Matobo rock art in its landscape

Understanding role(s) of rock art in Later Stone Age foragers territoriality

Biography

Léa Jobard is a CNRS PhD student at TRACES laboratory (University of Toulouse) and at the IFAS-Research (Johannesburg). She conducts her research under the supervision of François Bon and Camille Bourdier on the function of rock art in the territorial structuring of Later Stone Age communities.

As archaeology developed with new discoveries, it quickly appeared that similarities between the remaining artifacts found in sites separated by many kilometres were signs of recurrent visits by the same human group or of exchanges between different groups. In Southern Africa, settlement studies at all scales and based on different materials have flourished since the 1970s [START_REF] Parkington | Follow the San: An Analysis of Seasonality in the Prehistory of the South Western Cape[END_REF]. In the Matobo Hills, Nick Walker proposed a model of settlement patterns in the longue durée by Middle and Later Stone Age populations, mostly based on socio-economic behaviour [START_REF] Walker | Late Pleistocene and Holocene Hunter-Gatherers of the Matopos[END_REF]. However, knowing that rock art sites are highly significant for Southern Africa foragers [START_REF] Deacon | The Power of a Place in Understanding Southern San Rock Engravings[END_REF], we are led to wonder what role rock art played in the settlement pattern, what social function it had and if it changed through time and how we can answer those questions. In this article, I will give a general overview of my PhD research, addressing each of these issues.

The Matobo Hills paintings and Nick Walker's model

Located in Matabeleland South (Zimbabwe), a 2000 km² granite massif forms the Matobo Hills. Erosion caused by the presence of many streams and rivers created a hilly landscape of valleys and inselbergs, kopjes, block fields, rock shelters and caves [START_REF] Scharsich | Analysing land cover and land use change in the Matobo National Park and surroundings in Zimbabwe[END_REF][START_REF] Walker | Late Pleistocene and Holocene Hunter-Gatherers of the Matopos[END_REF] fig. 1 and2). Nowadays, the climate is generally temperate, with a four-month rainy season followed by warm, then cold and dry seasons [START_REF] Nhamo | Characterizing Hunter-Gatherer Rock Art: An Analysis of Spatial Variation of Motifs in the Prehistoric Rock Art of Zimbabwe[END_REF][START_REF] Walker | Late Pleistocene and Holocene Hunter-Gatherers of the Matopos[END_REF].

Despite variations in time and space, rainfalls are generally higher in the Matobo than in the surrounding areas and many natural reservoirs keep water even after the end of the rainy season [START_REF] Walker | Late Pleistocene and Holocene Hunter-Gatherers of the Matopos[END_REF]fig. 1). After the cold and dry conditions of the Late Glacial Maximum (23 000-18 000 BP), climate conditions gradually improved. The Younger Dryas event brought colder and drier weather between 12 700 BP and 10 200 BP [START_REF] Chevalier | Southeast African Records Reveal a Coherent Shift from High-to Low-Latitude Forcing Mechanisms Along the East African Margin Across Last Glacial-Interglacial Transition[END_REF]. Then, from 8 500 BP, temperatures rose, reaching a maximum around 7 000 BP before decreasing, and rainfall constantly increased until 2 000 BP [START_REF] Chevalier | Southeast African Records Reveal a Coherent Shift from High-to Low-Latitude Forcing Mechanisms Along the East African Margin Across Last Glacial-Interglacial Transition[END_REF].

Currently, the vegetation is typical of the Zimbabwean highveld, mostly covered with forest mixed with shrub and grassland, in contrast with dry savanna of the surrounding lowveld [START_REF] Scharsich | Analysing land cover and land use change in the Matobo National Park and surroundings in Zimbabwe[END_REF][START_REF] Walker | Late Pleistocene and Holocene Hunter-Gatherers of the Matopos[END_REF]. Moreover, the alternation between alluvial valley and rocky hills creates a particular patched environment with a rich and diverse biodiversity not found in the lowveld around the Matobo Hills (fig. 1). Past environment of the Matobo is poorly known, but it is trusted that it was as diverse as today [START_REF] Walker | Late Pleistocene and Holocene Hunter-Gatherers of the Matopos[END_REF]. Cave, Nswatugi Shelter, etc.). First noticed by geologists, this density of archaeological remains attracted a few researchers between the 1920s and 1980s. Amongst them, Nick Walker spent twenty years surveying a large part of the Matobo National Park and excavating several prehistoric sites. He noticed that in the area of Nswatugi Cave, where numerous shelters are available, only a third of them were used by foragers, suggesting that they carefully selected their sites [START_REF] Walker | Late Pleistocene and Holocene Hunter-Gatherers of the Matopos[END_REF]. His rock art survey, as well as other researcher's [START_REF] Garlake | The Painted Caves[END_REF][START_REF] Cooke | The painting sequence in the rock art of southern Rhodesia[END_REF], showed a high variability on many levels: site dimension and morphology (from overhang to 10 meters wide shelter), altitude (from valley to hilltop), images composition (from a few to hundreds overlaid), colours (reds, violet, white and black), painting techniques (mono-, bi-or polychrome, outlined, striped), styles (realistic or schematic), motifs (many animal species, humans, plants and geometric forms), archaeological context (from no associated activity to aggregation site). [START_REF] Walker | Late Pleistocene and Holocene Hunter-Gatherers of the Matopos[END_REF]. Before 9 600 BP, a few LSA groups visit wide shelters such as Pomongwe, only during the autumn, and then during the summer as well. Around 9 100 BP, foragers are present all year round and groups seem numerous considering the number of sites and deposits during this period. As Walker states, this hypothesis can't be correlated with mobility patterns of Lowveld or Kalahari sites in surrounding vicinities as they are poorly studied. In the Matobo, different groups settle in a foraging range organised around one or two main dwelling sites and around smaller, more specialised sites. They increasingly assert their identity and territory according to stylistic differences between artifacts found in different sites. Concomitant with this regionalisation, most of the rock art is produced during this specific period, given the high quantity of ochre present in the deposits.

Walker suggested that rock art was performed during group ritual practices in response to higher population density and stronger social structure [START_REF] Nhamo | Characterizing Hunter-Gatherer Rock Art: An Analysis of Spatial Variation of Motifs in the Prehistoric Rock Art of Zimbabwe[END_REF][START_REF] Walker | Painting and ceremonial activity in the Later Stone Age of the Matopos, Zimbabwe[END_REF]. After 8 500 BP, and even after 7 800 BP when climate conditions deteriorated, the population density decreases while fewer and smaller sites are occupied. But during this same period, there is also evidence of a periodic aggregation of several groups. From 2 200 BP, some foragers start rearing sheep and making ceramics. Around 300 AD, farmers settle in the Matobo, paint rock art, and foragers are pushed back into zones less suitable for agriculture or cattle [START_REF] Walker | Painting and ceremonial activity in the Later Stone Age of the Matopos, Zimbabwe[END_REF][START_REF] Walker | Late Pleistocene and Holocene Hunter-Gatherers of the Matopos[END_REF].

Walker's model provides insight into regional settlement tendencies that could be efficiently re-examined and enriched with the study of rock art and its location in the landscape.

Environmental study of rock art

To understand the complexity of past communitiestheir social structure, their economy, their adaptation strategies, their relations with other communities, etc.it is essential to investigate their territoriality: the social link between this community, its culture and the physical space where its activities take place [START_REF] Delanda | A New Philosophy of Society: Assemblage Theory and Social Complexity[END_REF]. This kind of study can be conducted at different scales, notably at intra-site, local or regional scales [START_REF] Hyder | Location Analysis in Rock Art Studies[END_REF].

Moreover, this concept of territoriality does not necessarily refer to the western notion of property, where an individual or a group owns a homogeneous piece of land. Indeed, several ethnographic and ethnohistoric reports show that the idea of possession, territory or border can be quite different according to communities in Southern Africa [START_REF] Smith | Dangerous Ground: A Critique of Landscape in Rock-Art[END_REF]. A group's territoriality is determined by the landscape in which they live. We refer here to Ingold's definition of this notion: rejecting the idea of an outer physical space existing independently from human existence, he sees the landscape as an ensemble of entities which exist in relation to one another's entities [START_REF] Ingold | The temporality of the landscape[END_REF]. In other words, the landscape of a forager group refers to the physical world they inhabit, as they perceive, see and relate to it.

To better understand, Ingold gives the example of borders: a river or mountain range is a border only if the people living around feel that it is one. As a complementary notion, the environment is similar to the landscape and puts emphasis on the processes between the elements of the worldwhereas landscape focuses on the form [START_REF] Ingold | The temporality of the landscape[END_REF].

Another useful concept to study human territoriality is the taskscape, a notion referring to the ensemble of relations (social, economic, spatial) between tasks done by a group, in which every task carried out by an individual makes sense cognisant of the tasks carried out by the community [START_REF] Ingold | The temporality of the landscape[END_REF][START_REF] Robinson | Land Use, Land Ideology: An Integrated Geographic Information Systems Analysis of Rock Art Within South-Central California[END_REF]. As a task necessarily takes place in a chosen spacewe choose, to a certain extent, what we do and wheretaskscape is linked to the landscape. To understand past societies' space organisation, we have thus to consider their taskscape and its relation to the landscape, through every archaeological artifact or activity remnant and palaeoenvironment analysis.

Even if most settlement studies concentrate on socio-economic activities (intra-site organisation, resource supplying strategies, group mobility, long-distance exchanges, etc.), more and more researchers investigate rock art through this concept of taskscape, as it is as much a task as any other actionable event [START_REF] Robinson | Land Use, Land Ideology: An Integrated Geographic Information Systems Analysis of Rock Art Within South-Central California[END_REF][START_REF] Wienhold | GIS in Rock Art Studies[END_REF].

Rock art has the advantage of being non-portable, and we find it in the exact same place where it has been produced and used, contrary to other artifacts [START_REF] Nash | The Figured Landscapes of Rock Art: Looking at Pictures in Place[END_REF].

Thanks to ethnohistorical records, we know that painted places were highly significant for foragers, who knew their environment perfectly [START_REF] Deacon | The Power of a Place in Understanding Southern San Rock Engravings[END_REF]. Settlement studies tend to undervalue the influence of parameters such as the territoriality of other groups, symbolism, danger or superstition linked to places, over the presence or absence of resources [START_REF] Wheatley | Spatial Technology and Archaeology. The Archaeological Applications of GIS[END_REF]. Economic, social and symbolic realms should not thus be separated, but integrated into the same studies in order to offer a richer and more comprehensive vision of taskscape, social relationships, and ideological behaviours [START_REF] Hyder | Location Analysis in Rock Art Studies[END_REF][START_REF] Robinson | Land Use, Land Ideology: An Integrated Geographic Information Systems Analysis of Rock Art Within South-Central California[END_REF].

In a diachronic perspective, when searching for causes of change or long lasting-practices, it is relevant to include rock art as it plays a significant social role and is ruled by cultural standards [START_REF] Heyd | Art, Rock Art and Climate Change[END_REF]. Rock art could be part of adaptation strategies to new environments or social contexts [START_REF] Mcdonald | Rock Art and Social Identity: A Comparison of Holocene Graphic Systems in Arid and Fertile Environments[END_REF] whilst, possibly, the economic system remains the same through time, supporting the necessity for researchers to cross every archaeological domain.

Environmental study of Matobo Hills' rock art

In Matobo Hills, where some elements to understand settlement patterns are already available but do not take into account the hundreds of LSA rock art sites, it seems necessary to make an environmental study of rock art. Indeed, analysing the choices performed in rock art sites, such as the location or the iconography, enable investigation of (1) their socio-cultural functions (ritual, identity, pedagogic, etc) as the iconographic and topographic variability suggests diverse functions (Lenssen-Erz, 2004);

(2) the role of rock art in organising space and structuring territory (Lenssen-Erz, 2004), a question usually explored through an economic angle, in order to propose a multi-proxy vision of the social geography of these groups;

(3) the evolution of those territorial dynamics through time and the agents of continuity or change, especially at unstable climate periods in order to test the adaptation strategies of the foragers.

In order to address those issues, our PhD research includes different scales of analysis: the images, the site and the landscape [START_REF] Hyder | Location Analysis in Rock Art Studies[END_REF]. The first objective is to set up a database, including information about paintings (theme, style, technique), support (place and rock attributes), archaeological context (artifacts found in the deposit, other activities remnant, possible site use, dates), physical characteristics of the site (topography and geomorphology), and its place in the landscape (altitude, accessibility, view from and visibility of the site, notable elements of the landscape). This database is linked with a geographic information system (GIS) that enables us to easily visualise and analyse space data (for example, viewshed, network modelling; [START_REF] Wienhold | GIS in Rock Art Studies[END_REF]. Data is collected from unpublished documents by previous researchers, such as Nick Walker, partly curated by the Zimbabwe Museum of Human Sciences (Harare) and the Zimbabwe Museum of Natural History (Bulawayo). In order to test the relevance of the results, a control sample, constituted of unpainted rock shelters, will be recorded and analysed through the same criteria.

Based on MATOBART chrono-stylistic sequencing, each site will be attributed to one or several chronological phases, assuming that the sites offering similar themes, styles, and techniques are more or less contemporaneous. From statistical analyses of the database and GIS processing, such as viewshed or least cost path calculation, the second objective is to highlight the main selection criteria for a site, and to find out if some images, or associations of images, are linked with some particular elements of the site, landscape or taskscape. We also plan to study networks between rock art and any other archaeological sites, how they work together as a system and assess socio-cultural function(s) of rock art sites [START_REF] Lenssen-Erz | The Landscape Setting of Rock Painting Sites in the Brandberg[END_REF].

Our third goal is to compare the results in diachrony (chrono-stylistic phases), analysing variability of site physical criteria and networks through time, and determining which are the elements of stability or change. Finally, our last objective is to compare our model with climate change in eastern Southern Africa, questioning change or stability agents at stake in rock art use.

As a conclusion, this research has a strong conceptual ambition and is focused on a spatial and interdisciplinary approach confronting rock art, socio-economy activities and environmental conditions. Hopefully, in future, it will provide a more dynamic anthropological perspective on the production and use of images within forager societies.

Figure 1 :

 1 Figure 1: Landscapes of the Matobo Hills. Top: View on the Matobo National Park on a rainy day (User:Digr -CC BY-SA 3.0 -https://commons.wikimedia.org/); Bottom left: View from World's View Hill; Bottom right: Balancing rocks (Gerhard Hubert -CC BY-NC 4.0https://global-geography.org/)

Figure 2 :

 2 Figure 2: Various site's morphologies and dimensions. Top left: View of Inanke Cave (image: C. Dudognon); Top right: View of Bambata Cave (image: C. Dudognon); Bottom left: View of Nswatugi Cave (image: C. Dudognon); Bottom right: View of Honey Hunter Shelter (image: C. Bourdier)


Léo Théodon
email: l.theodon@emse.fr

Jérôme Laurencin
email: jerome.laurencin@cea.fr

Maxime Hubert
email: maxime.hubert@cea.fr

Peter Cloetens
email: cloetens@esrf.eu

Johan Debayle
email: debayle@emse.fr

A stochastic geometrical 3D model for time evolution simulation of microstructures in SOC-electrodes

Keywords: 3D microstructure model, Nickel-YSZ electrode, Solid oxide fuel cell, stochastic geometry

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In a context of climate change and increasing scarcity of fossil fuels, the Solid Oxide Cells (SOCs) technology has attracted considerable attention in the recent years. Indeed, this high temperature electrochemical device could be used to match the fluctuations between the electricity production and consumption when using renewable energy sources. Thanks to their flexibility, the SOCs can be operated in both fuel cell (SOFC for solid oxide fuel cell) and electrolysis (SOEC for Solid Oxide Electrolysis Cell) modes. They are composed of a dense electrolyte sandwiched between two porous electrodes. The socalled hydrogen electrode is typically a cermet composed of Yttria Stabilized Zirconia and Nickel (Ni-YSZ). Within the electrode, the dihydrogen and steam gaseous molecules are transported through the percolated porosity while the ionic and electronic conductivity for the oxygen ions and electrons is ensured by the YSZ and Ni phases, respectively. The electrochemical reactions takes place at the triple phase boundary (TPB) lines where the gas, electronic and ionic phases meet. Therefore, the SOCs electrochemical performances are strongly dependent on the electrode microstructural properties [START_REF] Prakash | Properties and development of Ni/YSZ as an anode material in solid oxide fuel cell: A review[END_REF], and in particular some specific geometrical characteristics [START_REF] Holzer | Redox cycling of ni-ysz anodes for solid oxide fuel cells: Influence of tortuosity, constriction and percolation factors on the effective transport properties[END_REF][START_REF] Gaiselmann | Quantitative relationships between microstructure and effective transport properties based on virtual materials testing[END_REF][START_REF] Effori | Reaction mechanism and impact of microstructure on performances for the lscf-cgo composite electrode in solid oxide cells[END_REF]. Moreover, in operation at high temperature, the hydrogen electrode is subjected to a phenomenon of Ni migration and agglomeration, which tends to decrease the density of TPB lines and hence the cell electrochemical performances [START_REF] Hubert | Impact of nickel agglomeration on solid oxide cell operated in fuel cell and electrolysis modes[END_REF][START_REF] Zekri | Nickel depletion and agglomeration in sofc anodes during long-term operation[END_REF].

The characterization of the complex electrode microstructure is thus essential to better understand the initial electrode performances as well as the degradation mechanisms. For this purpose, different characterization techniques such as Focused Ion Beam coupled with a Scanning Electron Microscope (FIB-SEM) [START_REF] Wilson | Threedimensional reconstruction of a solid-oxide fuel-cell anode[END_REF][START_REF] Matsui | Anode microstructural change upon longterm operation for the cathode-supported tubular-type sofc[END_REF][START_REF] Prokop | A threedimensional heterogeneity analysis of electrochemical energy conversion in sofc anodes using electron nanotomography and mathematical modeling[END_REF], X-ray absorption tomography [START_REF] Shearing | Analysis of triple phase contact in ni-ysz microstructures using non-destructive x-ray tomography with synchrotron radiation[END_REF][START_REF] Nelson | Three-dimensional microstructural changes in the ni-ysz solid oxide fuel cell anode during operation[END_REF][START_REF] Chen-Wiegart | Effect of ni content on the morphological evolution of ni-ysz solid oxide fuel cell electrodes[END_REF] or X-ray holotomography [START_REF] Hubert | Impact of nickel agglomeration on solid oxide cell operated in fuel cell and electrolysis modes[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] Monaco | Degradation of ni-YSZ electrodes in solid oxide cells: Impact of polarization and initial microstructure on the ni evolution[END_REF] have been used to image the Ni-YSZ electrode in its pristine state or after long-term operation. Despite the possibility to reconstruct volumes at high resolution, this approach requires the manufacturing and testing of several cells in different operating conditions. The number of volumes are usually limited since the 3D characterization is time consuming. In this approach, the experimental dataset is in general insufficient to draw precise correlations on the impact of the electrode microstructure on SOCs performances and durability in different operating conditions. In order to overcome this problem, the development of model allowing the generation of synthetic microstructures has met a growing interest. Thus, many models have been already proposed in recent years, allowing the generation of representative microstructures for which the geometrical characteristics match the real electrode properties. Some of these models are based on non-overlapping hard spheres packing [START_REF] Moussaoui | Particle-based model for functional and diffusion layers of solid oxide cells electrodes[END_REF][START_REF] Metcalfe | Connected three-phase boundary length evaluation in modeled sintered composite solid oxide fuel cell electrodes[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] Golbert | Microstructural modeling of solid oxide fuel cell anodes[END_REF][START_REF] Westhoff | Generation of virtual lithiumion battery electrode microstructures based on spatial stochastic modeling[END_REF]. In this case, the microstructure is generated using an iterative process that can be relatively time-consuming. Other models are based on random Gaussian fields [START_REF] Abdallah | Morphological modelling of threephase microstructures of anode layers using SEM images[END_REF][START_REF] Moussaoui | Stochastic geometrical modeling of solid oxide cells electrodes validated on 3d reconstructions[END_REF][START_REF] Prifling | Stochastic 3d microstructure modeling of anodes in lithium-ion batteries with a particular focus on local heterogeneity[END_REF][START_REF] Moussaoui | Stochastic geometrical and microstructural modeling for solid oxide cell electrodes[END_REF][START_REF] Moussaoui | 3D geometrical characterization and modelling of solid oxide cells electrodes microstructure by image analysis[END_REF] or even random graphs [START_REF] Neumann | Stochastic 3d modeling of complex three-phase microstructures in sofc-electrodes with completely connected phases[END_REF]. They allow generating microstructures in a shorter time since no iterative procedures are involved for these simulations. All the proposed models offer rather good results, in particular for the modeling of globally homogeneous microstructures. Nevertheless, when the microstructure presents some local heterogeneities, some characteristics such as the microstructure factor, the geometrical tortuosity or the constrictivity, along with the continuous Phase Size Distributions (PSD), can be difficult to approach with high accuracy.

It is worth noting that the microstructure tends to be more and more heterogeneous after operation at high temperature due to the microstructural evolutions occurring in the electrode. Thus, taking into account this temporal evolution of the microstructure in order to optimize the performance and durability of SOCs is essential. Some physically based models have been already proposed to simulate in 3D this time evolution of the microstructure. In this frame, the phase-field approach has been recently used to simulate the Ni coarsening in operation [START_REF] Takaki | Phase-field modeling and simulations of dendrite growth[END_REF][START_REF] Lei | Phase field simulation of anode microstructure evolution of solid oxide fuel cell through ni(oh)2 diffusion[END_REF][START_REF] Chen | Simulation of coarsening in three-phase solid oxide fuel cell anodes[END_REF][START_REF] Jiao | Simulation of nickel morphological and crystal structures evolution in solid oxide fuel cell anode using phase field method[END_REF]. However, this type of models requires very long computation time, limiting both the number of simulations and the volume of the studied microstructure. In this context, the development of geometrical models able to reproduce correctly the microstructural evolution upon operation is necessary and fully complementary with the physically based models. Indeed, the geometrical approach does not require large computational resources. Besides, the model does not depend on the material physicochemical properties and can be thus applied to any three-phase microstructure. However, to the best of our knowledge, no geometrical model has been proposed yet to simulate the microstructural evolution of SOC electrodes.

In this context, a Stochastic Time Evolution Microstructure (STEM) model has been developed for three-phase material and applied to the Ni-YSZ cermet for SOC application. The model is composed a of a static component, later called "static model", able to simulate the initial and final state of the same microstructure, and a dynamic component, able to simulate the temporal evolution of the microstructure between these two static bounds. This model is based on a degenerate random graph structure, similar to the one proposed by M.Neumann et al. [START_REF] Neumann | Stochastic 3d modeling of complex three-phase microstructures in sofc-electrodes with completely connected phases[END_REF], in which a temporal dimension has been added.

In a first part, the real electrode reconstructions and the microstructural properties measured on the 3D volumes are presented. In a second part, the static model is defined, followed by the STEM model, as well as examples of synthetic microstructure simulations. The optimization process used to identify the model parameters is also explained and validated. The static model is then calibrated and validated on two real Ni-YSZ microstructures obtained before and after ageing. The ability of the STEM model to reproduce the microstructural evolution is validated thanks to two other electrode reconstructions obtained at two intermediate times. The results including the evolution of the geometrical characteristics over the time are presented for the synthetic microstructures. Finally, the performances and limitations of the STEM model are discussed.

Image reconstruction and characterization

Image reconstruction

In this article, the Ni-YSZ cermet of a typical SOC were reconstructed by synchrotron X-ray holotomography [START_REF] Villanova | 3d phase mapping of solid oxide fuel cell ysz/ni cermet at the nanoscale by holographic x-ray nanotomography[END_REF][START_REF] Laurencin | Characterisation of solid oxide fuel cell ni-8ysz substrate by synchrotron x-ray nano-tomography: from 3d reconstruction to microstructure quantification[END_REF] before and after operation. This technique allows the acquisition of 3D volumes with a large field of view (50 µm) by keeping a resolution of a few tenth of a nanometer [START_REF] Hubert | Efficient correction of wavefront inhomogeneities in x-ray holographic nanotomography by random sample displacement[END_REF]. Fig. 1 shows three 3D reconstructions of real microstructures for three different operating times, with a voxel size of 50 nm. The protocol used to record the data and the procedure to reconstruct the electrode microstructures are thoroughly described in [START_REF] Hubert | Efficient correction of wavefront inhomogeneities in x-ray holographic nanotomography by random sample displacement[END_REF][START_REF] Hubert | Durability of solid oxide cells : an experimental and modelling investigation based on synchrotron x-ray nano-tomography characterization[END_REF]. It is worth noting that the studied images were obtained from cells manufactured in the same conditions and operated at T = 850°C for t = 0, 1000, 1500 and 2000 hours. All the details about the operating conditions as well as the reconstructions have been already published in [START_REF] Hubert | Impact of nickel agglomeration on solid oxide cell operated in fuel cell and electrolysis modes[END_REF].

Image characterization

The characterization of a 3D image consists in measuring a number of properties on it. A 3D volume representing the complex electrode microstructure is thus considered fully characterized once these properties are known. The most frequently measured properties are detailed in [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 microstructures in sofc-electrodes with completely connected phases[END_REF] whereas the method for their computation is described in [START_REF] Laurencin | Characterisation of solid oxide fuel cell ni-8ysz substrate by synchrotron x-ray nano-tomography: from 3d reconstruction to microstructure quantification[END_REF][START_REF] Usseglio-Viretta | Quantitative microstructure characterization of a ni-ysz bi-layer coupled with simulated electrode polarisation[END_REF]. The properties used in this article are listed below.

1. The phase volume fraction ε i is the proportion of phase i to the total volume. It is simply computed by dividing the number of voxels belonging to phase i by the total number of voxels. 2. The proportion of connected phase δ i is determined by computing the proportion of phase connecting two opposite sides of the total volume [START_REF] Usseglio-Viretta | Quantitative microstructure characterization of a ni-ysz bi-layer coupled with simulated electrode polarisation[END_REF]. 3. The density of Triple Phase Boundary length (TPBl) ξ TPBl is computed by identifying each voxel edges at the triple contacts. A correction factor must be applied to take into account the cubic discretization of the domain in the form of voxels [START_REF] Laurencin | Characterisation of solid oxide fuel cell ni-8ysz substrate by synchrotron x-ray nano-tomography: from 3d reconstruction to microstructure quantification[END_REF]. 4. The specific surface areas S i p and interfacial specific surface areas S i, j p correspond to the normalized area of phase i and the normalized area at the interface of phases i and j. They are computed as described in [START_REF] Legland | Computation of minkowski measures on 2d and 3d binary images[END_REF]. 5. The tortuosity is a property fully described in [START_REF] Clennell | Tortuosity: a guide through the maze[END_REF]. The geometric tortuosity τ geo i computed for each of the phases i is equal to the ratio of the minimum distance to connect one side to the opposite side of the volume within the phase, to the length of the volume in that direction. In other words, if the length of the volume in a given direction is denoted l and l m is the minimum length to go from one side of the volume to the other in that direction within phase i, then the following applies:

τ geo i = l m /l. (1)
The minimum length l m is computed using Dijkstra's algorithm on a Fibonacci heap structure [START_REF] Thulasiraman | Graphs: theory and algorithms[END_REF][START_REF] Fredman | Fibonacci heaps and their uses in improved network optimization algorithms[END_REF][START_REF] Cecen | Tortuosity / percolation / connected path calculation for 3d structures, GitHub[END_REF]. 6. The continuous Phase Size Distributions (PSD) or granulometry is computed by performing a sequence of morphological operations, namely morphological openings, using a homothetic sphere as structuring element as described in [START_REF] Laurencin | Characterisation of solid oxide fuel cell ni-8ysz substrate by synchrotron x-ray nano-tomography: from 3d reconstruction to microstructure quantification[END_REF][START_REF] Serra | Image analysis and mathematical morphology[END_REF]. 7. The constrictivity β i is a property that measures bottleneck effects of a given microstructure [START_REF] Gaiselmann | Quantitative relationships between microstructure and effective transport properties based on virtual materials testing[END_REF][START_REF] Holzer | The influence of constrictivity on the effective transport properties of porous layers in electrolysis and fuel cells[END_REF]. It is based on the PSD, and therefore depends on the granulometry of the microstructure. It is defined as follows:

β i = r i,min /r i,max 2 (2)
with r i,max the maximum radius of the spheres allowing to fill at least 50% of phase i, i.e. the median of the cumulative PSD, noted d i p , and r i,min the maximum radius of the spheres allowing to fill at least 50% of phase i by intrusion in a given transport direction. The quantity r i,min can be computed using a Mercury Intrusion Porosimetry (MIP) simulation described in [START_REF] Holzer | The influence of constrictivity on the effective transport properties of porous layers in electrolysis and fuel cells[END_REF][START_REF] Münch | Contradicting geometrical concepts in pore size analysis attained with electron microscopy and mercury intrusion[END_REF] or derived from the PSD. 8. The microstructure-factor M i of phase i in a given transport direction is defined as follows:

M i = σ e i /σ 0 i ( 3
)
where σ e i and σ 0 i are respectively the effective and intrinsic conductivities for the percolated phase i. Note that for Ni-YSZ cermet electrodes, an empirical law has been proposed to relate the M-factor to the volume fraction, the constrictivity and the geometrical tortuosity [START_REF] Neumann | Quantifying the influence of microstructure on effective conductivity and permeability: Virtual materials testing[END_REF] in the form:

M i = ε 1.67-0.48β i i / τ geo i 5.18 . (4)
9. The covariance function C i (r) of a phase i in a given direction contains information on the spatial structure and regularity of the microstructure, and in particular the volume fraction of the given phase [START_REF] Serra | Image analysis and mathematical morphology[END_REF]. It is defined as the probability that two points separated by a distance r belong to the same phase i. In particular, the following applies:

C i (0) = ε i .
It can be noticed that many properties such as the tortuosity are dependent on the transport direction if considering anisotropic media. Nevertheless, it has been found that the values of tortuosity, M-factor, constrictivity, or covariance, had no preferred direction for typical electrode microstructures studied throughout this paper. As an illustration, Fig. 2 shows the covariance functions computed on four different 3D reconstructions of real microstructures along the three main axis. It appears clearly that these values do not depend on the considered axis proving that the real microstructures are isotropic. Finally, the choice of the studied characteristics are motivated by their significance with respect to the effective transport properties of each phase in the heterogeneous microstructure. This is particularly the case for volume fraction, tortuosity and constrictivity [START_REF] Holzer | Redox cycling of ni-ysz anodes for solid oxide fuel cells: Influence of tortuosity, constriction and percolation factors on the effective transport properties[END_REF][START_REF] Gaiselmann | Quantitative relationships between microstructure and effective transport properties based on virtual materials testing[END_REF]. The density of TPBls is also a crucial parameter controlling the electrochemical reaction in the Ni-YSZ cermet [START_REF] Monaco | Electrode kinetics of porous ni-3ysz cermet operated in fuel cell and electrolysis modes for solid oxide cell application[END_REF].

Proposed stochastic geometrical model

Static model

The static model, which is proposed in this work to simulate three-phase electrode microstructures, is based on three independent point processes. The sets of points X P , X YZS related to the porosity and YSZ phase are generated using two Poisson processes [START_REF] Kingman | Poisson Processes[END_REF] of intensity λ P and λ YS Z respectively. For the Ni phase, the set of points X Ni is obtained using a Neyman-Scott point process [START_REF] Neyman | Statistical approach to problems of cosmology[END_REF]. This method allows generating sets of clustered points [START_REF] Chiu | Stochastic Geometry and Its Applications[END_REF][START_REF] Milne | Point processes and some related processes[END_REF] controlled by three parameters. Indeed, the method depends on (i) an intensity parameter λ Ni for the centers of the clusters, (ii) a parameter σ for the standard deviation of a normal distribution that controls the dispersion around the centers of the clusters, and (iii) a parameter λ c for the Poisson distribution providing the number of points per cluster. The Neyman-Scott point process has been selected to produce aggregated sets of points, which should be well adapted to reproduce accurately the phenomena of Ni agglomeration and depletion in the microstructure. The choice of this method for the Nickel phase is thoroughly justified and discussed in section 3.3.

After the generation of the random point patterns, it is worth noting that no graph related to the three electrode phases is built, in contrast to the method proposed by M.Neumann et al. [START_REF] Neumann | Stochastic 3d modeling of complex three-phase microstructures in sofc-electrodes with completely connected phases[END_REF]. In our case, each voxel in the volume V is directly associated to the set of points X P , X YZS or X Ni in such a way that the distance between the voxel position and the set of points is minimal. In other words, the voxel is assigned to the phase related to the closest set of points using the method described in [START_REF] Felzenszwalb | Distance transforms of sampled functions[END_REF] to compute the three distance maps. Finally, a two-step Gaussian pseudo-smoothing parameter θ described in [START_REF] Neumann | Stochastic 3d modeling of complex three-phase microstructures in sofc-electrodes with completely connected phases[END_REF] is applied to the microstructure, which allows in particular to control the density of TPBl. The proposed static model thus depends on six parameters.

Examples of synthetic microstructures are shown in Fig. 3. It appears clearly that it is possible to control the local heterogeneity of the microstructure by adjusting respectively the density and size parameters, λ Ni and λ c , of the clusters for the Nickel phase (white phase).

STEM model

To represent the temporal evolution of the microstructure, the Stochastic Time Evolution Microstructure (STEM) model is based on the interpolation between two different states obtained with the static model (cf. previous section 3.1). The sets of parameter used to generate these two bounds are denoted ω 0 = {λ 0

P ; λ 0 YS Z ; λ 0 Ni ; λ 0 c ; σ 0 ; θ 0 } and ω 1 = {λ 1 P ; λ 1 YS Z ; λ 1 Ni ; λ 1 c ; σ 1 ; θ 1 } while the interpolation parameter evolving between these two states is denoted t ∈ [0; 1].
Three sets of points X P (t), X YS Z (t) and X Ni (t) can be generated for each t ∈ [0; 1] by interpolation as follows. A set X 0 (respectively X 1 ) is generated using the density max(λ 0 P , λ 1 P ) (respectively max(λ 0 YS Z , λ 1 YS Z )). Assuming that λ 0 P > λ 1 P , the states X P (t = 0) and X P (t = 1) are defined such that X P (t = 0) = X 0 and #X P (t = 1) ∼ P(λ 1 P ), with P(λ 1 P ) a Poisson distribution of parameter λ 1 P . The set X P (t = 1) is built from X P (t = 0) by removing randomly a sufficient number of random points. The number of elements between X P (t = 0) and X P (t = 1) decreases linearly so that X P (t) ⊆ X P (t = 0), ∀t ∈ [0; 1]. The process is symmetrical in the case where λ 0 P < λ 1 P and is identical for the definition of X YS Z (t). Thus, there is no spatial interpolation in the construction of the sets X P (t) and X YS Z (t).

Then, for the construction of the set X Ni (t), two sets X m (t = 0) and X m (t = 1) are generated using the cluster densities λ 0 Ni and λ 1

Ni respectively. These two sets represent the positions of the cluster centers of the Nickel phase in the microstructure for the two bounds at t = 0 and t = 1. Using the Auction Algorithm [START_REF] Blackman | Design and Analysis of Modern Tracking Systems[END_REF], it is possible to match the sets X m (t = 0) and X m (t = 1). The sum of the distances that must be covered by the center of the cluster to go from one state to the other is therefore minimal. The states X m (t) are finally defined by spatial interpolation between X m (t = 0) and X m (t = 1). The elements of X m (t = 0) that could not be matched with any element of X m (t = 1), and vice versa, are removed randomly according to the value of t ∈ [0; 1] so that #X m (t) varies linearly.

Finally, for each subset of points clustered around an element of X m (t), the same procedure as described above for the sets X P (t) and X YS Z (t) is performed, using the densities λ 0 c and λ 1 c . The standard deviation of the distances of each of the generated point clusters around each center is itself adjusted so that it is equal to σ(t) = t × σ 1 + (1t) × σ 0 . A set of points X Ni (t) can thus be defined as the set of points surrounding the set of cluster centers X m (t).

By definition, the three sets X P (t = 0), X YS Z (t = 0) and X Ni (t = 0) (resp. X P (t = 1), X YS Z (t = 1) and X Ni (t = 1)) allow generating a microstructure with the same characteristics as the one generated with the parameter set ω 0 (resp. ω 1 ), once the smoothing parameter θ 0 (resp. θ 1 ) is applied. Therefore, it is possible to produce any intermediate state of the microstructure for t ∈ [0; 1] using the X P (t), X YS Z (t) and X Ni (t) sets and applying a smoothing of parameter evolving with the time as follows:

θ(t) = t × θ 1 + (1 -t) × θ 0 . (5)
Fig. 4 illustrates the results obtained by applying the STEM model to two states of the cermet microstructure in order to simulate the coalescence of the Nickel phase (white). It can be seen that it is mainly the Nickel phase that is moving by model design, which makes the transformation realistic.

Discussion

There are several reasons why it was decided to use a Neyman-Scott point process for the Ni phase rather than a Poisson point process. Initially, it was decided to combine graph matching techniques described in [START_REF] Guo | A quotient space formulation for statistical analysis of graphical data[END_REF][START_REF] Guo | Statistical shape analysis of brain arterial networks (BAN)[END_REF] for the dynamic aspect of the model with the graph structure described in [START_REF] Neumann | Stochastic 3d modeling of complex three-phase microstructures in sofc-electrodes with completely connected phases[END_REF] for the static part. Nevertheless, it soon became apparent that it was possible to generate microstructures with satisfactory properties while using an underlying graph structure with an extremely small number of edges. Indeed, comparative tests involving synthetic microstructures generated from full graphs on the one hand and from their set of vertices only on the other hand have shown relative error differences of the order of 0.1% at most for all the measured geometrical properties. Therefore, the choice was made to leave the graph structure aside and to build the model only on sets of points, i.e. the sets of vertices of degenerate graphs (cf. description of the static model 3.1). Thanks to this choice, it was possible to reduce the number of model parameters from ten to four. On the other hand, contrary to the microstructures studied in [START_REF] Neumann | Stochastic 3d modeling of complex three-phase microstructures in sofc-electrodes with completely connected phases[END_REF], those emulated with the STEM model are not necessarily locally homogeneous or fully connected [START_REF] Prakash | Properties and development of Ni/YSZ as an anode material in solid oxide fuel cell: A review[END_REF][START_REF] Jiao | Simulation of nickel morphological and crystal structures evolution in solid oxide fuel cell anode using phase field method[END_REF]. These characteristics are essential to simulate the phenomenon of the Nickel coarsening and agglomeration. Indeed, it seemed natural to use an aggregated point process to structure the Nickel phase, knowing that the analytical relationships between the model parameters and the geometrical properties of the microstructure could not be preserved.

Another reason that motivated the use of the Neyman-Scott point process is the evolutionary aspect of the microstructure. In the first version of the model, with an underlying graph structure, the different graphs representing a kind of skeleton of the microstructure were matched by solving a quadratic assignment problem [START_REF] Koopmans | Assignment problems and the location of economic activities[END_REF], using the FAQ (Fast Approximate Quadratic assignment problem) algorithm [START_REF] Vogelstein | Fast approximate quadratic programming for graph matching[END_REF][START_REF] Frank | An algorithm for quadratic programming[END_REF]. However, the computation time can be very long on large volumes. Without the graph structure, the proposed model consists in solving a fast linear assignment problem [START_REF] Kuhn | The Hungarian Method for the Assignment Problem[END_REF]. Moreover, by using a Neyman-Scott point process, it was possible to further optimize the computation time by matching the sets of cluster centers rather than the whole sets of points associated with the Nickel phase, thus reducing the size of the sets to be matched by at least one order of magnitude.

Finally, it should be noted that the set matching process only applies to the Nickel phase. This choice not only reduces the computation time, but it is also motivated by the objective of obtaining a more realistic transformation. Indeed, by modifying only slightly the sets of points associated with the porous and ceramic phases of the microstructure, these phases themselves will only be marginally affected by the temporal evolution and the interpolation process. This behavior is more in line with reality, as can be seen in Fig. 4.

Models fitting and validation

Cost function and optimization process

The fitting process requires the definition of a cost function. Among the most interesting geometrical properties for optimizing the SOC performances, the choice was made to retain the density of TPBl, the interfacial specific surface areas and the volume fractions. For a set of parameter values ω for the STEM model and a set of geometric properties measured on the 3D reconstruction of an electrode, the proposed cost function is defined as follows.

F k (ω) = 4 × 2 i=0 ∆ (ε i ) + 0≤i< j≤2 ∆(S i, j ) + 2 × ∆(ξ TPBl ) (6)
where ∆ is the relative error on the geometric properties measured on the 3D synthetic microstructure generated by the STEM model with the set of parameter values ω and the geometric properties contained in the set k.

It should be noticed that the specific surface areas can be deduced by linear combination of the interfacial specific surface areas, the reverse not being true. For this reason, the interfacial specific surface areas have been chosen as parameters in the cost function.

The fitting process consists of applying the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm [START_REF] Auger | A restart cma evolution strategy with increasing population size[END_REF][START_REF] Auger | Tutorial cma-es: evolution strategies and covariance matrix adaptation[END_REF] to the cost function F k for a set of properties k. To validate the approach, a synthetic microstructure was generated by the STEM model using a set of known parameter values ω 0 . The necessary geometric properties were measured on this microstructure. The fitting process was then run, and a set of parameter values ω 0 was obtained. Table 1 shows the relative errors between the elements of ω 0 and ω 0 . As the differences are extremely small, the fitting process is considered as validated.

Static model validation

In order to validate the static model, the fitting procedure was performed using data from several real microstructures. The quality of the model was mainly evaluated with respect to the errors made on the geometrical properties of the generated synthetic microstructure that are not taken into account in the cost function, such as the tortuosity or the constrictivity.

The rendering volumes for the reconstructions of the real microstructures and the related synthetic microstructures generated by the static model after the optimization step are displayed in Fig. 5. The two real microstructures correspond to the same type of Ni-YSZ cermet obtained for the pristine cell (t = 0 h) and after an operation of t = 2000 h. The measurements made on the corresponding real and synthetic microstructures are provided in Tables 2 and3.

When comparing the geometrical properties of the real and synthetic microstructures, it appears that the results are in good agreement regarding the volume fractions and the density of TPBl (i.e. the properties that have the most weight in the cost function). The results are also satisfactory for the interfacial specific surface areas, even if the discrepancy on S 0,2 p is higher. However, it can be noticed that the error on the specific surface areas between the real electrodes and the synthetic microstructures remains very limited. This implies a probable limitation of the model, which fails to take into account precisely the interaction between the phases. This error on S 0,2 p is more pronounced for the aged electrode for which the Nickel phase is aggregated around clusters leading to enhance the local heterogeneity in the microstructure. As for the other properties, a good agreement 6 between the synthetic microstructures and the real reconstructions is found for the proportion of connected phase, the geometrical tortuosity, the constrictivity and the M-factor, at least for the cell at t = 0 h. It can be noted that the error is higher for the M-factor of the aged electrode (t = 2000 h). This limitation on the microstructural factor is a recurrent problem that has been already reported in literature [START_REF] Moussaoui | Particle-based model for functional and diffusion layers of solid oxide cells electrodes[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 microstructures in sofc-electrodes with completely connected phases[END_REF]. Finally, as a general matter, it can be concluded that the results are globally better with the first microstructure (t = 0 h) than with the second (t = 2000 h), in particular for the constrictivity, the tortuosity and the M-factor. As mentioned before, the model is more suited to microstructures with low local heterogeneity, which is not the case for the second microstructure, due to the agglomeration of the Nickel phase.

(µm -1 ) δ(%) ε(%) d p (µm) τ geo (-) β(-) M-factor (-) ξ T PBl (µm -2 ) S 0,1 p S 0,2 p S
(µm -1 ) δ(%) ε(%) d p (µm) τ geo (-) β(-) M-factor (-) S p (µm -1 ) δ(%) ε(%) d p (µm) τ geo (-) β(-) M-factor (-) Real µ-

STEM model validation

The STEM model is able to simulate the evolution of the microstructure by interpolation between two known states that can be emulated with the static model. In this study, electrode reconstructions are available for cells operated at the same tem-perature at different ageing times t ∈ {0 h; 1000 h; 1500 h; 2000 h}. Although it is possible to use the static model to generate microstructures fitted for these four operating times, the STEM model is designed to predict the microstructure evolution between the two bounds for which it was calibrated (i.e. with t ∈ [0; 2000]). To validate the STEM model, the microstructural properties measured on the synthetic volumes at different operating times must be compared to the ones extracted from the reconstructions. Fig. 6 shows the temporal evolution of a 3D microstructure generated by the STEM model for t ∈ [0; 2000]. In Fig. 7, some properties taken from the synthetic volumes are plotted as function of time and compared with the real data. At the same time, Tables 4 and5 compile all the properties measured on the real microstructures and the microstructures generated by the STEM model. They also include the properties extracted from 3D microstructures obtained with the static model by fitting its parameters to the real data.

First, Fig. 7 shows that over a set of six STEM simulations, the standard deviation of the measured properties is extremely The properties are computed over an average of 6 simulations. The dotted lines correspond to the standard deviation. Measurements made on 3D reconstructions of real microstructures are also indicated. small, except for the geometrical tortuosity of the Nickel phase. This last result is probably correlated with the fact that only the set of points providing the structure of the Nickel phase is mobile due to the agglomeration. Regarding the tortuosity, it can be noted that the one of the YSZ phase is almost constant over time, which is explained by the stability of the ceramic phase in operation. The STEM model is not without limitations if considering the comparison between the volume fractions, specific surface areas and interfacial specific surface areas for the synthetic and real microstructures at t = 1000 h and t = 1500 h. However, it is worth noting that the variations of these properties, sometimes significant and unintuitive for the real microstructures, come from the fact that the data are extracted from four different microstructures, and not from the same SOC whose properties were measured at four different times in its lifespan. Tables 4 and5 show in more details the relative discrepancies that arise between the properties of real microstructures and those that the STEM model can generate. It can be seen that some parameters such as the density of TPBl, the specific surface areas and, to a lesser extent, the interfacial specific surface areas, are rather well predicted by the model. The volume fractions are also sometimes a little off the mark, especially for the YSZ phase, although the deviation is rarely greater than 10%. The structural properties such as the M-factor, tortuosity, con-strictivity or mean phase diameter are also rather well approximated if the errors are put in perspective with those generally reported in literature [START_REF] Moussaoui | Particle-based model for functional and diffusion layers of solid oxide cells electrodes[END_REF][START_REF] Moussaoui | Stochastic geometrical modeling of solid oxide cells electrodes validated on 3d reconstructions[END_REF][START_REF] Prifling | Stochastic 3d microstructure modeling of anodes in lithium-ion batteries with a particular focus on local heterogeneity[END_REF][START_REF] Neumann | Stochastic 3d modeling of complex three-phase microstructures in sofc-electrodes with completely connected phases[END_REF].

Electrode at t

(µm -1 ) δ(%) ε(%) d p (µm) τ geo (-) β(-) M-factor (-) ξ T PBl (µm -2 ) S 0,1 p S 0,2 p S
Overall, the curves representing the temporal evolution of the microstructural properties generated by the STEM model (Fig. 7) are rather smooth and continuous, with the notable exception of the tortuosity of the Nickel phase. As a general comment, the deviations from the real data are usually in the range of 10% at the most, with the exception of the specific case of S 0,2 p already mentioned in section 3.1. This rather good agreement between the evolutions of real and synthetic microstructures tends to validate the model. Indeed, the STEM model is able to generate three-phase microstructures at each moment of the SOC lifetime between two known boundary states. Although the properties of the synthetic microstructures sometimes deviate slightly from the real data, their evolution is rather smooth, with the understanding that the model can still be greatly improved. Indeed, it should be noted that an area for improvement would be to reconsider the way in which the interpolation between the two extreme bounds is performed. In fact, it has been stressed that many parameters, such as the number of elements of the sets of points structuring the three phases, the cluster dispersion parameter of the Neyman-Scott point process or the smoothing parameter of the microstructure, should evolve in a linear way. This choice is obviously questionable and more accurate evolutions based on physical considerations could be implemented in the STEM model. Nevertheless, the STEM approach constitutes a relevant framework to obtain geometrical models able to reproduce correctly the microstructural evolution of SOC electrodes.

Conclusion

In this paper, an original model named STEM has been proposed to simulate the evolution of a SOC microstructure by interpolation between two known states. These bounds are simulated using a static model based on sets of random points. Special care was paid to optimize the process of microstructure generation and its evolution over time. Indeed, the generation of a microstructure with the static model takes only a few seconds, while the simulation of its temporal evolution with the STEM model takes only a few minutes. The static model showed that it is able to generate virtual microstructures whose geometrical characteristics are close to the real microstructures. The STEM model is able to simulate the continuous evolution of these characteristics. The most important discrepancies can be explained (i) by the simplicity and lack of flexibility of the static model, (ii) the improvements that still need to be made to the STEM model, and (iii) the fact that the experimental data at our disposal do not represent the temporal evolution of a single SOC (but were taken from four different SOCs of the same batch operated under the same conditions at different times).

Figure 1 .

 1 Figure 1. 3D reconstruction of three real microstructures. Cubes of 12.5 µm side (250 3 voxels). From left to right, the operating time of the cell is 0 h, 1500 h and 2000 h. The porous, YSZ and Ni phase are respectively black, gray and white.

Figure 2 .

 2 Figure 2. Covariance functions computed on 3D reconstructions of real microstructures. From left to right, the operating time of the SOC is t = 0 h, t = 1000 h, t = 1500 h and t = 2000 h.

Figure 3 .

 3 Figure 3. 3D rendering volume of three synthetic microstructures. Cubes of 17.5 µm side (350 3 voxels). From left to right, the parameters λ P = λ YZS = 2.10 -3 , θ = 3 and σ = 5 are constants, whereas λ Ni ∈ {2.10 -3 ; 1.10 -4 ; 5.10 -5 } and λ c ∈ {1; 20; 40}. The porous, YSZ and Ni phase are respectively black, gray and white.

Figure 4 .

 4 Figure 4. Evolution of the synthetic 3D microstructure between two states with the STEM model. Cubes of 12.5 µm side (250 3 voxels). Boundary states are given by ω 0 = {2.10 -3 ; 2.10 -3 ; 1.10 -4 ; 15; 10; 3} and ω 1 = {2.10 -3 ; 2.10 -3 ; 5.10 -5 ; 50; 5; 3}. The porous, YSZ and Ni phase are respectively black, gray and white.

Figure 5 .

 5 Figure 5. From left to right, 3D reconstruction of a real microstructure before use, after 2000 h of use, and synthetic microstructures generated by the static model after optimization (t = 0 h and t = 2000 h). Cube of 17.5 µm side. The porous, YSZ and Ni phase are respectively black, gray and white. Electrode at t = 0 h Porous phase Solid phase (YSZ) Properties S p (µm -1 ) δ(%) ε(%) d p (µm) τ geo (-) β(-) M-factor (-) S p (µm -1 ) δ(%) ε(%) d p (µm) τ geo (-) β(-) M-factor (-) Real µ-structure 1.49 0.95 0.22 0.56 1.61 0.59 0.010 2.39 0.99 0.49 0.59 1.17 0.61 0.166 Virtual µ-structure 1.55 0.97 0.22 0.52 1.61 0.65 0.011 2.4 0.99 0.48 0.63 1.18 0.58 0.153 Error (%) 3.6% 2.9% 0.1% 7.1% 0.1% 10.6% 4.4% 0.1% -0.8% -0.8% 6.8% -0.7% 4.5% -7.9% Solid phase (Ni) Properties S p (µm -1 ) δ(%) ε(%) d p (µm) τ geo (-) β(-) M-factor (-) ξ T PBl (µm -2 ) S 0,1

Figure 6 .

 6 Figure 6. Evolution of the microstructure generated by the STEM model by interpolation between two extreme states fitted to real data. Cubes of 12.5 µm side (250 3 voxels). The porous, YSZ and Ni phase are respectively black, gray and white.

Figure 7 .

 7 Figure 7. Evolution of the properties of the synthetic microstructure generated by the STEM model throughout the lifecycle of the SOC (from t = 0 h to t = 2000 h).The properties are computed over an average of 6 simulations. The dotted lines correspond to the standard deviation. Measurements made on 3D reconstructions of real microstructures are also indicated.

Table 1 .

 1 Comparison of the values used to generate a synthetic microstructure with those returned by the fitting process.

	Parameters	Original values (ω 0 ) Fitted values ( ω 0 )	Error
	λ P	16 µm -3	16.12 µm -3	0.7%
	λ YS Z	16 µm -3	15.91 µm -3	-0.6%
	λ Ni	0.8 µm -3	0.79 µm -3	-0.6%
	λ c	20	19.81	-0.9%
	σ	0.2 µm	0.21 µm	3.4%
	θ	3	3.02	0.7%

Table 2 .

 2 Geometric properties of the 3D reconstructions of the real and synthetic microstructures.

	1,2 p

Table 3 .

 3 Geometric properties of the 3D reconstructions of the real and synthetic microstructures.

	structure	1.41	0.99 0.31 0.72	1.35	0.58	0.041	2.25	0.99	0.47	0.6	1.16	0.59 0.163
	Virtual µ-structure 1.42	0.99 0.28 0.7	1.44	0.62	0.026	2.06	0.99	0.51	0.75	1.16	0.58 0.141
	Error (%)	∼ 0%	0.6% -8% 2.8%	7.3%	5.8% -36%	-8.7%	∼ 0% 8.7% 25%	0.2%	1.7% -13.5%
		Solid phase (Ni)
	Properties	S p	S 0,2 p	S 1,2 p
	Real µ-structure	1.05	0.92 0.22 0.8	1.65	0.56	0.009	1.11	1.3	0.11	0.94
	Virtual µ-structure 0.94	0.89 0.21 0.84	1.82	0.66	0.005	1.09	1.26	0.15	0.79
	Error (%)	-10.1%	-3% -8% 5%	10.2% 16.5% -39%	-1.8%	-3.4% 42.8% -16%

p (µm -1 ) δ(%) ε(%) d p (µm) τ geo (-) β(-) M-factor (-) ξ T PBl (µm -2 ) S 0,1

Table 4 .

 4 Geometric properties of the 3D reconstructions of the real and synthetic microstructures. Comparison between the properties of the real microstructure, those of the microstructure generated by the static model, and those of a microstructure generated by the STEM model.

		= 1000 h
		Porous phase						Solid phase (YSZ)
	Properties	S p (µm -1 ) δ(%)	ε(%) d p (µm) τ geo (-) β(-)	M-factor (-) S p (µm -1 )	δ(%)	ε(%) d p (µm) τ geo (-) β(-) M-factor (-)
	Real µ-structure 1.52	0.99	0.29	0.41	1.4	0.5	0.029	2.32	0.99	0.45	0.36	1.18	0.48 0.134
	Static model	1.46	0.99	0.29	0.44	1.42	0.53	0.028	2.2	0.99	0.45	0.44	1.18	0.49 0.134
	STEM model	1.63	0.98	0.28	0.35	1.45	0.49	0.023	2.28	0.99	0.5	0.43	1.17	0.51 0.165
	Error static (%) -3.9%	0.2% -0.9% 7.3%	-1%	5.7% -5.4%	-4.8%	∼ 0% -0.2% 19.4% -0.2% 0.9% 0.3%
	Error STEM (%) 3.4%	-0.5% -2.3% -14.6% 2.8%	-2% -21%	-1.5%	∼ 0% 11%	18%	-0.8% 5.6% 22.8%
		Solid phase (Ni)
	Properties	S p (µm -1 ) δ(%)	ε(%) d p (µm) τ geo (-) β(-)	M-factor (-) ξ T PBl (µm -2 ) S 0,1 p	S 0,2 p	S 1,2 p
	Real µ-structure 1.22	0.95	0.26	0.51	1.58	0.53	0.014	1.68	1.3	0.21	1.0
	Static model	1.24	0.97	0.26	0.48	1.63	0.63	0.013	1.68	1.21	0.25	0.98
	STEM model	1.18	0.91	0.23	0.46	1.73	0.57	0.007	1.73	1.35	0.26	0.91
	Error static (%) 1.8%	3.1% -0.9% 5.9%	-3.2% 19.1% -9.2%	0.1%	-7.4% 17.4% -1.5%
	Error STEM (%) -2.7%	-3.6% -13% -10% 9.8%	8.4% -46%	2.9%	3.6% 24.6% -8.7%
	Electrode at t = 1500 h
		Porous phase						Solid phase (YSZ)
	Properties	S p (µm -1 ) δ(%)	ε(%) d p (µm) τ geo (-) β(-)	M-factor (-) S p (µm -1 )	δ(%)	ε(%) d p (µm) τ geo (-) β(-) M-factor (-)
	Real µ-structure 1.5	0.99	0.27 0.61	1.41	0.62	0.028	2.41	0.99	0.45	0.56	1.2	0.59 0.128
	Static model	1.38	0.98	0.27 0.69	1.5	0.61	0.020	2.06	0.99	0.45	0.71	1.21	0.6	0.123
	STEM model	1.53	0.99	0.28 0.59	1.44	0.61	0.026	2.21	0.99	0.5	0.71	1.19	0.65 0.158
	Error static (%) -8%	-0.7% 0.9% 13.1% -6.3% 1.1% -28%	-14.7%	∼ 0%	-0.2% 26.8% 1.1%	1.8% -3.8%
	Error STEM (%) 2.1%	0.7% 3.7% -3.6% 2.1%	-0.8% -6.3%	-9.1%	∼ 0%	11%	26.7% -0.8% 9.9% 23%
		Solid phase (Ni)
	Properties	S p

Table 5 .

 5 Geometric properties of the 3D reconstructions of the real and synthetic microstructures. Comparison between the properties of the real microstructure, those of the microstructure generated by the static model, and those of a microstructure generated by the STEM model.

	1,2 p

Acknowledgment

The author(s) acknowledge(s) the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-18-CE05-0036 (project ECOREVE).