PhD Cătălin Bărboianu

Qualitative analysis of the reflection of the mathematical dimension of gambling in gaming online content -second technical report

Introduction

Mathematics is strongly connected to gambling through the mathematical models underlying any game of chance. Mathematics is reflected not only in games' design/characteristics and their outcomes, but also in gamblers' perception and knowledge of the mathematics-related facts of gambling -which influence their gambling behavior. The math-indispensability principle [START_REF] Bărboianu | Mathematician's call for interdisciplinary research effort[END_REF] applies not only in problemgambling research, but also in the gambling industry. The structural, informative, strategic, psychological, pathological, and ethical aspects of gambling have been identified to be grounded in the mathematics of games and gambling [START_REF] Griffiths | Fruit machine gambling: The importance of structural characteristics[END_REF][START_REF] Bărboianu | Is the secrecy of the parametric configuration of slot machines rationally justified? The exposure of the mathematical facts of games of chance as an ethical obligation[END_REF][START_REF] Bărboianu | Mathematical models of games of chance: Epistemological taxonomy and potential in problem-gambling research[END_REF]Turner & Hobay, 2004;Harrigan, 2009, and others).

Gambling cognitive distortions, language, and miscommunication

Gambling-specific cognitive distortions (in the form of misconceptions, misunderstandings, reasoning fallacies, biases, false or irrational beliefs, or illusions, alone or mixed) are believed to be an important cause of the development of problem gambling and are considered as major risk factors [START_REF] Lambros | Numerical reasoning ability and irrational beliefs in problem gambling[END_REF]Leonard & Williams, 2016, and others). We have analyzed these cognitive distortions in relation to the mathematical dimension of gambling and found that most of them are mathematically related (Bărboianu, 2022, pp. 219-221). An important element that shapes and influences the aspects of gambling mentioned above, especially cognitive distortions, is language. The language of gambling can be intentionally or innocently misleading, confusing or conflicting, largely due to the mathematical nature of the essential concepts governing gambling, but also to the nature of language itself. The language of gambling unavoidably uses mathematical and mathematically-related terms and as such is a mixed language and therefore predisposed to semantic conflicts. This language may aim toward descriptions of the games, of their associated strategies, for communication between gamblers and between gamblers and experts, and to express any observations or research results in regard to this phenomenon. The fact that some specialized terms belong or are tightly related to probability theory accounts for their conflicting potential in the gambling language, since the concepts of probability theory are sensitive to interpretation, despite their mathematical nature (Bărboianu, 2022, pp. 203-218). The problematic gambling language manifests in the activity spheres of developers, operators, gambling communities, information providers, experts (including problemgambling experts), and gamblers' relationships with these people. This language affects the descriptions of games and gambling that the players actually use to become informed about the phenomenon; also affected are the communication between gamblers, and between gamblers and people from the gambling industry or experts as well as gamblers' own conceptual judgments related to gambling.

Goals and outcomes of the study

In this theoretical framework, research is able to derive concrete norms and criteria to adequately reflect the mathematical dimension of gambling in the communication and texts associated with the gambling industry. These norms and criteria of adequacy will be further communicated to policy and decision makers in both governmental and private sectors, with the recommendation for implementation. Our study aims to evaluate qualitatively the reflection of the mathematical dimension of gambling in the content of gambling websites. This analysis is necessary in order to have an objective and concrete image of the actual state of this matter in the online industry and of the challenges that such research and application would face in the real world of gambling. A minimum number of 120 gambling websites will be reviewed annually for their content in that respect. A statistical analysis will record the presence of the mathematical dimension of gambling and its forms in the content of participating websites, and a qualitative research will analyze and assess the quality of the content with respect to that dimension.

Methods and technical description of the instruments

The current study is a combination of quantitative and qualitative analysis, in which the latter is predominant and is given the central role.

The participants in the study (gambling websites, through their webmasters) were recruited through online advertising and direct invitations. Given the focus on the qualitative aspect of the study, the sample is not representative for the entire population. Besides, representativeness cannot be established with respect to the specificity of the population (gambling websites) and of the targets of the study. The criteria of eligibility for participation that we have applied are:

-not having legally prohibited content or advertising; -meeting the gambling legal requirements; -having informative content besides the games and games' rules -being fully operational and navigable.

The quantitative analysis will use basic descriptive-statistics methods, summarizing the data recorded from the sample by standard statistical indicators, with the following main specific variables: v₁ -the presence of structural descriptions of the games in parametric terms (valued yes/no); v₂ -the presence of informative sections ('How to' articles, blogs, guides) (valued yes/no); v₃ -the presence of sections dedicated to odds/probability/math (valued yes/no); v₄ -the usage of essential math terms specific to gambling (odds/probability, expectation, average/mean, etc.) (valued yes/no); v₅ -the presence of the definitions of the math terms used (valued yes/no); v₆ -the correctness of the math definitions used (valued on a scale from 0 to 5) v₇ -the presence of game strategy topics (valued yes/no); v₈ -the presence of math-based game strategy topics where applicable (using notions of probability theory, statistics, and game theory) (valued yes/no); v₉ -the presence of systematic-learning or advanced content of gambling math (lessons, academy-style sections, in-depth guides, etc.) (valued yes/no); v₁₀ -the mentioning of author for math-related articles (valued yes/no); v₁₁ -the match between the math-related articles and their authors' declared expertise (valued on a scale from 0 to 3); v₁₂ -in-text presence of awareness on possible misconceptions, fallacies, and irrational beliefs in regard to gambling (valued yes/no); v₁₃ -the correlation of the above awareness with the mathematical aspects of gambling (valued on a scale from 0 to 3).

The values are conditional on each other as follows: v₃, v₇, v₉ conditional on v₂; v₅ conditional on v₄; v₆ conditional on v₅; v₈ conditional on v₇; v₁₁ conditional on v₁₀; v₁₃ conditional on v₁₂.

The qualitative analysis will use as methods discourse analysis, content analysis, thematic analysis, conceptual interpretation, semantic analysis, doubt about sense, and analysis of arguments. It will have a strong component of linguistic-conceptual-logical analysis, targeting the following main elements: 1 -the usage of terms with non-uniform semantics; 2 -the contextual usage of math terms; 3 -the conceptual linkages relative to the relevance for the topic; 4 -the soundness of arguments based on applied math; 5 -the association between game strategy and the concepts of probability theory and game theory; 6 -the presence and contextual impact of "mathematically prohibited" or misleading terms (such as 'winning strategy', 'how to win', etc.).

Although the qualitative analysis is independent of the quantitative one, correlations will be made between the conclusions of the former and the variables of the latter. Representation of the mathematical dimension of gambling may or may not be adequate in the content of gambling websites. The main goal of the qualitative analysis is to establish the disciplinary areas, as well as their individual roles, which can contribute to the theoretical framework that will derive the norms and criteria for such adequacy in the content of the websites and of the gambling communication. These disciplinary areas entitled for involvement include mathematics, psychology, linguistics, philosophy of language, epistemology, and philosophy of mathematics.

Content and roles of the technical reports

Monthly technical reports describing the partial results of the qualitative analysis will be published on academic channels, preceding the main publication at the end of the study. Each technical report will cover the review and recording of data from a minimum of ten websites. Some of the websites reviewed during the project are nominated in the section titled Appendix, along with brief descriptions from their owners. We found such technical reports necessary, first because the current study is atypical in what concerns the statistical analysis and the qualitative analysis, as well as the objects under investigation. Therefore, the main role of the technical reports is to detect and define any methodological and technical difficulties encountered during this study and any challenges they may pose, for them to be analyzed and surmounted in both the continuation of the current study and any future similar research. Besides presenting these difficulties and challenges, the technical reports will also contain unpredicted observations regarding the analyzed content that might require the revision or change of the methods and instruments used. The results of the technical reports will be gathered, and general conclusions will be drawn in the main publication.

Observations and conclusions from the review of the second sample

The partial results of the statistical analysis confirmed most of our expectations in regard to content, although conclusive results for various hypotheses cannot yet be stated due to the small size of the sample reviewed thus far.

The only variables receiving zero (no) value for all reviewed websites (100%) are v₁ and v₉. It worth noting that even those websites with gambling-math content detected do not have structural descriptions of the games in parametric form, although this is a requirement for explaining how the math was applied to describe them, the behavior of their outcomes, or the effectiveness of any gambling strategy presented. Even the simplistic descriptions for the game of slots did not refer to their structural parameters, except the number of reels (symbol distribution, weighting of the reels, etc.), which account for each version of the game before considering the mathematical facts associated with them. Although we did not expect much systematic-learning content (variable v₉), as the partial statistics confirmed, we expected more strategic content, even in a "how to win" approach, not necessarily math related. About 30.7% of the reviewed websites have strategic sections (variable v₇), of which 75% have math-based game strategy topics (variable v₈). Of the reviewed websites, 23.07% touched on the subject of gambling misconceptions and fallacies, of which 66.66% related them to some extent to gambling mathematics.

All of the websites reviewed thus far have links to online casinos and are affiliated with the games in one form or another, so it is fair to hypothesize that making profit from these games is their main aim. This profit motive might be a reason for these websites to have little interest in informative content, although content is a factor in SEO performance. As we know, sites dedicated exclusively or mostly to gambling information do exist, and such sites are higher ranked in Google searches than the small-to-medium gambling affiliates. Our advertising approach for the current program has targeted forums and online communities most suited for this latter category of gambling websites, and hence the informative sites have not been reached. However, as we stated, the main aim of our study is a qualitative analysis of the content, and while randomness is somewhat ensured by the method of collecting participants, the sample is not required to be representative for the entire population. In light of this principle, we propose to review separately a number of sites in the category previously described just in the qualitative aspects without extending the current statistical analysis to include them.

For the few gambling websites detected with gambling-math content thus far, we noticed that their articles are not systematized relative to each other regarding delivery of expert knowledge to the general public -they are conceived to be read independently. The general tendency that we detected in regard to organization of content and content itself is a struggle for SEO performance. This approach obviously affects content as to its expert dimension and epistemic goals. Just to mention a few factors: the excessive repetition of some keywords, unnecessary descriptions in the detriment of technical content with specialized terms, incomplete arguments, etc. We have even detected articles claimed as "expert", some of them with gambling-math content, showing "mathematically prohibited" or misleading terms (such as 'how to win'), which hypothetically can be related to SEO strategies. Some general predominant tendencies from the first sample get contoured with this second sample: -the SEO approach at the expense of poor quality of content; -the lack of qualified experts for some topics; -not indicating (let alone explaining) the importance of the roles of the mathematical dimension of gambling.

Most of the reviewed websites have links to responsible-gambling organizations. Some of them have responsible-gambling sections. The presence of such links or sections was not a variable to measure in the statistical analysis, but it is worth analyzing it in relation to other variables (v₇, v₁₂, v₁₃). One site has reflected in their responsible-gambling page the mathematical dimension of gambling relative to a healthy attitude about gambling, including ethical aspects.

Appendix -Selective list of the websites reviewed during the project ________________________________________________________________________ casinovertailu.com Finnish website reviewing casinos of all kinds of brands. It aims to give detailed information for the players to help them to find the right places for them to gamble.

instantwithdrawals.org

Expert casino reviews and ratings based on withdrawal speed and stability. Details of fastest cashout casinos per country, software and payment method. allfreechips.com Online gaming informational site delivering both user and professional reviews about online gaming. casino-professor.com/en Online casino comparison domain that lists licensed casino sites from around the world. Featuring reviews with real tests, videos, user comments, guides, and in-depth content.

gamblermaster.com Site featuring a wide variety of guides, reviews, and articles about gambling online with real money, primarily in the online casino space. onlinecasinosfinder.com Provides reviews of a selection of online casinos, online casino gambling guides and tips for beginners and seasoned pros, and free online casino games. gamble-usa.com Licensed and trusted source for all things gambling. The site reviews and compares online casino operators and Sportsbooks in every US state. Its gambling guides include tips, tricks, and strategies. casinorankings.com Casino reviews and rankings provided by experts. Beginner's guide to casino gambling in both local and online casinos. jeffbet.com Fully licensed online casino and sportsbook offering a wide range of games and betting markets. The casino educates its players with gambling guides and game reviews.

bitcasinorank.com Provides information on everything related to gambling with cryptocurrency. Specializing in reviewing and ranking cryptocurrency online casinos.


Dr Alexandre Manon

Alexandre Nau

Dr Thomas Belin

Arnaud Mazurier

Dr Jean Marc Bassat

Dr Nicolas Bion

Dr Clément Comminges
email: clement.comminges@univ-poitiers.fr

Jean Marc Bassat

Dr A Nau

Dr A Mazurier

Influence of Electrode Potential on Oxygen Mobility Probed by Polarized Isotopic Exchange in Solid Oxide Electrolyser Cells: Insights for Electro-Assisted Oxidation Reactions

Keywords: Polarized Oxygen Isotopic Exchange, SOEC, Oxygen mobility, MIEC, Oxygen catalysis

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Coupling catalytic oxidation reactions with gas phase electrocatalysis by employing Solid Oxide Electrolyser Cells (SOEC) is a topic of raising interest. Indeed, SOEC systems are investigated for decades for high temperature water electrolysis (HTE) to produce hydrogen [1] or syngas (via CO2 / H2O coelectrolysis) [2] at the cathode. In these systems, the anodic reaction consists of releasing oxygen gas via the oxygen evolution reaction (OER). Attempts to valorise this oxygen through coupled catalytic oxidation reactions at the anode of the SOEC have been realized to generate valuable molecules. By introducing CH4 in the anodic chamber, valuable CH4 oxidation products can be obtained (syngas by partial oxidation of methane or ethylene via oxidative coupling of methane) [3] , while undesirable CO2 resulting from CH4 combustion is to be avoided [4] . To perform these catalytically assisted electrolyses, one must develop anodic electrocatalysts that fulfils the classical prerequisites in electrochemistry (electronic conduction, O 2-ionic conduction) as well as in gas phase catalysis (porosity, specific surface area, active sites for CH4 adsorption, C-H bond activation, oxygen surface / bulk diffusion…). The control of the oxygen mobility is therefore crucial for the development of this technology. Reference materials used as anodes in SOEC are classically doped perovskites [5] or Ruddlesden-Popper phases [6] . There is therefore a great challenge both in the material development as well as in the understanding of oxygen mobility in these materials. Among the fundamental questions that can be raised, the effect of the electrode potential on the oxygen mobility is particularly relevant. Only few studies related to oxygen operando isotopic exchange with a control of the electrode potential have been performed. Sobyanin et al. performed homomolecular oxygen isotope exchange on a Pt/10-YSZ/Pt electrochemical cell at 500°C and concluded that the electrode polarization had not effect on the oxygen homomolecular exchange rate with respect to its open circuit voltage. [7] Wachsman et al. developed another strategy that enables the possibility of performing Polarized Oxygen Isotopic Exchange (POIE) directly on catalytic powders. They investigated the influence of a cathodic polarization on the oxygen surface exchange coefficient (k*) of several catalysts (LSM, LSCF, LSC and LSF) and concluded that k* is greatly increasing with the applied overpotential (η). This study is confirming the findings of Hopper et al. which investigated epitaxial thin film heterostructures of LSCF/GDC/YSZ with in situ synchrotron X-ray diffraction. [8] We recently developed the polarized isotopic exchange method that enables the possibility to perform Oxygen Isotopic Exchange (OIE) experiments in situ on symmetrical cells and to identify the rate determining step by evaluating the kinetic constant dependency on the applied electrode potential [9] . The goal of this study is to understand the influence of the electrode potential on the oxygen mobility on three different oxide materials classically used as oxygen electrodes in SOFC or SOEC cells: i) La0.8Sr0.2MnO3 (LSM), a perovskite with partially filled band considered as electronic conductor [10] . Due to its very poor oxygen ionic conductivity (Dbulk = 3.1.10 -16 cm².s -1 at 700°C) [11] , active sites for oxygen reduction and incorporation are located at the triple phase boundary (TPB) [12] , although at high overpotential oxygen ions can diffuse in the bulk oxide to reach the LSM/YSZ interphase. [13] ii) La0.6Sr0.4Co0.2Fe0.8O3- (LSCF), an oxygen deficient perovskite with a band gap, considered as a p-type semi-conductor and possessing a Mixed Ionic and Electronic Conduction (MIEC) [10] . At temperatures below 600°C, the ionic conductivity is rather low and the classical TPB model applies, as for LSM. [14] Above 600°C, LSCF behaves as a MIEC and the oxygen reduction reaction (ORR) is stimulated by the formation of oxygen vacancies in the oxide while increasing the electrode overpotential. Interestingly, it has been shown on LSCF epitaxial thin films [8] as well as on LSCF polarized powder [15] that increasing the electrode overvoltage (cathodic or anodic) induces a significant increase in the surface exchange coefficient (k*). iii) La2NiO4+ (LNO) is a material which belongs to the so-called Ruddlesden-Popper series Lnn+1NinO3n+1 (here with Ln = La and n =1). [6a] This compound shows a K2NiF4type layered structure, with alternating LnNiO3 perovskite layers and LnO rocksalt layers within which reside the additional interstitial oxygen atoms. This oxygen over stoichiometry is responsible of a large oxygen conductivity, which combined to reasonable p-type electronic conduction properties [16] leads to obtain an oxide with interesting MIEC properties. [17] More precisely, the oxygen conductivity mechanism is labelled as interstitially, because involving a pushpull mechanism between the interstitial and apical oxygens (being the last ones located in the perovskite layer). [18] Thanks to the stacking along the crystallographic c-axis of the structure, such MIEC properties are anisotropic, being the conductivities about 1000 times larger in the (a,b) plane compared to the stacking caxis. [19] It is to note that the oxygen over-stoichiometry (+δ) is influenced by the temperature and the oxygen partial pressure [16,20] , such oxide "breathing" in a reversible way oxygen above a limit located at low temperature (around 400°C). In first approximation, a decrease of the over-stoichiometry diminishes the ionic conductivity [21] and electronic conductivity, as it lowers at the same time the hole concentration. [16,20] It is generally accepted that dissociative adsorption of oxygen is the rate determining step for LNO electrodes. [22] Such kind of nickelates (also including Ln = Pr and Nd) are extensively studied as SOFC oxygen electrode materials. [23]

Results and Discussion

OIE on Powder Samples

Diffusion coefficients and rate constants for heteroexchange were calculated by fitting the curves of heterolytic exchange with O2. As detailed by Klier et al. [24] and Muzykantov et al. [25] , three types of isotope exchange were considered in this numerical approach: (i) homomolecular exchange type (named r0) with no participation of atoms from the solid (Equation 1): 18 O2 + 16 O2 → 2 16 O 18 O

(1) (ii) heterolytic exchange (named r1) between one atom from the gas phase and one atom from the solid. Oxygen atoms from the solid surface (Os) are involved in this reaction (Equations 2 and 3): 18 O2 + 16 Os → 16 O 18 O + 18

Os

(2) 16 O 18 O + 16 Os → 16 O2 + 18 Os (3) (iii) 2-atoms heterolytic exchange (named r2) between the gas O2 molecule and the oxygen atoms from the solid (Equations 4-6): 18 O2 + 2 16 Os → 16 O2 + 2 18 Os (4) 16 O 18 O + 2 16 Os → 16 O2 + 16 Os + 18 Os (5) 16 O 18 O + 2 18 Os → 18 O2 + 16 Os + 18 Os (6) A model derived from Klier et al. was used to describe the main processes [24] : both exchange rate and bulk isotope diffusion were involved. Sample particles were modelled as one dimension spherical particles of radius L (m) (90 nm, 112 nm and 1 µm for LSM, LSCF and LNO respectively). The oxygen amount in the sample (Nox atoms) was defined as the sum of oxygen atomic amount in the solid except the surface (Nv atoms) and those located at the surface (Ns atoms). It was determined using crystallographic considerations: 1.3101x10⁻², 1.3017x10⁻² and 1.0125x10 -2 mol.g⁻¹ for LSM, LSCF and LNO respectively. The homomolecular exchange reaction in the gas phase (r0 constant rate in s -1 ) was considered as negligible due to the low temperature used for the isothermal isotopic exchange experiments. In the gas phase, the variation of the 18 O fraction αg was related to the adsorbed amount of oxygen on the oxide following Equation 7:

𝜕𝛼 𝑔 𝜕𝑡 = 𝑏 ⋅ 𝑟 𝐻 ⋅ (𝛼 𝑠 -𝛼 𝑔 ) (7)
where b is the ratio of the number of oxygen atoms at the surface Ns of the solid compared to the gas phase Ng, 𝑟 𝐻 the rate constant for the heteroexchange reactions (rH = 0.5r1 + r2 in s -1 ) and αs the 18 O fraction at the oxide surface. At the gas/solid interface, the oxygen atoms were assumed to be identical to the bulk (homogeneous surface) and the number of adsorbed oxygen was considered negligible compared to the number of oxygen atoms in the oxide surface. The variation of the atomic fraction of 18 O at the surface of the sphere αs was given by the quantity adsorbed on this surface from which was subtracted the quantity which diffuses in the solid (Equation 8):

𝜕𝛼 𝑠 𝜕𝑡 = 𝑟 𝐻 ⋅ (𝛼 𝑔 -𝛼 𝑠 ) -3 ⋅ 𝑏 𝑠 ⋅ 𝐷 𝐿 ⋅ 𝜕𝛼 𝜕𝑅 | 𝑅=𝐿 (8)
where 𝑏 𝑠 is ratio between Nv which is the oxygen atomic amount in the solid except the surface and Ns the oxygen atomic amount at the surface. D (m 2 s -1 ) is the effective diffusion coefficient inside the particle. In the solid, the second Fick's law was considered for the diffusion according to Equation 9:

𝜕𝛼 𝜕𝑡 = 𝐷 𝑅 2 ⋅ 𝜕 𝜕𝑅 ⋅ (𝑅 2 𝜕𝛼 𝜕𝑅 ) (9)
where α is the 18 O fraction in the solid except the surface, D the already defined diffusion coefficient and R the coordinate along the oxide particle radius (m). The variation of the 16 O 18 O gas fraction x34 is given by Equation 10:

𝜕𝑥 34 𝜕𝑡 = 𝑏{𝑟 0 [2𝛼 𝑔 (1 -𝛼 𝑔 ) -𝑥 34 ] + 𝑟 1 [𝛼 𝑔 (1 -𝛼 𝑠 ) + 𝛼 𝑠 (1 -𝛼 𝑔 ) - 𝑥 34 ] + 𝑟 2 [2𝛼 𝑠 (1 -𝛼 𝑠 -𝑥 34 )]} (10)
Using the definition of the 18 O fraction αg (Equation 11):

𝛼 𝑔 = 1 2
𝑥 34 + 𝑥 36 (11) and the sum of all oxygen atomic fractions (Equation 12): 𝑥 32 + 𝑥 34 + 𝑥 36 = 1 (12) the calculated 16 O2 and 18 O2 gas fraction could be determined as a function of 18 O fraction αg and the 16

The initial conditions were set as a known amount of 18 O oxygen in the gas phase and none at the surface nor in the oxide (the natural abundance of 18 O in oxide was neglected). The boundary conditions were expressed by a finite concentration at the core of the oxide particle (Equation 15):

𝜕𝛼 𝜕𝑅 | 𝑅=0 = 0 ( 15
)
and by an identical atomic fraction at the surface and in the oxide volume (Equation 16): 𝛼| 𝑅=𝐿 = 𝛼 𝑠 (16) Using an homemade software written in Python, this system of 3 equations (in adimensional forms) was solved by the method of lines (70 points along the characteristic length L were used) [START_REF] Krank | The mathematics of diffusion[END_REF] . The number of points used was adjusted in order to obtain invariance of the results. Fraction profiles ( 16 O2, 16 O 18 O and 18 O2 in the gas phase) as a function of time were obtained with the numerical solution of this system. Then, the physicochemical parameters were determined from the experimental gas fraction curves using the global particle swarm optimization method (PSO) [START_REF] Poli | [END_REF] . This method is a very popular optimization which has been successfully applied to the kinetic parameters estimation problem [28] . The following parameters were applied: 20 swarm particles over 1000 iterations and 1x10 -8 as minimal change in the swarm's best objective value. The sum of squared residuals was used as the minimizing function (Equation 17): 𝜎 = ∑ (𝑥 32 𝑒𝑥𝑝 -𝑥 32 𝑐𝑎𝑙𝑐 )

where x32, x34 and x36 are mole fractions of oxygen molecules of three different isotopic compositions 16 O2, 16 O 18 O and 18 O2 in the gas phase, indexes exp and calc are related to experimental and calculated values of oxygen mole fractions. Figure 1 shows the temperature programmed oxygen isotopic exchange (TPOIE) over LSM, LSCF and LNO. Commercial powders of LSM and LSCF were used, whereas LNO was synthetized by solid-state reaction. The powder X-ray diffraction of the synthesised LNO is plotted in Figure S1. Single phase was obtained without detecting any secondary phase. [17] Two regimes are observed in the case of LSM. The first at low temperature (ca. 360°C) and the second one at higher temperature (ca. 500°C). This behaviour is similar to the one observed for LaMnO3. [29] Those two regimes are attributed to surface and subsurface diffusion at low temperature, whereas bulk diffusion is triggered at higher temperature. Oxygen exchange on LSCF starts at low temperature (ca. 250°C) and quickly reaches its equilibrium value at 420°C indicating that all oxygen atoms from the gas phase are easily exchangeable with oxygen atoms from the oxide. For LNO, oxygen exchange begins at 360°C and reaches equilibrium at 500°C. Fitting of g with the model at different temperatures (Figure 2) allows determining r1, r2, rH and their corresponding activation energies, the results are shown in Table 1. In the case of LSM the one atom heteroexchange is the dominant mechanism for oxygen exchange at all investigated temperatures as shown by the value of the relative proportion X1. In the case of LSCF, the one atom heteroexchange mechanism is favoured at low temperature and tends to decrease in favour of the two atoms heteroexchange (the X2 value increases with temperature). This corresponds to the temperature where the O 2-ionic conductivity becomes appreciable. Therefore, we can attribute the one atom heteroexchange to the electronic properties whereas the two atoms heteroexchange is mostly due to the ionic conductivity of the oxide. LNO, which is also a MIEC, behaves differently from LSCF since only the 2 atoms heteroexchange is identified over the temperature range. The overall heteroexchange properties are reflected by rH (determined from Equation 7). Activation energies for rH are 76 kJ.mol -1 , 74 kJ.mol -1 and 17 kJ.mol -1 for LSM, LSCF and LNO respectively. These activation energies for heteroexchange are quite different from literature values. For LSM, the activation energy for the surface exchange coefficient k* is 128 kJ.mol -1 when determined from Isotope exchange / depth profile on a dense sample. [11] For LSCF, a value of 45.4 kJ.mol -1 below 500°C was obtained from isothermal isotope exchange on powder sample. [30] Concerning LNO, apparent activation energies for surface exchange are generally ranging from 120 to 140 kJ.mol -1 , although much smaller values were reported. It is difficult to explain these discrepancies, although it is known that the sample morphology (powder, dense sample, porous electrodes (where the gas diffusion can affect the mass transport) as well as the employed method (Electrical conductivity relaxation (ECR), oxygen isotopic exchange (OIE), electrochemical impedance spectroscopy (EIS)) can give quite different results. [31] Electrochemical Characterizations of Symmetrical Cells Symmetrical cells were prepared by screen printing inks of LSM, LSCF and LNO on dense 8-YSZ substrates 248 µm thick (see experimental section). Interfacial layers of ceria doped gadolinium oxide (CGO) were employed for the LSM and LSCF cells to inhibit the reactivity between the catalysts and YSZ electrolyte. As preliminary tests, no CGO layer was employed for LNO symmetrical cells. The porous electrode layers obtained is typically 23 µm thick as measured by X-ray tomography (Figure 3) on the LSM symmetrical cell. The CGO interfacial layer (average of 9 µm thickness) is not totally flat nor smooth and the agglomerated particles present some voids in the dense YSZ coverage (Figure S2). Consequently, one cannot exclude the possibility of some remaining LSM/YSZ and LSCF/YSZ direct interface. Figure 4 shows the electrochemical impedance spectra, linear scan voltammetries and corresponding Tafel plots obtained for the three symmetrical cells at 600°C. Nyquist plots measured at open circuit voltage (OCV) at 600°C are shown for LSM (Figure 4a), LSCF (Figure 4d) and LNO (Figure 4g) symmetrical cells. Typical spectrum of porous LSM is obtained at 600°C with a large low frequency arc related to gas diffusion in the porous structure (mass transfer resistance) particularly pronounced at 20 mbar O2 and a high frequency arc attributed to electrode kinetics. [5] The LSCF electrode displays also the expected electrochemical behaviour with mostly three visible contributions at OCV. The high frequency contribution is attributed to the LSCF/CGO interphase charge transfer, the mid frequency to oxygen surface exchange and the low frequency contribution to gas phase O2 diffusion in pores. [32] The LNO electrode is characterized by two depressed semi circles. Its ohmic resistance is higher than that of LSM and LSCF cells at the same temperature, LNO being known to react with the YSZ electrolyte material forming insulating La2Zr2O7 or SrZrO3 particles at the electrode/electrolyte interphase when sintered at high temperature. [17] Two processes are observed at medium and low frequency with a higher impedance for the process at mid frequency, as reported in the literature at low PO2 [22c] . The contribution at mid frequency (kHz) is associated to interfacial ionic transfer impedance between the electrode material and the YSZ electrolyte, and the low frequency contribution (Hz) to the electrode reaction process (molecular oxygen dissociation and adsorption phenomena). [17, 22c] Table 1. Physicochemical parameters obtained from the refinement of oxygen gas fraction curves using global particle swarm optimization method. Heteroexchange rate constants (r1, r2 and rH) and diffusion coefficient related to the characteristic length (D/L²) are given with the coefficient of determination (R 2 ). Relative proportions of r1 (X1) and r2 (X2) for the whole exchange process are calculated. Assuming an Arrhenius law within the temperature range explored for each material, values for activation energy are also tabulated. - [a] X1 (-) [b] 0.9095 0.9770 0.9921 0.7959 0.6901 0.6278

-[a] -[a]
- [a] Ea (r1) (kJ.mol -1 ) 79.97 68.02 - [a] r2 (× 10 [a] The calculated values are negligible (< 1×10 -16 s -1 ). [b] The relative proportion Xi is defined as Xi = r1 / (r1 + r2).

Despite the possible formation of La2Zr2O7 or SrZrO3 at the LNO/YSZ interphase (which can partially hinder the O 2-ionic diffusion from LNO to YSZ), no additional processes were observed on the EIS spectra at 600°C (Figure 4g) nor at 700°C (Figure S3) so the cell was used for further experimentations.

Polarization curves (without and with ohmic drop correction) are shown on Figure 4b, 4e and 4h for LSM, LSCF and LNO respectively. The current density evolution for LSM cell (Figure 4b) and LNO cell (Figure 4h) at low cell voltage (-0.4 V < E < 0.4 V) is rather flat. In this domain, the main process involved is O2 reacting directly from the gas phase to the TPB, as the number of these actives sites is rather limited compared to the whole porous electrode surface. Then, the current density increases exponentially which indicates that the electrochemical process involves the surface diffusion path in addition to the gas/TPB path.

It is worth noting that the current density is rather small on LNO, possibly due to zirconate formation at the LNO/YSZ interphase.

The LSCF electrode (Figure 4e) shows rapid electrochemical kinetics with the highest current densities among the three electrodes investigated. Corresponding Tafel plots are shown in Figures 4c, 4f and 4i at 600°C for LSM, LSCF and LNO respectively where log I is plotted as a function of electrode overpotential (η). Since no reference electrode is used in the experimental setup, one must estimate the overpotential magnitude. As the polarization curves are reasonably symmetrical (Figures 4b, 4e and 4h), we can consider that the total driving force (E = ηanodeηcathode) is equally split in two parts, half for the oxygen evolution reaction (OER) at the anode and half for the ORR at the cathode. In other words, we consider by this hypothesis that  symmetry coefficient (in the sense of Marcus Hush) is equal to 0.5, which is not rigorously exact as experimentally demonstrated by Monaco et al. on LSCF symmetrical cells. [33] To evaluate this hypothesis, we performed polarization curves at 700°C on a LNO symmetrical cell including a platinum reference electrode (Figure S4). It seems that this hypothesis is reasonable for estimating the anodic overvoltage, but leads to an underestimation of the cathodic overvoltage. In this way, the overvoltage η at one electrode can be calculated with Equation 18:

𝜂 = Δ𝐸-𝑅 𝐻𝐹 𝐼 2 ( 18
)
Where η is the overvoltage (V), E is the applied cell voltage (V), RHF is the sum of all ohmic resistances (electrolyte ionic conductivity, electrical contacts, cables) determined as the high frequency intercept with the real axis in the Nyquist diagram () and I the current (A). The term E-RHFI corresponds to the ohmic drop correction (in Figure 4b, 4e and 4h). Exchange current densities (j°) are determined from the intersection of Tafel cathodic and anodic branches at an overvoltage close to η = 0 V. The exchange current density is proportional to the surface exchange coefficient k * according to Equation 19: [14] 𝑗°=

4𝐹𝑘 * [𝑂 2-] 𝐴 ( 19
)
Where j° is the exchange current density (A.cm -2 ), F is the Faraday constant (C.mol -1 ), k * is the surface exchange coefficient (cm.s -1 ), [O 2-] is the bulk concentration of oxide ions in the oxide electrode (mol.cm -3 ) and A is electrode area (cm²). Therefore, comparing j° values is a convenient way to evaluate the evolution of k * . Table 2 shows the values of j° for the three electrodes at different temperatures as well as corresponding activation energies. For LSCF, the obtained values of j° are very well matching with those obtained by Esquirol et al. [14] As expected from the polarization curves, j° is much higher in the case of LSCF than LNO and LSM. Apparent activation energies were estimated from j° values for the three electrode materials investigated. These are quite different from those obtained from OIE experiments on powders for rH (Table 1), indicating that heteroexchange activation energy (rH) (or to the surface exchange coefficient k*) is sensitive to the material morphology (powder vs. porous electrode).

Polarized Oxygen Isotopic Exchange (POIE) on Symmetrical Cells

Figure 5 shows typical POIE experiments along with the fitting of g vs. time. One can distinguish two different behaviours: i) a dominant one oxygen atom heteroexchange mechanism in the case of LSM and LNO (Figures 5a and 5e respectively). This is exemplified by the majority formation of the 16 O 18 O isotopologue (m/z = 34) (Equation 2) and ii) a dominant two oxygen atoms heteroexchange mechanism in the case of LSCF (Equation 4) confirmed by the majority formation of 16 O2 isotopologue (m/z = 32). This observation is in agreement with the dominant mechanism observed on LSM and LSCF powders, where at high temperature the one atom heteroexchange is the main mechanism observed for LSM whereas the two atoms heteroexchange is the main mechanism observed for LSCF (Table 1). However, LNO displays a dominant one atom heteroexchange on the POIE experiments (Figure 5e) whereas the two atoms heteroexchange is the main mechanism on powder OIE (Figure 2c). The two atoms heteroexchange mechanism is most probably related to the MIEC properties of the catalyst, where two adsorbed 18 O atoms coming from the dissociative adsorption of 18 O2 can be more easily incorporated in the bulk of the MIEC (the bulk path) than diffusing to the TPB and then incorporating in the YSZ electrolyte. The latter mechanism (TPB way or surface path) is more likely to proceed via a one atom heteroexchange. This discrepancy between powder LNO and LNO symmetrical cell can arise from the formation of the insulating pyrochlore or zirconate layer at the electrode/electrolyte interphase which in turn blocks the O 2-incorporation into YSZ, thus favouring the TPB pathway. In all POIE experiments, the electrical current is quite stable with time indicating that electrodes reach steady state during the polarized isotopic exchange experiments. The 18 O gas atomic fraction (g) is determined from Equation 11 and its evolution with time is plotted on Figures 5b, 5d and 5f for LSM, LSCF and LNO respectively. Kinetics of oxygen exchange are assumed to be first or pseudo-first order reactions, implying that molecular oxygen is involved in the rate determining step (rds). [34] Two distinct kinetic models are proposed. In the case of LSM and LNO (where the one atom heteroexchange prevails), a simple first order decay of g as a function of time is observed and can be described with Equation 20:

𝑎 𝑔 = 𝐴 𝑒𝑥𝑝 -𝑘𝑡 ( 20
)
where A is the pre-exponential factor (same unit as g and is close to 100 %), k is the first order rate constant of the rds (s -1 ) and t is time (s). The rate constant describes the surface path. In the situation of LSCF (where the two atoms heteroexchange prevails), the simple first order decay shown in Equation 20does not fit correctly the data (Figure 5d) where it seems that two kinetic constants are observed. The kinetic model used for LSCF is therefore involving a competitive two first order reactions (Equation 21): 𝑎 𝑔 = 𝐴 𝑓𝑎𝑠𝑡 𝑒𝑥𝑝 -𝑘 𝑓𝑎𝑠𝑡 𝑡 + 𝐴 𝑠𝑙𝑜𝑤 𝑒𝑥𝑝 -𝑘 𝑠𝑙𝑜𝑤 𝑡 (21

)
Where the pre-exponential factors Afast and Aslow are the weight (or population) of oxygen atoms following the fast and slow pathway respectively (%), kfast and kslow are the corresponding first order rate constants. This model describes the two different possible paths that can occur on such a MIEC electrode: the surface path and the bulk path. [33] As the two atoms heteroexchange prevails, we therefore assign kfast to the bulk path and kslow to the surface path. By fitting the g = f(t) curves obtained for LSM, LSCF and LNO with the appropriate kinetic model, one can evaluate the evolution of first order rate constants as a function of temperature as well as the overvoltage (η). It is worth noting that these kinetic constants are related to surface reactions and are expressed in s -1 . Usually, the surface exchange coefficient (ks * expressed in cm.s -1 ) is used to characterize the surface exchange properties and is expressed as a first order kinetic constant normalized to a characteristic length (x). The cm.s -1 unit was not chosen in the present study since the choice of this characteristic length requires the knowledge of the rate determining step, i.e. the distance over which an electron is transferred from substrate to adsorbate in the case of a rds involving an electron transfer, what can be approximated to be the cation-anion distance in the oxide. [10] Obtained kinetic constants as well as pre-exponential factors can be found in Tables S1,S2, and S3 for LSM, LSCF and LNO respectively. Figure 6 shows the evolution of k (surface path, Equation 20) in the case of LSM as a function of temperature and overvoltage. It can be observed that k is increasing with the applied overvoltage, and the magnitude of this increase is higher at high temperature. This behaviour is similar to the one observed for POIE realized on a Pt/YSZ/Pt symmetrical cell, where only the surface path is allowed. [9] Equation 20 describes well the experimental data at the exception of the largest overpotential value at 600°C (Table S1). In this situation, oxygen vacancies formation in the bulk of LSM are stimulated by the high overvoltage thus inducing an ionic conductivity. [13] Figure 7a shows the evolution of kslow (surface path) as a function of temperature and overvoltage for LSCF. This evolution is very similar to the one observed for LSM (Figure 6) and for Pt [9] which gives an additional confirmation that the rate constant for the surface path is significantly enhanced by the applied overvoltage. A possible explanation for it can be that the surface exchange coefficient (k*) is positively dependent on  and this increase is more pronounced as the temperature increases, as also suggested by other authors, the magnitude of increase being close to that study. [8,15] In contrast, kfast (bulk path) is almost independent of  in the temperature range investigated (Figure 7b), although at 600°C a slight increase is observed at high .

Equation 21 used to extract kslow and kfast fits correctly the experimental data although a slight deviation is observed at 600°C (Table S2). As proposed by Laurencin et al. [35] , "the relative proportion of surface to bulk paths can be expressed through the  indicator defined as the ratio between the kinetic rates of direct oxidation (or reduction) at TPB (the surface path) and oxygen incorporation / excorporation from LSCF (the bulk path)". In our situation, we discriminate between the rate constants for surface and bulk path (kslow and kfast) and between the oxygen populations that follow the surface or bulk mechanism (the Aslow and Afast preexponential factors in Equation 21, Table S2). Doing so, the ratio Aslow/Afast must be reflecting the as defined  indicator.  as a function of overvoltage () is plotted in Figure 8 at 400°C, 500°C and 600°C. It can be noticed that at low temperature (400°C), the surface path is favoured ( > 1). That is not surprising since the ionic conductivity of LSCF is rather low at this temperature. When the temperature comes around the critical temperature where the ionic conductivity is triggered (in this case around 500°C),  is close to 1 (i.e. the surface and bulk paths are balancing). Above this critical temperature (500°C), the bulk path is favoured ( < 1). It is interesting to note that at all investigated temperatures,  is decreasing with overvoltage (), thus  is favouring the bulk path in terms of oxygen population (and not on the basis of rate constants), whereas an increase in  favours the kinetic constant for surface exchange (Figures 6, 7a and [9] ). This surprising result is probably resulting from the two orders of magnitude between kfast and kslow (Figures 7a and7b). It is important to point out that these results were obtained at low oxygen partial pressure (PO2 = 15 mbar), which favours the formation of oxygen vacancies in the bulk of LSCF. Our observations are in contradiction with the work of Monaco et al. [33] In their study, they evidenced a transition from bulk to surface path when an anodic polarization is applied at temperatures between 700 and 800°C at PO2 = 210 mbar. They used a three-electrode configuration that allows the observation of one electrode independently in the OER configuration (anodic polarization) whereas we used a twoelectrode configuration and we estimate the electrode overvoltage by assuming that OER and ORR are symmetrical processes (Equation 18), while the overall process OER + ORR is investigated. This might explain these discrepancies. In the case of LNO, where the surface path is dominant (possibly due the formation of an insulating layer at the LNO/YSZ interphase which partially blocks the O 2-transfer from bulk LNO to YSZ), a surprising inverted bell like evolution of k (surface path) with  is observed, showing an initial decrease to reach a minimum and then a significant increase at high  as expected for the surface path (Figure 9). This trend is observed both at 700°C and 600°C (the experiment was further confirmed with another symmetrical cell at 700°C, Figure S5). The initial inhibition of k at small overpotential ( < 200 mV) may be explained by a significant modification of the surface properties of LNO cathode (ORR) induced by a decrease of the oxygen overstoichiometry with overvoltage compared to the measure at the OCV. Since the overstoichiometry (+) is responsible for the ionic conduction properties as well as electronic conduction properties a decrease of the overstoichiometry would lower k; it can be that the cathode (ORR) exhibits a lower value of  compared to the anode (OER) at low overpotential as shown in Figure S4). Further investigations are necessary to clarify this behavior. Scheme 1 resumes the different pathways for oxygen incorporation in the electrolyte, namely the surface and bulk paths. Globally, it was demonstrated that the rate constant of the surface path is significantly enhanced by increasing the overpotential, which might indicate that the surface exchange coefficient is increased with the overpotential. Contrarily, the rate constant for the bulk path seems to be independent of . The rate constant for the surface path is ca. two orders of magnitude lower than the one for the bulk path. Therefore, the balance surface/bulk is drove by the temperature as well as the applied overvoltage.

Conclusion

Three state of the art materials used in Solid Oxide Electrolyser Cells were investigated in terms of oxygen activation, surface and bulk diffusion. Oxygen Isotopic Exchange on powders was modelled in a way that heteroexchange reactions are split between a one or two atoms reactions. The one atom heteroexchange mechanism is attributed to the electronic properties of the oxide and to the surface diffusion, whereas the two atoms heteroexchange mechanism is attributed to the ionic conductivity of the oxide and to the bulk diffusion. Polarized Oxygen Isotopic Exchange on symmetrical cells was employed to evaluate the influence of electrode overpotential on the surface and bulk pathways. It is demonstrated that the surface path is greatly enhanced by increasing the overpotential, where its influence on the bulk path seems to be negligible. These findings provides mechanistic information on the different pathways for oxygen mobility in a SOEC and gives insights for orientating this mobility through the surface or bulk paths. This orientation can be realized by tuning the balance between electrical and ionic conductivity. Moreover, the applied overvoltage is an additional parameter that can be used for orientate oxygen mobility in the electrode material. This may provide insights for the choice of an adequate electrode material in SOEC systems involving coupled catalytic oxidation reactions such as natural gas assisted CO2 + H2O co-electrolysis to generate valuable CH4 oxidation products such as ethylene or syngas.

Experimental section

Electrode catalysts

La2NiO4+δ was synthesized by solid-state reaction. Prior to weighting, La2O3 (Sigma-aldrich 99.9%) was dried at 1000°C and NiO (Thermofisher scientific, 99%) was dried at 300°C for two hours. The oxides were mixed with acetone in an agate mortar, pressed with a uniaxial press and sintered at 1050°C for 10h in air. The resulting pellet was ground with an agate mortar into powder, pressed and sintered at 1200°C for 10h in air. The pellet was ground into a powder for further characterization. The powder X-ray diffraction was done with a Bruker D8 advance Davinci 2 with Cu Kα1 = 1.5406 Å and Cu Kα2 = 1.5439 Å, 2θ from 10° to 100° with a step of 0.03° and 1.5s per step.

Oxygen isotopic exchange on powders

Isotopic oxygen exchange experiments were carried out in a closed recycling reactor system. A recirculating pump was used in order to avoid any diffusion and mass transport effects in the gas phase that affect partial pressures of different isotopologues measured by mass spectrometry (Pfeiffer Vacuum). The used reactor system is described in more detail in [36] . Temperatureprogrammed oxygen isotopic exchange (TPOIE) were carried out on 20.0 mg of LSCF, 19.9 mg of LSM and 25.4 mg of LNO. Samples were then activated with pure 16 O2 at 700°C for 1 h before being cooled down for TPOIE. This activation step released the material surface from adsorbed species such as carbonates that could influence oxygen exchange activity. The system was then purged (using vacuum) and the 18 O2 molecule charged (53 mbar for LSCF and LSM and 79 mbar for LNO; 99.9% purity, supplied by Isotec). For each experiment, the ratio between the introduced amount of 18 O atoms in the gas phase and the number of 16 O contained in the sample, is kept constant.

In the case of LNO the increased amount 18 O2 introduced (necessary to prevent air entering the system) is compensated by a larger sample mass. The ramp rate of the temperature was 5°C / min. The masses 32, 34, 36 m/z were monitored as a function of time to follow the exchange. The m/z values of 18, 28, 44, 46 and 48 were also recorded to follow the possible presence of H2 16 O, C 16 O2, C 16 O 18 O, C 18 O2 molecules in the gas phase.

Symmetrical cells screen printing

GDC powder (Fuellcellmaterials) was screen printed on YSZ button cell (fuel cell material) on both sides and heat treated at 1300°C to obtain GDC/YSZ/GDC button cell. LSM/GDC/YSZ/GDC/LSM and LSCF/GDC/YSZ/GDC/LSCF symmetrical cells were obtained by screen printing LSM powder (Fuellcellmaterials) or LSCF powder (Fuellcellmaterials) followed by a heat treatment at 1100°C. For the La2NiO4+δ / YSZ / La2NiO4+δ symmetrical cell preparations and oxygen isotopic exchange on powder, the LNO powder was milled with a high-energy ball mill SPEX sample prep 8000M with YSZ jar and balls. The ink was screen printed on both sides of YSZ substrate (Fuellcellmaterials) and heat treated at 1150°C for two hours in air.

X-ray tomography

X-ray tomographic acquisition of the LSM cell was performed on an EasyTom XL Duo device (RX-solutions, France) with a cooled nanofocus x-ray source equipped with a LaB6 cathode (Hamamatsu L10711) coupled to a flat panel detector (Varian PaxScan 2520DX) and according to the following parameters: 100 kV (tube voltage), 15 µA (target current), 250 nm focal spot size, 1440 projections (1840x1456 pixels; 16 bits), 1 frame per second (imager readout speed), 20 frames averaging, source-toobject and source-to-detector distances of 2.70 mm and 678.38 mm respectively, voxel resolution of 506 nm. Slice reconstructions were done with XAct software (RX-Solutions) with a filtered back projection algorithm, attenuation of beam hardening artefact, corrections of x-ray spot drift and ring artefacts. Image and measurements were performed with Avizo v. 2022.2 (Thermo Fisher Scientific-FEI).

Polarized isotopic exchange

The setup for isotopic exchange under polarisation consists of a coupling of a classic oxygen isotopic exchange setup for powder [36] and a button Cell Test Fixture (ProboStat A normal system, SS/S; NorECs) already described [9] . Pt current collectors were used, connected to a Multi Autolab (Metrohm PGSTAT302N) for electrochemical measurements. Linear scan voltammetries were measured at 20 mV.s -1 . Electrochemical Impedance Spectra were recorded in the potentiostatic mode with an AC amplitude of 10 mV rms between 10 5 and 10 -1 Hz. In the case of LNO, ohmic resistance was determined by fitting with the following electrical equivalent circuit: L-R-(R1/Q1)-(R2-Q2).

Where L is an inductance, R, R1 and R2, are resistances, and Q1 and Q2 are constant phase elements. Gas phase analyses were done with a mass spectrometer (Pfeiffer Vacuum QMS 200 Prisma) with a pressure in the ionization chamber kept at 10 -6 mbar. The symmetrical cell chambers were vacuumed for 10 minutes before the introduction of 18 O. Between each isotopic exchange a flow of pure 16 O2 passed through the symmetrical cell chambers until the amount of 18 O detected by the mass spectrometer was negligible. A pressure of 20 mbar was used for LSM and LSCF, for LNO a higher pressure of 50 mbar was used to prevent air entering the system.

Figure 1 .

 1 Figure 1. Temperature programmed oxygen isotopic exchange on powder samples of LSM with 53 mbar 18 O2 (red), LSCF with 53 mbar 18 O2 (blue) and LNO with 79 mbar 18 O2 (black). 18 O2 pressures were chosen to keep a constant 18 O(g)/ 16 O(s) ratio for all measurement. Temperature ramp = 5°C.min -1 .

  [22b]

Figure 2 .

 2 Figure 2. Isothermal OIE for a) LSM at 500°C, b) LSCF at 500°C, c) LNO at 475°C and d) Arrhenius plot of the bulk oxygen diffusion coefficient.

Figure 3 .

 3 Figure 3. X-ray tomography realized on the LSM symmetrical cell.

Figure 4 .

 4 Figure 4. Electrochemical characterizations performed on symmetrical cells at 600°C: Nyquist plots at open circuit voltage (OCV) for a) LSM with 20 mbar O2, d) LSCF with 15 mbar O2, and g) LNO with 50 mbar O2. Red circles indicate decades of frequency. Linear scan voltammetries (20 mV.s -1 ) measured on symmetrical cells at 600°C for b) LSM with 20 mbar O2, e) LSCF with 15 mbar O2, and h) LNO with 50 mbar O2. Black curves correspond to raw measurement, red curves are ohmic drop corrected voltammetries. Corresponding Tafel plots for c) LSM, f) LSCF and i) LNO.

Figure 5 .

 5 Figure 5. Polarized oxygen isotopic exchange experiments and corresponding kinetic modelling of alpha vs. time. a-b) LSM at 600°C -η = 132 mV, c-d) LSCF at 500°C -η = 343 mV, e-f) LNO at 700°C -η = 147 mV.

Figure 6 .

 6 Figure 6. First order kinetic constant k (surface path) as a function of applied overvoltage () obtained for LSM during POIE at PO2 = 20 mbar at 400°C (blue), 500°C (black) and 600°C (red). Error bars estimated at 10 %.

Figure 7 .

 7 Figure 7. First order kinetic constants as a function of applied overvoltage obtained for LSCF during POIE at PO2 = 15 mbar at 400°C, (blue), 500°C (black) and 600°C (red), a) kslow (surface path), b) kfast (bulk path). Error bars estimated at 10 %.

Figure 8 .

 8 Figure 8. Plot of the surface to bulk ratio () as a function of the overvoltage () at 400°C (blue), 500°C (black) and 600°C (red) for LSCF (PO2 = 15 mbar).

Figure 9 .

 9 Figure 9. Kinetic constant k (surface path) as a function of applied overvoltage obtained for LNO during POIE at PO2 = 50 mbar at a) 700°C and b) 600°C. Error bars were estimated at 10 % based on the variation of the cell polarization resistance (Rp) measured at OCV prior each experiment (Figure S3).

  O18 O gas fraction x34 with the following Equations 13 and 14: 𝑥 32 = 1 -𝛼 𝑔 -

	𝑥 36 = 𝛼 𝑔 -	𝑥 34 2	𝑥 34 2	(13)

Table 2 .

 2 Exchange

	Temperature (°C)	LSM	LSCF	LNO
	400	0.0135	0.0464	n.d.
	500	0.0560	0.748	n.d.
	600	0.253	14.05	0.712
	700	n.d.	n.d.	3.978
	Ea(j°) kJ.mol -1	71	139	122

current densities (j° / mA.cm -2 ) extracted from Tafel plots and corresponding activation energy (Ea / kJ.mol -1 ).

Acknowledgements

The authors acknowledge financial support from the European Union (ERDF), Région Nouvelle Aquitaine (Project CASTOR), ANR (project CHEEC ANR-20-CE05-0023). This work pertains to the French government program "France2030" (EUR INTREE, reference ANR-18-EURE-0010).

Entry for the Table of Contents

The multi-step process for oxygen mobility in solid oxide electrolyser cells including dissociation and incorporation in the electrolyte is investigated with polarized oxygen isotopic exchange. The fastest way is O 2-incorporation in the electrode catalyst if it shows mixed ionic and electronic properties, whereas the much slower surface path can be activated by the electrode overpotential ().

Supporting Information

The supporting information includes XRD of LNO, X-ray tomography of the LSM cell, EIS of LNO cell at 700°C, polarization curves of a LNO cell comprising a reference electrode, polarized isotopic exchange performed on another LSM cell, and tables for the calculation of kinetic constants during POIE for LSM, LSCF and LNO.

Supporting information for: