑆𝑃 = 3.580. Therefore, it can be claimed that there is a very good consistency between the two sets of scaling parameters fitted on the RF and SP microstructures. It is worth reminding that the two numerical models, which have been validated on one single electrode reconstruction, must provide two approximations of the real electrodes when changing the 'basic' parameters of the microstructure such as the porosity. From that point of view, the two sets of calibrated semi-analytical correlations should converge toward two independent estimations of the reality. As the scaling parameters are very close to each other, it means that the two predictions remain fully coherent when considering the two approximations given by the RF and SP models. As a consequence, the semi-analytical correlations should provide two accurate predictions over a large range of real electrodes microstructures. In order to check this assertion, the predictions of the correlations have been compared to the properties measured on various electrode reconstructions.

Model validation on diverse electrodes reconstructions

In Table I, the properties measured on the reconstructions of electrode_1, electrode_2 and electrode_3 (Fig. 2a) are compared to the data calculated with the semi-analytical correlations.

As a general comment, it can be seen that a good agreement is found between the properties measured on the 3D volumes and the ones calculated with the calibrated semi-analytical equations. Indeed, whatever the electrodes or the calculated properties, the error on the parameters estimations does not exceed 12% with an average of 5.4%. More precisely, the inspection of the data reported in Table I reveals that the correlations calibrated on the SP and RF microstructures provide in average almost the same estimation with respect to the real electrode properties (mean error of 5.5% and 5.3% for the SP and RF models, respectively).

Therefore, the difference in the parameters coming from the two calibrations on the RF and SP microstructures is quite negligible, confirming that both sets of correlations provide a good approximation of the reality.

Electrode_2

(Ni-YSZ)

Electrode_2

(Ni-YSZ) The semi-analytical equations are thus able to accurately predict the properties for the typical Ni-YSZ cermets of Electrode_1 and Electrode_2 as well as for the LSCF-CGO composite of Electrode_3. In the latter case, the electrode microstructure is much finer than the two cermets with lower mean diameters for the ionic and electronic conducting phases (cf. microstructures displayed in Fig. 2a with {〈𝐷 𝐿𝑆𝐶𝐹 〉 = 0.17 µ𝑚; 〈𝐷 𝐶𝐺𝑂 〉 = 0.16 µ𝑚 for Electrode_3} < {〈𝐷 𝑁𝑖 〉 = 0.34 µ𝑚, 〈𝐷 𝑌𝑆𝑍 〉 = 0.28 µ𝑚 for Electrode_1}). This finer microstructure for Electrode_3 results in a significantly higher density of TPBl that is well captured by the semi-analytical relation. As a conclusion, it can be inferred from this analysis that the proposed correlations are able to accurately predict the density of TPBl and the interfacial surface areas for different kinds of SOC electrodes (manufactured by the classical manufacturing routes for ceramics).

Model validation on the loss of TPBl after Ni agglomeration

For Electrode_1, some reconstructions have been also acquired after ageing at 850°C. The experimental conditions of the long-term testing and the microstructural characterization have been detailed in [START_REF] Hubert | Impact of Nickel agglomeration on solid oxide cell operated in fuel cell and electrolysis modes[END_REF]. It was found that the Ni phase undergoes a substantial particle coarsening in operation that was modeled by a standard Ostwald ripening power-law equation:

[〈𝐷 𝑁𝑖 (𝑡)〉 𝑛 -〈𝐷 𝑁𝑖 (𝑡 = 0)〉 𝑛 ] = 𝑘 𝑝𝑜𝑤 × 𝑡 [START_REF] Moçoteguy | A review and comprehensive analysis of degradation mechanisms of solid oxide electrolysis cells[END_REF] where 𝑡 is the time, 𝑘 𝑝𝑜𝑤 the rate constant and 𝑛 an exponent which is characteristic of the predominant mass transport in the sintering process. The grain-size growth in the cermet was measured on the reconstructed electrodes and the best adjustment of eq. ( 21) on the experimental data was found for 𝑛 = 8. It was also highlighted that the Ni phase coarsening leads to a substantial decrease in TPBl whereas the volume fractions and the mean phase diameters for the complementary phases remain quite unaffected by the agglomeration.

To check the possibility to use the present microstructural modeling to reproduce the loss of TPBl, eq. ( 21) has been combined with the correlation (4) calibrated with the SP microstructures. Without any supplementary fitting, the loss of TPBl given by the semianalytical relations (4) and eq. ( 21) is compared in Fig. 8 to the values measured on the reconstructions. Even if the scattering on the experimental data is rather significant, the two equations are able to predict the evolution of TPBl over the time. This result brings a supplementary evidence on the reliability of the correlation. Besides, it can be mentioned that the system of eqs. ( 4) and ( 21 Density of active TPBl measured on 3D reconstructions [START_REF] Hubert | Impact of Nickel agglomeration on solid oxide cell operated in fuel cell and electrolysis modes[END_REF] Correlation (4) combined with eq. ( 21)

Finally, some authors have also investigated the evolution of the cermet microstructural properties before and after Ni agglomeration by 3D electrode reconstructions [START_REF] Faes | Nickel-zirconia anode degradation and triple phase boundary quantification from microstructural analysis[END_REF][START_REF] Lee | Degradation of nickel-yttria-stabilized zirconia anode in solid oxide fuel cells under changing temperature and humidity conditions[END_REF]. For instance, Faes et al. [START_REF] Faes | Nickel-zirconia anode degradation and triple phase boundary quantification from microstructural analysis[END_REF] have reported a percentage decrease of 25% in the density of TPBl (after 1130h for 'Stack A') whereas Lee et al. [START_REF] Lee | Degradation of nickel-yttria-stabilized zirconia anode in solid oxide fuel cells under changing temperature and humidity conditions[END_REF] have found, in their experimental conditions, a percentage drop of 55% (after 50h at 1200°C). The change in the 'basic' microstructural parameters provided in the two articles have been used to calculate the TPBls decrease with the correlation (4). The model leads to an estimation of the percentage loss of 26% and 50% (instead of 25% and 55%, respectively). The good agreement between the published data and the estimated values also highlights the model validity.

Conclusion

The concept of "extended" microstructure has been applied to express the correlations relating the key microstructural properties, which control the electrochemistry, to the 'basic' parameters of the microstructure. This approach was initially employed in [START_REF] Gokhale | A stochastic geometry model for total triple phase boundary length in composite cathodes for solid oxide fuel cells[END_REF] to establish the relation linking the density of TPBl as a function of (i) the porosity, (ii) the electrode composition, (iii) the geometrical shape and size distribution of particles for the electronic and ionic conducting phases. Following the same approach, the method has been applied in the present work to derive the equations for the ionic/pore, electronic/pore and ionic/electronic interfacial specific surface areas.

The correlations have been studied for the classical SOCs electrodes for which the phases exhibit a continuous shape without remanence of the initial powder particles after sintering. In that case, the three parameters associated to the geometry of the particles and their contacts in the final microstructure have been considered as simple scaling factors.

These scaling parameters have been determined by fitting the semi-analytical relations on a large dataset of properties measured on representative synthetic microstructures generated by two different mathematical methods (i.e. the Gaussian Random Field and the Sphere Packing models). In both cases, the semi-analytical correlations have been found to capture all the evolutions in terms of porosity, electrode composition and mean phase diameters. As long as the phases are well percolated (i.e. for volume fractions higher than 18%), it has been found that the correlations remain valid.

The validity of the calibrated correlations has been discussed with respect to the two numerical methods used for the microstructural generation. It has been shown that the correlations were able to accurately fit the two datasets that provide two independent but close estimations of the real microstructures. To go further in the model validation, the properties calculated by the semi-analytical equations have been also compared to the data measured on several 3D electrode reconstructions. It has been found that the correlations provide accurate predictions for typical Ni-YSZ cermets and LSCF-CGO composite electrodes. Indeed, the estimated average on the error between the estimated properties and the real data are lower than 5.4%, demonstrating the reliability of the proposed semi-analytical correlations. Finally, it has also been shown that the TPBl correlation combined with a power-law model describing the particle growth due to Ni coarsening is able to capture the loss of active TPBl in operation.

Fig. 2 :

 2 Fig. 2: (a): 3D rendering volumes for the reconstructions of Electrode_1 (Ni-YSZ), Electrode_2 (Ni-YSZ) and Electrode_3 (LSCF-CGO). (b) and (c): the synthetic microstructures of Electrode_1 generated by the Gaussian random field and the sphere packing models are shown with a sensitivity analysis on the porosity.

1

 1 Correlations for the electronic/pores and ionic/pores interfacial specific surface area

Fig. 4 :

 4 Fig. 4: Ionic/pores and electronic/pores interfacial specific surface areas estimated with the semi-analytical eqs. (16b) and (16a) are plotted as a function of the data measured on the set of RF and SP synthetic microstructures. The scaling parameters 𝛽 𝑖𝑜 and 𝛽 𝑒𝑙 have been fitted on the RF and SP microstructures to: 𝛽 𝑖𝑜 𝑅𝐹 = 3.369, 𝛽 𝑖𝑜 𝑆𝑃 = 3.476, and 𝛽 𝑒𝑙 𝑅𝐹 = 2.476, 𝛽 𝑒𝑙 𝑆𝑃 = 2.454.

Fig. 6 :

 6 Fig. 6: Electronic/ionic interfacial specific surface area and density of TPBl estimated with the semi-analytical eqs (20) and (4) are plotted as a function of the data measured on the set of RF and SP synthetic microstructures. Correlation calibrated with 𝛽 𝑖𝑜 𝑅𝐹 = 3.369, 𝛽 𝑒𝑙 𝑅𝐹 =

  computed on the synthetic volumes by considering only the percolated phases. The dataset has been compared to the prediction of the semi-analytical correlation (4) depending on the phase volume fractions. The results are illustrated in Fig 7 changing the ionic content in the electrode.

Fig. 7 :

 7 Fig. 7: Comparison between the density of TPBl calculated with the correlation (4) and the ones measured on the total and connected phases on the SP synthetic microstructures as a function of the ionic content in the electrode (𝛽 𝑖𝑜 𝑆𝑃 = 3.476, 𝛽 𝑒𝑙 𝑆𝑃 = 2.454).

  fraction θ io(-)

  ) constitutes a physically-based model to compute the loss of TPBl induced by the Ni agglomeration.

Fig. 8 :

 8 Fig. 8: Comparison between the loss of active TPBl measured on several 3D reconstructions after operation and the prediction combining the correlation (4) with the power-law model (21) for the Ni particle coarsening (𝛽 𝑖𝑜 𝑆𝑃 = 3.476, 𝛽 𝑒𝑙 𝑆𝑃 = 2.454).

  𝛼 (where 𝜃 𝑖 and represents the phase volume fraction with 𝜃 𝑖𝑜 = 𝑉 𝑖𝑜

	𝜃 𝑖𝑜 𝜃 𝑒𝑙	= 𝑉 𝑡𝑜𝑡	𝑎𝑛𝑑 𝜃 𝑒𝑙 =	𝑉 𝑒𝑙 𝑉 𝑡𝑜𝑡

Table I .

 I Comparison between the properties measured on the 3D reconstructions and the predictions given by the correlations.

			LTPB (µm -2 )			Sio/el (µm -1 )
		Measured on	Correlation (4)	Correlation (4)	Measured on	Correlation (20)	Correlation (20)
		the	calibrated with SP	calibrated with	the	calibrated with	calibrated with
		reconstruction	model	RF model	reconstruction	SP model	RF model
	Electrode_1 (Ni_YSZ)	4.75	4.28 (Error=-9.8%)	4.19 (Error=-11.8%)	1.59	1.59 (Error=+0.0%)	1.59 (Error=+0.0%)

Acknowledgement

The research leading of these results has received funding from the European Horizon 2020 -Research and Innovation Framework program (H2020-JTI-FCH-2015-1) under grant agreement n°735918 (INSIGHT project) and n°699892 (ECO project). The work has also been partially supported by the French National Agency (ANR) (ECOREVE project) and the Carnot Institute Energies du Futur (DURASOC project).

List of symbols

Roman symbols

𝑎

Scaling parameter in the correlations for Sio/el (-)

〈𝑑 𝑐 〉 Mean value of the contact diameter between the ionic and electronic particles (m)


A Rousseau

S Darbon

P Troussel

T Caillaud

J L Bourgade

G Turk

E Vigne

M Hamel

J Larour

D Bradley

V Smalyuk

P Bell

Development of a hardened imaging system for the Laser MegaJoule

The Laser MegaJoule (LMJ) facility will host inertial confinement fusion experiments in order to achieve ignition by imploding a Deuterium-Tritium microballoon. In this context an X-ray imager is necessary to diagnose the core size and shape of the DT-target in the 10-100 keV band in complement of neutron imaging system. Such a diagnostic will be composed of two parts: an X-ray optical system and a detection assembly. Each element will be affected by the harsh environment created by fusion reactions.

INTRODUCTION

X-ray imaging diagnostics are fundamental tools for inertial confinement fusion experiments in order to achieve ignition. Reasons for failure may be investigated by evaluating the quality of implosion symmetry. The diagnostic will record the core size and shape of a cryogenic Deuterium-Tritium imploding target in harsh environment. Perturbations will be induced by fluxes of neutrons and gamma rays which affect diagnostic performances. The diagnostic shall operate under a maximum neutron yield of 10 16 .

CONTEXT

DT fusion reactions yield a large number of neutrons whose energy distribution has its maximum at 14 MeV. Figure 1b reveals that the facility equipments, mainly the target chamber and diagnostics, will modify the initial neutron spectrum due to inelastic scattering. Gamma rays will be generated in addition to neutrons by two main reactions : DT fusion reactions emit quasi monochromatic 16.8 MeV gamma rays and the (n, n' ) reactions on surrounding materials will induce an increasing gamma ray production.

Due to a lower velocity than gamma rays, neutrons reach the target chamber wall after 100 ns increasing the dose dramatically as shown in figure 1a. The diagnostic will then benefit from a relatively quiet radiative environment in the first 100 ns of the experiment. Consequently, its architecture and its a e-mail: adrien.rousseau@cea.fr This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. implantation in the LMJ facility will take into account of this radiative surrounding and its temporal evolution.

Previous simulations and experiments on laser facilties have shown that the use of detectors inside the target chamber will at least reduce their performances, at worst destroy them [1]. So an alternative design represented in figure 2 is considered: an X-ray imaging system images the DT target on a scintillator which converts rapidly X-rays into visible light. The resulting image in the visible spectrum is then transfered through a 7 meter long optical relay to a shielded box where the detection is realized. The shielded box which can bear a maximum shielding weight of 28 tons will reduce significantly the radiative environment. The whole system shall have a final resolution of 10 mm within a 500 mm field of view (FOV).

DIAGNOSTIC DEVELOPMENT

Scintillators issues and solutions

The scintillator used for the diagnostic must meet several requirements. Among them: x-ray absorption, decay time and scintillation wavelength. In order to fit with the 100 ns time window of acquisition, the scintillator's decay time should not exceed 10 ns while absorbing hard X-rays in the spectral bandwidth of 10-100 keV without resolution degradation. Furthermore, in order to distinguish the scintillator's signal light from the parasitic light generated by Cerenkov effect in the optical relay, the scintillation wavelength shall be higher than 550 nm.

No commercial scintillator 1,2 meeting all these specifications is currently available. Thus, homemade scintillator developments are in progress.

Improvement of inorganic scintillators decay time is a very hard task due to their crystalline structure. On the contrary, organic scintillators with low decay time can be easily modified to achieve requirements. Indeed, by embedding a high Z number element, like lead, into the polymer chain, the effective Z number of the scintillator increases drastically from 6 to 53, improving its X-ray absorption which increases reasonably its decay time as shown in figure 3a. Such effective Z number implies that the photoelectric effect is dominant in absorption process, consequently intrinsic spatial resolution is not affected by X-ray scattering induced by Compton effect. The use of three different fluorophores dispersed into the polymer matrix shifts the emission wavelength to 550 nm. First attempt of homemade scintillator had low scintillation yield because of lead concentration that quenches scintillation mechanism. But recent improvements have been made. The best lead and fluorophores concentration must still be tuned to maintain good x-rays absorption and to limit quenching.

Optical relay developments

In order to reduce effects of the LMJ radiative environment, the detector is placed inside a shielded box. This area is located at 7 m from the back of the inserter. Thus, a 7 meter long optical relay is necessary to transfer the image from the scintillator to the detector. Such an optical relay must not degrade the quality of the image because of its optical performances and parasitic light generated by large amount of glass. Its intrinsic spatial resolution should not exceed 100 m within a 20 mm FOV. One way to limit glass quantity consists in using a catadioptric relay.

A first 4 meter long version [3], with commercial Maksutov objectives 3 , has been already tested under irradiation on the ELSA electron accelerator facility [2]. As shown in figure [2], the experimental setup has been designed to reproduce as much as possible the LMJ configuration. The integrated gamma rays dose was relevant to the first 100 ns of the nominal LMJ shot. Masks have been placed along the optical relay in order to determine which elements are the major contributors to the parasitic light. Results show that the parasitic light generated by the 4 meter long catadioptric relay only contributes 1 Crytur: YAG: Ce. 2 Saint-Gobain Crystals: BC-400. 3 Astro Rubinar MC 300 mm F/4.5. to 1 permil of the 16 bits back-illuminated CCD4 dynamic. Most of the parasitic light is generated by both the glass window of the inserter and the field lens. Nevertheless, this configuration has proved its robustness in term of vunerability but has poor optical qualities. An improved design of the Maksutov objective5 has been performed so as to fit with the LMJ configuration and its specifications. The expected nominal spatial resolution of the final catadioptric relay is 50 mm within a 20 mm FOV with an F/4 optical aperture.

13006-p.3

EPJ Web of Conferences

PROSPECTS

As the feasibility of both the scintillator and the catadioptric relay have been experimentally demonstrated, a prototype for the LMJ configuration are expected by 2012. It shall be tested in association with an X-ray imaging system on a laser facility or an X-ray generator. The X-ray imaging system is still to be performed in order to meet the final diagnostic requirement of a 10 mm within a 500 mm FOV. One concept of such X-ray imaging system consists in the association of an annular aperture and an X-ray microscope. None-periodic multilayer coating allowing a constant reflectivity over the domain 10-30 keV has been already simulated. Vulnerability of such a coating under neutrons and gamma rays irradiation must still be caracterized to validate the diagnostic concept.

  (a) Cumulated dose evolution versus time (b) Neutrons and gamma rays spectra

Figure 1 .

 1 Figure 1. Simulation of radiative environment at 4 m from TCC for a neutron yield of 10 16 .

Figure 2 .

 2 Figure 2. Diagnostic implantation in the LMJ facility.

Figure 3 .

 3 Figure 3. Developments towards low decay time and high efficiency scintillators.

Figure 4 .

 4 Figure 4. Behaviour of the catadioptric relay under irradiation on ELSA facility.

  Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20135913006EPJ Web of Conferences

Princeton Instruments, PIXIS 2048B-eXcelon.

These objectives should be made by 2012.13006-p.4