dans laquelle le terme $\phi(1 - \phi)^{\frac{4}{\xi}}$ représente l'aire d'interface grain/gaz par unité de volume de
matériau. C'est ainsi une grandeur accessible dans le cas des gros pores.

Deux séries d'échantillons sont été étudiées : des échantillons de (U, Pu)O₂ dont certains ont été dopés au chrome, d'autres des échantillons de UO₂. Ce sont des pastilles de 5,6mm de diamètre et de 0,7mm d'épaisseur.

Sur un échantillon de (U, Pu)O₂ peu fritté (1550°C), appelé "MIR55", la porosité est encore de taille importante. Elle a été étudiée à 3 longueurs d'onde : 6, 15 et 20 Å. On trouve, pour cet

<figure><img src="image_58.png" /><figcaption>FIGURE 2.3 – Étude par DNPA d'un échantillon de (U, Pu)O₂ (MIR55)</figcaption></figure>

échantillon, une relation linéaire entre $Iq^4$ et $q^4$ de type loi de Porod dont l'ordonnée à l'origine $y_0$ permet de calculer l'aire totale de ces interfaces dans le volume traversé par le faisceau de neutrons.

Un des échantillons de UO₂ présentait des pores de taille nanométrique. Dans ce cas de figure la DNPA permet d'obtenir comme information supplémentaire la taille des pores.

On a alors pu calculer la contrainte de frittage donnée par Reid 4.1 p. 27 et ainsi estimer la contrainte de frittage. La dérivée par rapport à la densité est faite par différence finie entre les valeurs de nos (quelques) différents échantillons, prélevés à différents instants du frittage, de notre série. On voit que dans le cas d'un échantillon dopé chrome la contrainte de frittage est faible dès 1600°C, signifiant qu'il ne reste qu'un faible moteur pour la poursuite de l'évolution dans la suite du traitement thermique.
<figure><img src="image_59.png" /><figcaption>FIGURE 2.4 – Évolution de l'aire de l'interface grain/gaz (DNPA) lors du frittage du MOX - effet de l'additif $Cr_2O_3$</figcaption></figure>

<figure><img src="image_60.png" /><figcaption>FIGURE 2.5 – Évolution de la contrainte de frittage lors du frittage du MOX - effet de l'additif $Cr_2O_3$</figcaption></figure>

## 2.2 Texture 3D

Une autre expérience a été menée au LLB sur la ligne 3T1 en vue d'évaluer le suivi du nombre de grains lors du frittage à l'aide d'une étude de la texture 3D aux neutrons. Elle permettrait de suivre l'évolution du volume de chacune des orientations cristallines différentes lors du frittage (diffraction de neutrons).

Cette technique consiste à tracer, à partir des intensités diffractées, les figures de pôles directes pour une orientation {$hkl$} donnée, comme indiqué dans la figure 2.6. Il s'agit de la projection stéréographique sur un plan de référence (surface de l'échantillon) de la distribution volumique des densités de pôles $f(\vec{g})$ (avec $\vec{g} = (h, k, l)$) dans un repère lié à l'échantillon) de tous les cristallites de l'échantillon sondé [MHM2008]. L'ensemble des cristallites dont les orientations ne
<figure><img src="image_61.png" /><figcaption>FIGURE 2.6 – Projection stéréographique [MHM2008]</figcaption></figure>

diffèrent que par rotation d'un angle $\gamma$ autour de $< hkl >$ s'additionnent dans la figure de pôle :

$$q_{hkl}(\chi, \phi) = \frac{1}{2\pi} \int_{0}^{2\pi} f(\vec{g}) d\gamma \qquad (2.8)$$

Sur les échantillons de $(U, Pu)O_2$ testés (pastilles d'un volume de l'ordre du $cm^3$, même après frittage à 1700°C, il restait suffisamment de directions cristallographiques différentes pour que la texture soit encore celle d'un matériau totalement désorienté comme le montre la figure 2.7.

<figure><img src="image_62.png" /><figcaption>FIGURE 2.7 – Figure de pôles {111} et {220} obtenues par diffraction de neutrons</figcaption></figure>
## 2.3 Autres techniques

La microscopie électronique avec diffraction d'électrons (Electron Back-scattered Diffraction - EBSD) pourrait apporter l'information de variation du nombre de grains au cours du frittage sur des coupes 2D d'échantillons prélevés à différents instants du cycle de frittage.
3

Conclusion

On a vu qu’à l’échelle de l’agglomérat de (U, Pu)O₂, la tomographie X haute résolution était un outil potentiel de suivi du frittage sur des échantillons prélevés tout au long du cycle de frittage.

Tandis qu’à cette même échelle, et avec une résolution supérieure, le MEB environnemental donnait des informations in situ mais en ne pouvant observer que la surface de l’échantillon. Cependant, la taille des échantillons étant de l’ordre de la profondeur de champ, on peut observer la demi-épaisseur de l’échantillon vue de l’extérieur.

À l'échelle de la pastille, la Diffusion de Neutrons aux Petits Angles est un outil qui vient compléter les observations optiques et de Microscopie Électronique en Balayage pour ce qui concerne la distribution de taille des pores associés, et ce sans problème d'échantillonnage du fait du volume de matériau sondé.
Conclusion - perspectives

À l'échelle sub-granulaire on peut considérer que le modèle développé apporte deux nouveaux par rapport aux autres modèles analogues : il prend en compte la diffusion en volume grâce à la continuité du champ de déplacement et donc du potentiel chimique dans le volume du grain. Il nécessite, cependant, encore du travail numérique et d'être éprouvé en le confrontant à un assez grand nombre de cas de figure de simulation. Du point de vue expérimental des données consolidées de coefficients de diffusion intergranulaires et en volume sur un même lot de matériau (ce qui est rarement le cas dans la littérature) pourraient être intéressantes afin de s'affranchir des variations inter-lots de teneurs en impuretés.

En vue de passer à une échelle supérieure (simuler le comportement d'une dizaine de grains au contact) un travail de parallélisation du modèle à l'échelle sub-granulaire serait à développer afin de mettre en œuvre l'algorithme de type Marini-Quateroni.

Les fonctionnalités du modèle à l'échelle sub-granulaire de frittage ont commencé à être éten-
dues au frittage Spark Plasma Sintering (SPS) en ajoutant un potentiel électrique calculé dans le
volume des grains et un champ de température variable dans l'espace du fait des sources locales
de chaleur induites par l'effet Joule du courant électrique de la machine SPS. On a pu retrouver,
par exemple, le fait que la température est plus élevée en périphérie du joint de grains là où
les lignes de courant se resserrent. La collaboration sur le sujet avec le SCPTS de Limoges se
poursuit toujours dans le cadre de la thèse de Radostin Blazhev (Annexe C, p.133).

Dans un deuxième temps, lorsque la modélisation à composition constante sera jugée satis-
faisante, l'adjonction au modèle de la diffusion chimique des éléments sera possible.

Dans l'optique de simuler le comportement d'un Volume Élémentaire représentatif d'un re-
trait de cœur de pastille, je co-encadre une thèse en cours, celle de Sylvain Martin intitulée
"Modélisation du frittage de la pastille de matériau", (début de la 3ème année de thèse), avec
comme directeurs de thèse : K. Saleh de l'Université de Technologie de Compiègne, J. Fortin
et M. Guessasma de l'Université de Picardie-Jules-Verne). La méthode des Éléments Discrets
(Dynamique des Contacts) y est mise en œuvre pour représenter les interactions entre un grand
nombre de grains. Les particules sont représentées par des sphères dont l'interpénétration est
autorisée pour représenter la densification. Plusieurs articles ont montré la pertinence de cette
approche ([LRO2010], [CLM2010]). La loi de contact est composée d'une force attractive et d'un
frein visqueux qui intègre les aspects cinétiques. La force attractive provient de la variation de
rayon de courbure entre la surface libre du grain et la surface du joint de grains qui traduit
un gradient de contrainte correspondant. Le frein visqueux représente la diffusion au joint de
grains qui, en créant un défaut de matière au point de contact, est à l'origine de la densification.
Pour des grains de l'ordre du micromètre, les forces d'interactions deviennent très grandes en
comparaison au poids des particules. De plus, les vitesses de densification peuvent varier de sept
ou huit ordres de grandeurs entre les contacts voisins selon l'avancement du frittage de chacun des contacts. La résolution d'un tel système pose des problèmes de convergence et de stabilité numérique.

Le principal avantage de la Dynamique des Contacts [MJ1999] est de permettre l'utilisation d'un pas de temps élevé, typiquement de l'ordre de $10^{-3}$s, grâce à un schéma de résolution implicite. A titre de comparaison, les méthodes explicites autorisent un pas de temps maximum de l'ordre de $10^{-15}$s dans le cas de particules métalliques de quelques dizaines de micromètres [LRO2010]. Le modèle a été implémenté dans le code MULTICOR développé à l'Université de Picardie Jules Verne [MJF2005]. Des simulations ont été effectuées pour des empilements monodisperses et polydisperses de 1500 particules en 2D.

Les aspects numériques sont étudiés, afin d'optimiser le temps de calcul et la précision des résultats, comme l'influence du pas de temps, l'initialisation des forces de contact, ou la recherche d'un critère d'arrêt des itérations le plus représentatif possible de l'état du système. Actuellement, l'évolution des paramètres caractéristiques, tel le rayon du cou, se superpose bien avec l'évolution affine caractéristique d'un frittage sans réarrangement. La version 3D a permis de valider les résultats, de façon plus précise, avec le suivi des paramètres caractéristiques tels la densité, la nombre de coordination et le réarrangement des grains.

Pour les études relatives à l'adjuvant $Cr_2O_3$, des points restent encore à éclaircir, notamment la démonstration d'une éventuelle phase amorphe aux joints de grains qui serait le vestige d'un film liquide de composition $CrO$ deux expériences sont prévues à cette fin : d'une part on a envoyé des échantillons à l'Institut des Transuraniens à Karlsruhe pour des observations au microscope électronique à transmission sur lames minces, d'autre part la spectroscopie XPS-UPS de fractures intra et inter-granulaires de $(U, Pu)O_2$ sont possibles au CEA de Valduc avec lequel on a pu établir en 2013 une convention de transport. Une thèse va être proposée pour poursuivre ces études.

L'étude des défauts ponctuels devrait pouvoir être effectuée en mettant en œuvre en boîte-à-gants, dans le cadre d'un stage post-doctoral, le dispositif de conductivité électrique qui a mis au point en inactif.

L'ensemble des travaux permet d'améliorer la compréhension que l'on a du frittage et, de là, s'en assurer une meilleure maîtrise.
A.1 Liste des publications

D. Proust, J. Léchelle, A. Meunier and A. Lajudie, "Hydrothermal reactivity of mixed-layer kaolinite/smectite and implications for radioactive waste disposal", European Journal of Mineralogy, 1990, 2, 313-325

D. Proust, J. Léchelle, A. Lajudie and A. Meunier, "Hydrothermal reactivity of mixed-layer kaolinite smectite : experimental transformation of high-charge to low-charge smectite", Clays and Clay Minerals, Vol 38, N°4, 415-425, 1990

R. Atabek, A. Lajudie, J. Léchelle and Roland Push, "Pilot Field Experiment with Canister-Embedding Clay under Simulated Repositories Conditions", Engineering Geology, 28 (1990) 291-302

R. Pusch, O. Karnland, A. Lajudie, J. Léchelle and A. Bouchet, "Hydrothermal field expe-
riment simulating steel canister embedded in expansive clay - Physical behavior of the clay". In.
Scientific Basis for Nuclear Waste Management XV (C. G. Sombret, ed.). Mat. Res. Soc. Symp.
Proc., 1992, 257, 547-556

R. Pusch, O. Karnland, A. Lajudie, J. Léchelle, A. Bouchet, "Hydrothermal field test with French candidate clay embedding steel heater in the Stripa mine", Svensk Kärnbränslehantering AB (Swedish Nuclear Fuel and Waste Management Co) Technical : SKB TR-93-02, 1993

J. Léchelle, M. Trotabas, "A mechanistic approach for the modelling of MOX fuel pellet sintering", in IAEA-TECDOC-1166, "Advanced methods of process/quality control in nuclear reactor fuel manufacture", proceedings of a technical committee held in Lingen, Germany, 18-22 october 1999, pp65-77

J. Léchelle, R. Boyer, M. Trotabas, "A mechanistic approach of the sintering of nuclear fuel ceramics", Materials Chemistry and Physics 67 (2001) 120-132, DOI:10.1016/S0254-0584(00)00429-6

H. Monchot and J. Léchelle, "Statistical non-parametric methods for the study of fossil po-
pulations", Paleobiology, 28(1) 2002 pp. 55-69
J. Léchelle, P. Bleuet, P. Martin, E. Girard, F. Bruguier, M.-A. Martinez, A. Somogyi, A. Simionovici, M. Ripert, F. Valdivieso, P. Goeuriot, "μ-XANES and X-ray μ-tomography study of oxidation state, morphology and chemistry evolution during nuclear fuel sintering", Nuclear Science Symposium Conference Record, 2003 IEEE, Volume 2, 19-25 Oct. 2003 Page(s) :772 - 775 Vol.2

E. Girard, F. Valdivieso, P. Goeuriot, J.-M. Chaix, G. Chiarelli, J. Léchelle, "Sintering with Heterogeneities - Characterization and Modelling of the Microstructural Evolution", Key Engineering Materials Vols. 264-268 (2004) pp. 197-200

J. Léchelle, P. Bleuet, P. Martin, E. Girard, M.-A. Martinez, A. Somogyi, A. Simionovici, M. Ripert, F. Valdivieso, "Micro-XANES and X-ray microtomography study of oxidation State, morphology, and chemistry evolution during nuclear fuel sintering", IEEE Transactions on Nuclear Science, Aug. 2004, Volume : 51, Issue : 4 pp 1657-1661

C. Guéneau, S. Chatain, S. Gossé, C. Rado, O. Rapaud, J. Lechelle, J.-C. Dumas, C. Chaitillon, "A thermodynamic approach for advanced fuels of gas-cooled reactors", Journal of Nuclear Materials, 2005, v. 344, iss. 1-3 [special issue], p. 191-197

J. Léchelle, L. Aufore and M. Reynaud, "Reactivity of mixed uranium and plutonium nitride with silicon carbide as materials for Gas Fast Reactors", in Recent Advances in Actinide Science, Edited by R. Alvarez, ND Bryan and I May, Royal society of Chemistry Publishing, The proceedings of the eighth actinide conference, Actinide 2005, held at the University of Manchester, UK, on 4-8 July 2005, p 328-330

M. Ajdour, J. Léchelle, F. Valdivieso, P. Goeuriot, K. Saikouk, R. Boyer, "Sintering Simulation at a Scale Lower than the Grain Size", Advances in Science and Technology 45 (2006) 516-521 - http://hal.archives-ouvertes.fr/hal-00293423/fr/

E. Girard, J.-M. Chaix, F. Valdivieso, P. Goeuriot and J. Léchelle, "Local study of defects during sintering of UO₂: image processing and quantitative analysis tools", Image Anal Stereol 2008;27:79-85

D. Prieur, A. Jankowiak, C. Leorier, N. Herlet, L. Donnet, P. Dehaudt, J. Léchelle, J.-P Laval, P. Blanchart, "Influence of the microstructure on the $U_{1-y}Am_yO_{2-x}$ ($y = 0.1; 0.15$) pellet macroscopic swelling", Advances in Science and Technology Vol. 73 (2010) pp 104-108, doi:10.4028/www.scientific.net/AST.73.104

D. Prieur, A. Jankowiak, J. Léchelle, N. Herlet, P. Dehaudt, P. Blanchart, "Reactive sintering of $U_{0.85}Am_{0.15}O_{2-x}$ blanket fuels in various oxygen potentials", Journal of Nuclear Materials, Volume 424, Issue 1, pp. 285-288.
doi: 10.1016/j.jnucmat.2012.02.002

J. Léchelle, S. Noyau, L. Aufore, A. Arredondo, F. Audubert, "Volume interdiffusion coefficient and uncertainty assessment for polycrystalline materials", diffusion fundamentals, 17 (2012) 2, pp. 1-39, (www.diffusion-fundamentals.org : : ISSN 1862-4138)
R.C. Belin, P.M. Martin, J. Léchelle, M. Reynaud, A.C. Scheinhost, "Role of cation interactions in the reduction process in plutonium-americium mixed oxides", Inorg. Chem. 2013 52(6), pp. 2966-72, doi:10/1021/ic3023776

S. Martin, M. Guessasma, J. Léchelle, J. Fortin, K. Saleh, F. Adenot, "Simulation of sintering using a non-smooth Discrete Element Method. Application to the study of rearrangement", Computational Materials Science, Volume 84, March 2014, pp. 31-39

J. Léchelle, S. Martin, R. Boyer, K. Saikouk, "A sub-granular scale model for solid state free sintering : result on the evolution of two grains", Journal of Chemical Technology and Metalurgy, 49, 3, 2014, pp. 263-274

M. Chollet, J. Léchelle, R.C. Belin, J.C. Richaud, "In situ X-Ray diffraction study of point defects in Neptunium dioxide at elevated temperatures", J. Appl. Cryst., 47, 2014, pp. 1008-1015, doi:10.1107/S1600576714007912

S. Berzati, S. Vaudez, R.C. Belin, J. Léchelle, Y. Marc, J.-C. Richaud and J.-M. Heintz, "Controlling the oxygen potential to improve the densification and the solid solution formation of uranium-plutonium mixed oxides", Journal of Nuclear Materials Volume 447, Issues 1-3, April 2014, pp. 115-124

S. Martin, R. Parekh, M. Guessasma, J. Léchelle, J. Fortin, K. Saleh, "Study of the sintering kinetics of bimodal powders. A parametric DEM study", Powder Technology, April 2014, in Press-corrected Proofs, http://dx.doi.org/10.1016/j.powtec.2014.03.057

## A.2 Participation à colloques, congrés

J.-M. Chaix, J. Lechelle, "Simulation numérique d'empilements aléatoires d'amas de particules (Numerical simulation of random packing of particles)", Congrès, Effets de la microstructure et de la porosité sur les propriétés mécaniques et physiques des matériaux MdP (Grenoble, 6-8 avril 1998) in Colloque national de métallurgie des poudres, Grenoble, France (06/04/1998), 1998, pp. 335-340, SF2M, Nanterre, France (1998) (Monographie) INIST-CNRS, Cote INIST : Y 31940, 35400007011803.0450

J. Léchelle, T. Albiol, M. Trotabas, "Modélisation mécaniste du MOX : code Salammbô", Congrès, Effets de la microstructure et de la porosité sur les propriétés mécaniques et physiques des matériaux MdP (Grenoble, 6-8 avril 1998) in Colloque national de métallurgie des poudres, Grenoble, France (06/04/1998), 1998, pp. 341-346, SF2M, Nanterre, France (1998) (Monographie) INIST-CNRS, Cote INIST : Y 31940, 35400007011803.0450

J. Léchelle, P. Martin, E. Girard, M.-A. Martinez, A. Somogyi, A. Simionovici, M. Ripert, F. Valdivieso, "μ-XANES and X-ray μ-tomography study of oxidation state, morphology and chemistry evolution during nuclear fuel sintering", Nuclear Science Symposium Conference Record, 2003 IEEE, Publication Date : 19-25 Oct. 2003, Volume : 2, On page(s) : 772-775 Vol.2, ISSN : 1082-3654, ISBN : 0-7803-8257-9, INSPEC Accession Number : 8066682

J.-M.Chaix, E. Girard, P. Goeuriot, J. Léchelle, F. Valdivieso, "Analyse de l'évolution des
défauts lors du frittage d'une céramique", 28ème Journée ISS France 3 février 2005

E. Girard, J.-M. Chaix, C. Carry, F. Valdivieso, P. Goeuriot, J. Léchelle, "Exploitation des mesures locales de microstructure et de défauts contrôlés pour analyser l'évolution au cours du frittage d'une céramique", PMF2005, site universitaire de Cherbourg 18-20 mai 2005

M. Ajdour, J. Léchelle, F. Valdivieso, P. Goeuriot, K. Saikouk, R. Boyer, "Sintering simulation at a scale lower than the grain size", Conférence CIMTEC 2006, Acireale, Sicile, 5-9 juin 2006

J. Léchelle, E. Belval-Haltier, R. Belin, "Temperature Reactivity of Mixed Uranium and Plutonium Carbide Fuels with Silicon Carbide as an Inert Matrix. Efficiency of Refractory Metals as Interdiffusion Barriers.", Conférence Plutonium futures - the science 2006, Asilomar, Pacific Grove, California, July 9-13, 2006UCRL-PROC-222342

M. Beauvy, P. Martin, S. Vaudez, L. Paret, V. Basini, J. P. Ottaviani, J. Léchelle, C. Dalmasso, "Basic science on advanced plutonium fuels : new investigations on structure, thermo-physical properties and radiation damage", Conférence Plutonium futures - the science 2006, Asilomar, Pacific Grove, California, July 9-13, 2006UCRL-PROC-222342

C. Guéneau, S. Chatain, J.-C. Dumas, J. Lechelle, C. Rado, F. Defoort, N. Dupin, B. Sundman, H. Noel, R. Konings, "Fuelbase : a thermodynamic database for advanced nuclear fuels", in Proceedings HTR2006 : 3rd International Topical Meeting High Temperature Reactor Technology; 01/10/2006 - 04/10/2006; Johannesburg; Afrique du Sud, pp 225-232

M. Ajdour, J. Léchelle, F. Valdivieso, P. Goeuriot, "3D modeling of sintering - Case of two grains", American Institute of Physics Conference Proceedings 908, Materials Processing and Design; Modeling, Simulation and Applications; NUMIFORM '07; Proceedings (2007) 1363-1368

C. Guéneau, S. Chatain, S. Gossé, J.-C. Dumas, J.-P. Piron, J. Léchelle, Y. Pontillon, G. Ducros, C. Rado, F. Defoort, K. Froment, N. Dupin, B. Sundman, H. Noel, R. Konings, H. Kinoshita, C. Utton, "Fuelbase : a thermodynamic database for advanced nuclear fuels", International Information Exchange Meeting Thermodynamics of Nuclear Fuels - 2006; 27/11/2006 - 01/12/2006; Saclay; France

C. Guéneau, S. Gossé, S. Chatain, N. Dupin, B. Sundman, C. Martial, J.-C. Dumas, C. Rado, J. Léchelle, H. Noel, C. Utton, H. Kinoshita, P. Gotcu, D. Manara, R. Konings, "Progress of the Fuelbase thermodynamic database for advanced nuclear reactors", 12th International Symposium on the Thermo-chemistry and Thermo-physics of Nuclear Materials (ST2 NM - 2008); 30/08/2008 - 03/09/2008; Pörtschach; Autriche

J. Léchelle, R. Boyer, K. Saikouk, "Modélisation du frittage à l'échelle granulaire application aux combustibles nucléaires", 3ème Séminaire des Matériaux Fluides à Hautes Températures (MFHT - 2008); 22/10/2008 - 24/10/2008; Gréoux-les-Bains; France

A. Berche, C. Utton, C. Rado, J. Léchelle, B. Deschamps, T. Alpettaz, S. Chatain, C. Guéneau, "Thermochemical compatibility between carbides fuels and SiC for the Gas-cooled Fast Reactor", 39èmes Journées des Actinides (JDA - 2009); 28/03/2009 - 31/03/2009; La Grande
Motte; France

C. Guéneau, A. Berche, S. Chatain, S. Gossé, C. Rado, J. Léchelle, J.-C. Dumas, C. Martial, N. Dupin, B. Sundman, "Compatibility at high temperature between candidate fuel materials for gas cooled fast reactors", 13th International IUPAC Conference on High Temperature Materials Chemistry (HTMC - 2009); 15/09/2009 - 18/09/2009; Davis (Californie); Etats-unis

A. Berche, C. Guéneau, T. Alpettaz, S. Gossé, S. Chatain, C. Rado, O. Dugne, J. Léchelle, F. Adenot, J.-C. Dumas, C. Martial, N. Dupin, B. Sundman, F. Hodaj, C. Chatillon, "Bilan des études de compatibilité des matériaux combustibles pour le réacteur RCG", Journées Plénières du GNR MATINEX - 2010; 13/01/2010 - 14/01/2010; Aix-en-Provence; France

A. Pieragnoli, J. Léchelle, R.C. Belin, G. Cécilia, P.M. Martin, P.J. Valenza and A. Scheinost, "PuCrO₃ compound : synthesis, structural and thermodynamic studies", Nuclear Materials Conference 2010, NUMAT2010, 4-7 octobre 2010, Karlsruhe, Germany

A. Berche, P. Benigni, J. Rogez, J. Léchelle, F. Adenot, "Optimal design of a nuclearised high temperature Knudsen Effusion Mass Spectrometry for nuclear applications", Workshop on Knudsen Effusion Mass Spectrometry (KEMS), April, 23 - 25, 2012, Juelich, Germany

S. Martin, M. Guessasma, J. Fortin, K. Saleh, J. Léchelle, F. Adenot, "Modélisation du frittage par une approche de dynamique des contacts", 11ième Colloque National en Calcul des Structures, CSMA2013, 13-17 mai 2013, Presqu'île de Giens, France

J. Léchelle, M.-H. Mathon, G. Cécilia, L. Aufore and E. Belval-Haltier, "Investigating the sintering stress of heavy Element materials by means of SANS", International Conference on Neutron Scattering (ICNS), 8-12 juillet 2013, Edinburgh, Écosse

R. Blazhev, I. Saykova, J. Léchelle, N. Pradeilles, A. Maitre, "Modeling and analysis of the electric field effects at macroscopic to mesoscopic scales in spark plasma sintering", Anniversary Scientific Conference with International Participation 60 Years UCTM, June 4-5, 2013, Sofia, Bulgaria

J. Léchelle, S. Martin, "Sintering modeling for nuclear fuel : from grains to mesoscopic scale", Anniversary Scientific Conference with International Participation 60 Years UCTM, June 4-5, 2013, Sofia, Bulgaria

R. Blazhev, J. Léchelle, N. Pradeilles, G. Antou, I. Seikova, A. Maître, "Numerical modeling of temperature gradient evolution during Spark Plasma Sintering process : a macroscopic and sphere-to-sphere approach", CERMODEL2013, 10-12 juillet 2013, Trento, Italie
B

Brevets

(WO2011080475) Cermet with improved heat conductivity and nuclear fuel comprising said cermet Pub. No. : WO/2011/080475 International Application No. : PCT/FR2010/052910 Publication Date : 07.07.2011 International Filing Date : 23.12.2010 Applicants : Commissariat à l'Energie Atomique et aux Energies Alternatives [FR/FR]; 25 rue Leblanc Bâtiment " Le Pontant D " F-75015 Paris (FR) (For All Designated States Except US). Matheron, Pierre [FR/FR]; (FR) (For US Only). Brothier, Méryl [FR/FR]; (FR) (For US Only). Lechelle, Jacques [FR/FR]; (FR) (For US Only). Derriennic, Tangui [FR/FR]; (FR) (For US Only) Inventors : Matheron, Pierre; (FR). Brothier, Méryl; (FR). Lechelle, Jacques; (FR). Derriennic, Tangui; (FR)

Ma contribution dans ce brevet a consisté à calculer la conductivité thermique équivalente d'un Volume Élémentaire Représentatif des propriétés thermiques du matériau composite contenant des amas ellipsoïdaux aplatis de $UO_2$ mono ou bi-modaux selon l'axe ou perpendiculairement à l'axe d'aplatissement.

Une différence de température est imposée entre deux faces en vis-à-vis du VER (cubique) et le flux de chaleur est calculé sur une de ces faces (cf figure B.1). On en déduit ainsi une conductivité équivalente.

<figure><img src="image_63.png" /><figcaption>FIGURE B.1 – Composites céramique-métal à amas ellipsoïdaux aplatis mono- et bi-modaux, calcul de flux de chaleur</figcaption></figure>
C

# Encadrement - jury de thèses
d'Université

## Thèses encadrées

"Influence de défauts structuraux sur l'évolution microstructurale de UO₂ au cours du frittage" thèse soutenue par Erick Girard le 12 février 2004 jury : J.P. Bonnet (Président), A. Hatzotte (Rapporteur), J.-M. Heintz (ICMCB), P. Goeuriot (co-Directeur de thèse), F. Valdivieso (co-Directeur de thèse), J.M. Chaix (Directeur de thèse), R. Castelli (AREVA-NC- invité), J. Léchelle (encadrant CEA - invité).

"Influence du dopant Cr₂O₃ sur l'homogénéisation de la solution solide (U, Pu)O₂ dans un combustible MOX - Contribution à la connaissance du système U-Pu-Cr-O", thèse soutenue le 12 décembre 2007 par Adrien Pieragnoli, commission d'examen : J.M. Heintz (Rapporteur), J.C. Niepce (Président - Rapporteur), J.P. Bonnet (Directeur de thèse), R. Castelli (Areva-NC- invité), P. Blanpain (Areva-NC - invité), Y. Guérin (CEA - examinateur), S. Rossignol (Examinateur), J. Léchelle (Encadrant CEA - Examinateur).

"Solubilité du chrome dans le combustible MOX", thèse soutenue le 17 juillet 2013 Régis Thomas, directeur de thèse : Jean-Marc Heintz, ENSCBP, commission d'examen : M. Pijolat (Rapporteur), A. Maître (Rapporteur), L. Brunel, J.M. Heintz (Directeur de thèse), J. Léchelle (encadrant CEA), M. Maglione (Président), P. Martin, J.F. Silvain (Co-Directeur de thèse), R. Castelli (Areva-NC invité), J.J.Vermoyal (EDF invité).

(en cours) "Modélisation du frittage de la pastille matériau", (simulation de la dizaine milliers de grains par la méthode des éléments discrets), thèse de S. Martin (3ème année), directeurs de thèse : K. Saleh de l'Université de Technologie de Compiègne et J. Fortin de l'Université de Picardie-Jules-Verne

## Thèses avec une contribution numérique de ma part :

"Maîtrise de microstructures MOX de type céramique-céramique", thèse soutenue le 12 juillet 2001 par G. Toury, Université de Limoges, commission d'examen : D. Bernache-Assolant (Président), J.M. Heintz (Rapporteur), J.C. Niepce (Rapporteur), J.P. Bonnet (Directeur de thèse), M. Trotabas (Cogema - examinatrice), M.J. Gotta (Encadrante CEA - examinatrice), Y. Guérin (CEA - examinateur), J. Léchelle (CEA - examinateur).
Contribution numérique : estimation du champ de contrainte dans la pastille hétérogène due à des frittages différentiels.

"Développement d'un code de calcul pour la simulation du frittage en phase solide ", thèse soutenue par Mounia Ajdour, décembre 2006, n° d'ordre 432 SGM, commission d'examen : R. Boyer (Président), S. Bordère (Rapporteur), P. Haldenwang (Rapporteur), F. Sidorov (Examenateur), J. Léchelle (CEA - invité), P. Goeuriot (Directeur de thèse), F. Valdivieso (co-directeur de thèse)

Contribution numérique : soutien à la thèse

"Élaboration de combustible à base d'oxydes d'uranium et d'américium : modélisation thermodynamique et propriétés des matériaux", thèse soutenue le 20 septembre 2011 par Damien Prieur, Université de Limoges, A. Maitre (Président), D. Chateigner (Rapporteur), J.M. Heintz (Rapporteur), J.P. Laval (examineur), P. Blanchart (Directeur de thèse), A. Jankowiak (Encadrant CEA - examineur), P. Dehaudt (CEA - invité), R. Konings (ITU-invité), P. Martin (CEA - invité).

Contribution numérique : Ajustement de constantes d'entahlpies libres de formation de solutions solides

"Étude des phénomènes d'autodiffusion et d'interdiffusion du plutonium dans des céramiques de type $U_{1-y}Pu_yO_{2-x}$, thèse soutenue le 3 juillet 2012 par Sébastien Noyau, Université de Limoges, J.M. Heintz (Rapporteur), D. Mangelinck (Rapporteur), M. Pijolat (Présidente), D. Wrin (CEA), A. Chotard (Areva-NP - invité), J. Léchelle (CEA - examineur), F. Audubert (Encadrante CEA - examinatrice), A. Maître (Directeur de thèse). n° : 020-2012 en diffusion restreinte jusqu'en 2017.

Contribution numérique : Calculs de coefficients d'interdiffusion en volume

"Étude des équilibres de phase en fonction de la température dans le système $UO_2 - PuO_2 - Pu_2O_3$ pour les céramiques nucléaires en fortes teneurs en plutonium", thèse soutenue le 28 février 2013 par Thibaut Truphémus, Université Aix-Marseille, commission d'examen : A. Maître (Président), R. Konings (Rapporteur), O. Tougait (Rapporteur), C. Guéneau (CEA - examineur), J. Rogez (Directeur de thèse - examineur), R. Belin (Encadrant CEA - examineur), T. Genèves (Invité), J. Léchelle (invité)

Contribution numérique : calcul des proportions de phases à deux teneurs en plutonium en recherchant des sous-populations dans une distribution de niveaux de teneurs en plutonium sur la base de cartographies X semi-quantitatives et de profils quantitatifs associés en microsonde électronique.

"Influence du potentiel d'oxygène sur la microstructure et l'homogénéité U - Pu des combustibles $U_{1-y}Pu_yO_{2-x}$", thèse soutenue le 2 décembre 2013 par de Ségolène Berzati, P. Blanchard (Rapporteur), J.-M. Chaix (Rapporteur), J.-M. Heintz (Directeur de thèse - Examinateur), S. Vaudez (Encadrant CEA - Examinateur), M. Maglione (Examinateur), F. Hodaj (Examinateur), J.-P. Ganne (Examinateur), J.F. Silvain (co-Directeur de thèse), n° : 4938 - Bordeaux 1.

Contribution numérique : formation à l'utilisation du logiciel SAGE pour le calcul du potentiel d'oxygène, calcul du facteur thermodynamique de diffusion pour le coefficient d'interdiffusion U/Pu sur la base du modèle de solution solide régulière de Lindemer et Besmann pour décrire la solution solide $(U, Pu)O_2$.
Soutenance prévue en décembre 2014 de la thèse de Radostin Blazhev sur l'étude des mécanismes du frittage SPS en co-tutelle entre l'Université de Sofia (Bulgarie) et de Limoges
Contribution numérique : enchaînement des calculs de champs de potentiel électrique, de température et de déformation de frittage pour simuler le frittage SPS.

## Jury de Thèse

"Modélisation et simulation du frittage de matériaux dopés et de multimatériaux à l'échelle de la microstructure", thèse soutenue le 6 décembre 2013 par Howatchinou Tossoukpè, École Nationale Supérieure des Mines de Saint-Étienne, spécialité : science et génie des matériaux, directeur de thèse : Prof. Ass. F. Valdivieso, co-directeur : Prof. Ass. J. Bruchon, co-encadrant : Prof. S. Drapier, rapporteurs : Pr. Christophe Martin, SIMAP-INPG, Grenoble et Dr. Marc Bernacki, CEMEF-ENSMP, Sophia-Antipolis, examinateur : J. Léchelle.

## Post-doc encadré

Avril 2013- Avril 2014 : Simon Quetel-Weben, Mise au point et utilisation en boîte-à-gants d'un dispositif de mesure de conductivité électrique sur des matériaux à base de radionucléides émetteurs-alpha.

## Encadrement de Stages Universitaires

1997 : Stage de maîtrise de mathématiques, CMI, Claire Négrel, Schéma de discrétisation par la méthode des Volumes Finis du modèle granulaire (mécanique et chimie), encadrants Universitaires R. Boyer et R. Herbin

1998 : Stage de DESS de maths applis, CMI, Farid Abdelfehta, Gestion de l'intersection de deux maillages

1999 : Stage de DESS de maths applis, Frédéric Atamian, CMI, Remplacement du remaillage par un barycentrage lors de la déformation de grains au contact

2000 :

- Stage de DESS de maths applis, CMI, Jérôme Régis (Utilisation de la méthode de Frank et Wolfe pour la résolution de l'égalité des potentiels des éléments chimiques aux interfaces entre deux grains pour le modèle de frittage à l'échelle sub-granulaire)

- Stage de DESS de maths applis Laurent Quaglia (Mise en œuvre de la décomposition de Domaines de type Dirichlet-Neumann sur un ensemble de 14 grains au contact - mécanique du modèle de frittage à l'échelle granulaire - algorithme de Marini-Quateroni)

2001 :

- Stage d'Ingrid Rouby 3ième année de l'ISITV - option Ingéniérie mathématique, Modifications du mailleur Netgen pour lui ajouter des primitives de type ellipsoides dans son format algebraic3D

- Stage de Nicolas Raoulx École N. S. de Physique de Grenoble, Mesure de coefficients d'interdiffusion sur des monocristaux $UO_2$ et $CeO_2$

2002 : Stage de Rachid Labdidi DESS Maths Applis, CMI
2003 : Stage 2ième année CPE Lyon, Diane Schambill, analyse d'images microsonde sur des cartographies microsondes de l'élément Pu, étude de la taille des hétérogénéités

2004 : Stage d'IUP Génie des Matériaux Inorganiques, Faculté de Saint Jérôme, Quentin Pasquet, Utilisation de la spectroscopie α haute résolution et du SIMS pour mesurer le coefficient de diffusion à dilution infinie de $ ^{235}U $ dans un matériau inerte

2005 : Stage d'IUP Génie des Matériaux Inorganiques, Faculté de Saint Jérôme, Estelle Rollin, Mesure du coefficient de diffusion à dilution infinie de $ ^{235}U $ dans un matériau inerte, Utilisation de la spectroscopie α haute résolution, du SIMS, et du MEB pour estimer la profondeur des cratères.

2007 : Stage de Master I Maths Appliquées de Strasbourg, Marc Wolff (encadrants universitaire K. Saikouk, R. Boyer) calcul de rayons de courbures locaux par différentes techniques numériques, étude du déplacement des nœuds de maillage en conservant la masse.

2010 : Support numérique au stage de Jeremy Teyssier, 3ième année CPE Lyon, Recherche de critères pour caractériser l'homogénéité d'une microstructure contenant les éléments $U$ et $Pu$ sur la base de cartographies microsonde.

2011 : Stage de Mahkmout Diouara, Programmation d'une fonctionnalité au logiciel de simulation de frittage pour obtenir une coupe 2D au format in2d de netgen, d'après un algorithme fourni par Robert Boyer.

2011 : Stage de Radostin Blazhev en thèse (discipline : chimie) en co-tutelle entre l'Université de technologie chimique et de métallurgie de Sofia (Bulgarie) et l'Université de Limoges SPCTS - UMR CNRS 7315, enchaînement des étapes de calcul de champs de potentiel électrique puis de température du fait de l'effet Joule et de frittage pour simuler le frittage SPS du cuivre.

2012 : Stage de 3ième année de l'École de Céramiques de Limoges, Mickael Grimoin, Conception et utilisation d'un dispositif de mesure de conductivité électrique pour un four en Alumine nucléarisé.

2013 : Second stage de Radostin Blazhev en thèse (discipline : chimie) en co-tutelle entre l'Université de technologie chimique et de métallurgie de Sofia (Bulgarie) et l'Université de Limoges SPCTS - UMR CNRS 7315, calcul du coefficient d'intensification locale du champ électrique par une microstructure diélectrique
D

# Enseignement - Animation Scientifique

## D.1 Enseignement

Cours de 3<sup>ième</sup> année d'école de Céramiques de Limoges, option nucléaire, module 352, propriétés des actinides (environ 5h00) :

- 5 janvier 2010,

- 11 janvier 2011,

- 9 janvier 2012,

- 15 janvier 2013,

- 14 janvier 2014.

Cours de Master 2 "MANUEN" Grenoble - module F2 - propriétés du combustible nucléaire (environ 4h30) :

- 02/12/2010,

- 01/12/2011,

- 06 et 10/12/2012.

- 05 et 10/12/2013.

Formations sur les propriétés du combustible à l'Ecole des Combustibles (CEA Cadarache) :
- mars 2012,

- septembre 2012.

## D.2 Animation Scientifique

En 2011, j'ai été animateur de l'axe modélisation du Groupement de recherche Spark Plasma Sintering (GDR-3165 SPS) sur le frittage SPS. Ce GDR n'est actuellement plus financé, en cas de reprise de son financement, il est prévu que je soit co-animateur de cet axe.
E

Rapport de soutenance de thèse
UNIVERSITE
PIERRE ET MARIE CURIE
-PARIS 6-

4 Place Jussieu 75252 PARIS CEDEX 05

PYRAMIDE DE LA SCOLARITE

BUREAU DES ENSEIGNEMENTS DU 3ème CYCLE

Cour 13-24
Tel. : 43 36 25 25

DOCTORAT DE L'UNIVERSITE PARIS 6
Spécialité :

SCIENGES DES MATERIAUX

RAPPORT de SOUTENANCE
de THESE

Thèse soutenue le 29 NOVEMBRE 1990

par Monsieur LECHELLE Jacques

Sujet de la thèse :

INFLUENCE DES IONS SUR LES INTERACTIONS SOLIDES-LIQUIDES DANS LES
ARGILES SMECTIQUES - APPLICATION AU STOCKAGE DES DECHETS RADIOACTIFS.

Composition du jury :

MR FRITZ
MR MEUNIER
MR BAFFIER
MR COLLONGUES
ME LEJUS
ME THEBY
ME ATABEK

RAPPORT de soutenance

M. Léchelle a présenté avec clarté et concision un travail original et
difficile sur un sujet de première importance. Il a répondu avec
assurance aux questions du jury, prouvant ainsi qu'il maîtrisait parfaitement son sujet.

Mention accordée au candidat
par le jury :

Tres itonorable

<figure><img src="image_64.png" /></figure>

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>

<figure><img src="image_67.png" /></figure>

* L'article 16 de l'arrêté du 5 juillet 1984 prévoit l'attribution des mentions :

- passable

- honorable

- très honorable

PARIS, le 29 NOVEMBRE 1990

Le président et les membres du jury :

<figure><img src="image_68.png" /></figure>

<figure><img src="image_69.png" /></figure>
4, Place Jussieu - 75252 Paris Cédex 05
SERVICE DE LA SCOLARITE
Bureau des Enseignements du 3e cycle
Pyramide de la Scolarité
Cour 13-24
Tél. : 43.36.25.25 poste :

Science des Matériaux

RAPPORT

de Monsieur Bertrand FRITZ

qualité ... Directeur de Recherche

Lieu d'exercice ... Centre de Géochimie de la Surface 1 rue Blessig STRASBOURG
sur la Thèse présentée par M R ... LECHELLE, Jacques

ayant pour sujet : Influence des ions sur les interactions solides ... liquides dans les argiles smectiques ... Application au stockage des déchets radioactifs.

Monsieur Jacques LECHELLE a réalisé un travail de recherche pour déterminer les propriétés physico-chimiques d'un matériau argileux particulier (Argile FOCA 7) dans les conditions d'un stockage de déchets nucléaires. Le manuscrit proposé pour la soutenance de thèse présente les résultats de ce travail en 82 pages de texte et 150 pages d'annexes de données analytiques et informatiques.

La première partie du travail (chapitres 1 et 2) fait le point des connaissances sur le matériau argileux envisagé pour les barrières de confinement dans les stockages de déchets nucléaires :

- Analyses chimiques

- Analyses minéralogiques

La composition du fluide intersticiel de CGS argiles est abordée dans le chapitre III à partir d'une étude expérimentale et de tests de saturations par des modèles à base thermodynamique. Cette étude expérimentale porte ensuite sur l'altération hydrothermale des argiles (chapitre IV) à 150,200 et 250° et sur leur comportement à l'interface avec une autre barrière en ciment (chapitre V) ou en verre (chapitre VII)
Les simulations informatiques sont utilisées pour mieux mettre en évidence et comprendre le comportement très complexe des phases argileuses. Les conclusions soulignent le travail qui reste à faire en ce domaine qui est très important et crucial pour la sécurité des stockages utilisant les barrières argileuses.

Le travail réalisé peut être présenté devant l'Université de PARIS VI pour l'obtention du titre de Docteur de l'Université Pierre et Marie CURIE.

A Strasbourg, le 15 NOVEMBRE 1990

<figure><img src="image_70.png" /><figcaption>B. FRITZ,
Directeur de Recherche CNRS</figcaption></figure>
4, Place Jussieu - 75252 Paris Cédex 05
SERVICE DE LA SCOLARITE

Bureau des Enseignements du 3e cycle
Pyramide de la Scolarité

Cour 13-24

Tél. : 43.36.25.25 poste :

DOCTORAT de l'UNIVERSITÉ PARIS 6

Spécialité :

Science des Matériaux.

RAPPORT

de Monsieur Alain MEUNIER

qualité Professeur...

Lieu d'exercice POITIERS...

sur la Thèse présentée par M. LECHELLE, Jacques...

ayant pour sujet : Influence des ions sur les interactions solides...

Liquides dans les argiles smectiques - Application au stockage des déchets radioactifs...

Le mémoire présenté par Monsieur Jacques LECHELLE pour l'obtention du grade de docteur de l'Université de Paris VI a pour objet l'étude des réactions minérales affectant spécifiquement l'argile Fo-Ca soumise à des conditions d'altération hydrothermale. Ce mémoire s'organise en 6 grands chapitres.

Le premier chapitre résume l'état des connaissances relatives à la chimie des argiles étudiées dans le cadre des stockages de déchets nucléaires. Relativement court, il présente néanmoins les phénomènes principaux de façon claire. L'auteur s'est appuyé sur une bibliographie choisie qui est loin d'être exhaustive. Le chapitre 2 est consacré à l'étude minéralogique du matériau Fo-Ca-7. S'appuyant sur les analyses chimiques et la diffraction de rayons X, l'auteur aboutit à une formule structurale raisonnable de l'interstratifié kaolinite/ smectite. Il propose une quantification des différents composants du mélange minéral basée sur la mesure des intensités des raies de diffraction. Bien que la précision affichée à deux décimales soit difficilement justifiable, cette quantification est cohérente avec la composition chimique du matériau global.

Les troisième et quatrième chapitres rendent compte des propriétés chimiques de l'eau intersticielle de Fo-Ca-7 et des réactions d'altérations hydrothermales à partir d'expériences dont l'interprétation est basée sur l'utilisation d'un code de calcul thermodynamique. Ces deux aspects sont manifestement bien maîtrisés par l'auteur. Le travail est original et les résultats obtenus, même s'ils semblent contradictoires avec ce que l'on connaît déjà, sont intéressants. Il est cependant dommage que la somme des informations recueillies à la fois dans les expériences et par les simulations théoriques ne soit pas résumée et commentée clairement dans une discussion séparée.
Les cinquième et sixième chapitres sont consacrés à l'étude du comportement des matériaux argileux Fo-Ca-7 dans le contexte plus précis du stockage dans lequel les solutions d'altération ont pu traverser les ouvrages cimentés et le verre nucléaire. Comme précédemment, les données expérimentales sont interprétées à l'aide d'un code de calcul. Les résultats obtenus sont, là aussi, très intéressants mais malheureusement discutés trop brièvement. Tel est le cas des isothermes d'échange cationique et du coefficient de sélectivité. La conclusion générale, très brève, ne rend pas compte de l'ensemble des résultats obtenus.

L'ensemble du travail m'apparaît digne d'être soutenu pour l'obtention du grade de docteur de l'Université de Paris VI. Cependant, la forme habituelle du manuscrit dans lequel figures et tableaux sont complètement séparés du texte et la présentation de la bibliographie ont constitué pour moi un obstacle à la lecture.

<figure><img src="image_71.png" /><figcaption>A. MEUNIER</figcaption></figure>
F

Sélection de trois articles représentatifs
de l'étude

F.1 Écriture du modèle initial - discrétisation associée
# A mechanistic approach of the sintering of nuclear fuel ceramics

J. Léchelle<sup>a,*</sup>, R. Boyer<sup>b</sup>, M. Trotabas<sup>c</sup>

## Abstract

The CEA and the COGEMA, as part of their effort to model the different stages of the MOX fuel fabrication process, have specifically worked, on the sintering stage. A physical mechanistic model of MOX fuel sintering is proposed, as well as the numerical schemes that will lead to the achievement of the corresponding software tool. The model takes into account surface diffusion, grain boundary diffusion, volume diffusion, exchanges between solid and gas, as well as the mechanical strains of grains. The scale at which concentrations and strains are taken into account is smaller than grain size (0.1 µm for green pellets and 10 µm for sintered ones). The numerical resolution schemes of this problem have been conceived, and are also presented. They mainly consist of relaxation procedures to uncouple partial derivative mass conservation equations, flux balances along interfaces, non-linear potential equality conditions at interfaces, and Navier–Lamé equations over each grain. A code (SALAMMBO) is under construction, based on the latter; once validated by experimental and parametric tests, this code will describe Pu distribution, grain and pore size distribution, and local density of the pellets as functions of the sintering conditions, thus enabling further developments of the fabrication process such as the obtention of advanced microstructures of improved MOX fuels.
© 2001 Elsevier Science B.V. All rights reserved.

Keywords: Sintering (B); Computer modelling and simulation (C); Ceramics (A); Finite element analysis (B); Computational techniques (C)

## 1. Introduction

To improve MOX fuel fabrication, a numerical modelling of the sintering stage was undertaken. In order to design a code which could be adapted to the needs (addition of elements other than U, Pu and O) and which can be predictive, a mechanistic approach has been adopted. A rough description of the model has already been given [1]. This document describes in detail the improved physical model that has been developed, as well as the corresponding numerical schemes.

The physical phenomena taken into account are volume diffusion, surface diffusion, grain boundary diffusion, solid/ gas exchanges, diffusion-reaction in gas phase and the development (or the relaxation) of mechanical stresses. The evolution of concentrations and strain are modelled on a scale which is smaller than grain size (0.1 µm for green pellets and 10 µm after sintering) and involves a set of grains large enough to be representative of a pellet area, master blend area, matrix area, or interface between matrix and master blend areas.

Initial conditions are the following: the outer limits of the grains are given [2], solid phase concentrations ($UO_{2\pm x}$ or $PuO_{2\pm x}$), gas phase concentrations (Ar, $H_2$, $H_2O$) are

known, as well as displacement distributions. The computed parameters are: the evolution of the grain outer limits, the evolution of grain composition, pore composition evolution, and the evolution of the stress field in the grains.

It is then possible to obtain local mean parameters such as density, grain size, pore size, or Pu homogenization. Simple numerical schemes have been chosen in bulk material (P1 finite elements for grains and pore concentrations and P2 finite elements for displacements), and along interfaces (first-order finite differences) so that computer time can be used to simulate larger sets of grains. The main effort was made on a procedure to uncouple the partial derivative equations which govern the concentration evolution and the mechanical displacements over the whole set of grains.

## 2. The physical model

### 2.1. *Mechanical behaviour*

#### 2.1.1. *Context*

Several reasons account for the green fuel pellet possessing internal stresses:

* The pressing stage stresses are partly eliminated during decompression, they are radial [3,4]. Stresses are again

<footnote><sup>a</sup> CEA/DRN/DEC/SPUA, CEA de Cadarache, 13108 Saint Paul lez Durance, France</footnote>

<footnote><sup>b</sup> CNRS–Université de Provence, CMI, 39, rue Joliot Curie, 13453 Marseille Cedex 13, France</footnote>

<footnote><sup>c</sup> COGEMA/BCR/DSDP/DSD/SDCR, Montigny le Bretonneux, 78192 St Quentin en Yvelines, France</footnote>

<footnote>* Corresponding author.</footnote>
eliminated during pellet ejection from the press matrix (rebound).

* When porosity becomes closed, before occluded gas can diffuse out of the pellet, pore inner pressure may become significant (about 1 atm).

* During sintering, a local reordering of grains may occur that can lead to stresses when two grains are brought into contact.

* Solid/vapour (γ<sub>SV</sub>) and solid/solid (γ<sub>GB</sub>) surface energy create stresses at grain surface [5]. Where grain surface is convex, the stresses are of squeezing type, and of tensile type everywhere else. These stresses induce a bulk stress gradient. It is the main sintering driving force.

### 2.1.2. Mechanical equations

The stress tensor $\bar{T}$ is given by the dynamics fundamental relation applied to a small volume element inside a grain. A volume element of size dx dy dz within a grain hardly moves during sintering (quasi-static equilibrium). Mechanical equilibrium is reached more quickly than thermodynamic equilibrium, it reads

$$ \forall i \in \{1, 2, 3\} \sum_{j=1}^{3} \frac{\partial T_{ij}}{\partial x_j} = 0 $$

At this point, grain orientation has to be taken into account, as the relationship between stress and strain takes a simple form in the crystal referential (crystal axes). In our case of interest, crystal symmetry is cubic (O<sub>h</sub> point group in Schönflies notations) and only three constants are independent: c<sub>11</sub>, c<sub>12</sub> and c<sub>44</sub> [6].

Grain (g) orientation is given by three angles θ<sup>g</sup>, φ<sup>g</sup> and χ<sup>g</sup>. The laboratory referential is R(O, ⃗e<sub>1</sub>, ⃗e<sub>2</sub>, ⃗e<sub>3</sub>), grain g (100) axes are oriented by vectors (⃗e<sub>1</sub><sup>g</sup>, ⃗e<sub>2</sub><sup>g</sup>, ⃗e<sub>3</sub><sup>g</sup>) usually denoted (ā, ā, ĉ) in crystallography. P<sup>g</sup> denotes the matrix of basis change from (⃗e<sub>1</sub>, ⃗e<sub>2</sub>, ⃗e<sub>3</sub>) to (⃗e<sub>1</sub><sup>g</sup>, ⃗e<sub>2</sub><sup>g</sup>, ⃗e<sub>3</sub><sup>g</sup>).

where S is the elastic strain tensor (in Voigt notations) and ū the displacement at point M in grain g.

In case of the presence of a temperature gradient, the equilibrium would read [7]

$$ \begin{aligned} & c_{11} \frac{\partial^2 u_1^g}{\partial x_1^g} + c_{12} \frac{\partial^2 u_2^g}{\partial x_1^g \partial x_2^g} + c_{12} \frac{\partial^2 u_3^g}{\partial x_1^g \partial x_3^g} \\ & + \frac{1}{2} c_{44} \left( \frac{\partial^2 u_1^g}{\partial x_2^g} + \frac{\partial^2 u_2^g}{\partial x_1^g \partial x_2^g} \right) + \frac{1}{2} c_{44} \left( \frac{\partial^2 u_1^g}{\partial x_3^g} + \frac{\partial^2 u_3^g}{\partial x_1^g \partial x_3^g} \right) \\ & - K \alpha \frac{\partial T}{\partial x_1} = 0, \\ & \frac{1}{2} c_{44} \left( \frac{\partial^2 u_1^g}{\partial x_1^g \partial x_2^g} + \frac{\partial^2 u_2^g}{\partial x_1^g \partial x_3^g} \right) + c_{12} \frac{\partial^2 u_1^g}{\partial x_1^g \partial x_2^g} + c_{11} \frac{\partial^2 u_2^g}{\partial x_2^g \partial x_3^g} \\ & + c_{12} \frac{\partial^2 u_3^g}{\partial x_2^g \partial x_3^g} + \frac{1}{2} c_{44} \left( \frac{\partial^2 u_2^g}{\partial x_3^g} + \frac{\partial^2 u_3^g}{\partial x_2^g \partial x_3^g} \right) - K \alpha \frac{\partial T}{\partial x_2} = 0, \\ & \frac{1}{2} c_{44} \left( \frac{\partial^2 u_1^g}{\partial x_1^g \partial x_3^g} + \frac{\partial^2 u_3^g}{\partial x_1^g \partial x_2^g} \right) + \frac{1}{2} c_{44} \left( \frac{\partial^2 u_2^g}{\partial x_2^g \partial x_3^g} + \frac{\partial^2 u_3^g}{\partial x_2^g \partial x_2^g} \right) \\ & + c_{12} \frac{\partial^2 u_1^g}{\partial x_1^g \partial x_3^g} + c_{12} \frac{\partial^2 u_2^g}{\partial x_2^g \partial x_3^g} + c_{11} \frac{\partial^2 u_3^g}{\partial x_3^g \partial x_2^g} - K \alpha \frac{\partial T}{\partial x_3} = 0 \end{aligned} $$

This system has been written in the referential of grain g. The simulated system is made of a set of grains with very different orientations. It is better to write down all relationships in a single referential: the laboratory one. The useful expression for the code being developed is the one in the

$$ P^g = \begin{pmatrix} \cos \chi^g \sin \theta^g + \sin \chi^g \cos \varphi^g \cos \theta^g & -\sin \chi^g \sin \theta^g + \cos \chi^g \cos \varphi^g \cos \theta^g & \sin \varphi^g \cos \theta^g \\ -\cos \chi^g \cos \theta^g + \sin \chi^g \cos \varphi^g \sin \theta^g & \cos \theta^g \sin \chi^g + \cos \chi^g \cos \varphi^g \sin \theta^g & \sin \theta^g \sin \varphi^g \\ -\sin \chi^g \sin \varphi^g & -\cos \chi^g \sin \varphi^g & \cos \varphi^g \end{pmatrix} $$

The system is supposed to be at an isothermal stage. This hypothesis associated with those of a thermo-elastic behaviour of the grain with a cubic symmetry reads in the crystal axes:

$$ \begin{bmatrix} T_1 \\ T_2 \\ T_3 \\ T_4 \\ T_5 \\ T_6 \end{bmatrix} = \begin{bmatrix} c_{11} & c_{12} & c_{12} & 0 & 0 & 0 \\ c_{12} & c_{11} & c_{12} & 0 & 0 & 0 \\ c_{12} & c_{12} & c_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{44} \end{bmatrix} \begin{bmatrix} S_1 \\ S_2 \\ S_3 \\ S_4 \\ S_5 \\ S_6 \end{bmatrix} - K \int_{T_0}^{T} \alpha \, d\theta \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} $$

with

$$ S_1 = \frac{\partial u_1}{\partial x_1}, \quad S_2 = \frac{\partial u_2}{\partial x_2}, \quad S_3 = \frac{\partial u_3}{\partial x_3}, \quad S_4 = \frac{\partial u_2}{\partial x_3} + \frac{\partial u_3}{\partial x_2}, \quad S_5 = \frac{\partial u_1}{\partial x_3} + \frac{\partial u_3}{\partial x_1}, \quad S_6 = \frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1} $$
laboratory referential. It is as follows:

$$ \sum_{i=1}^{3} \sum_{j=1}^{3} \sum_{k=1}^{3} A_{i,j,k,1} \frac{\partial^2 u_i}{\partial x_j \partial x_k} = 0, $$

$$ \sum_{i=1}^{3} \sum_{j=1}^{3} \sum_{k=1}^{3} A_{i,j,k,2} \frac{\partial^2 u_i}{\partial x_j \partial x_k} = 0, $$

$$ \sum_{i=1}^{3} \sum_{j=1}^{3} \sum_{k=1}^{3} A_{i,j,k,3} \frac{\partial^2 u_i}{\partial x_j \partial x_k} = 0 $$

Coefficients $A_{i,j,k,l}$ are not given here, they are functions of $c_{11}$, $c_{12}$, $c_{44}$, $\theta^g$, $\varphi^g$ and $\chi^g$. The boundary conditions for this partial derivative equation are given hereafter.

### 2.1.2.1. Case of an interface gaseous pore p/solid grain g.
At the surface level of grain g, the stress $\bar{\bar{T}}^g$ is hydrostatic and is due to total pressure p in the gaseous phase: $\bar{\bar{T}}^g(\bar{n}^g) = -(\gamma_{SV}(1/R_1^g + 1/R_2^g) + p)\bar{n}^g$, where $\bar{n}^g$ is the outer normal vector of the grain. $-\gamma_{SV}(1/R_1^g + 1/R_2^g)\bar{n}^g$ term is linked to the small grain size. Surface energy $\gamma_{SV}(\neq 0)$ and curvature radii $R_1^g$ and $R_2^g$ (very small) make the gas-applied pressure differ from grain-applied stress [5].

Gas is described by the Rand-Markin model [8] where 1-13 species have a partial pressure $p_i$, and a concentration $c_i^p$. They are U, Pu, O, Ar, H, UO, UO₂, UO₃, PuO, PuO₂, O₂, H₂, H₂O.

$p = \sum_{i=1}^{13} p_i^p = RT \sum_{i=1}^{13} c_i^p$, where T is the temperature and R the perfect gas constant. The superscript p denotes the number of the pore. This can be written in terms of displacements as

$$ \sum_{i=1}^{3} \sum_{j=1}^{3} B_{i,j,1}^{g} \frac{\partial u_{i}^{g}}{\partial x_{j}} - K n_{1}^{g} \int_{T_{0}}^{T} \alpha \, dT = - \left( \gamma_{SV} \left( \frac{1}{R_{1}^{g}} + \frac{1}{R_{2}^{g}} \right) + RT \sum_{i=1}^{13} c_{i0}^{p} \right) n_{1}^{g}, $$

$$ \sum_{i=1}^{3} \sum_{j=1}^{3} B_{i,j,2}^{g} \frac{\partial u_{i}^{g}}{\partial x_{j}} - K n_{2}^{g} \int_{T_{0}}^{T} \alpha \, dT = - \left( \gamma_{SV} \left( \frac{1}{R_{1}^{g}} + \frac{1}{R_{2}^{g}} \right) + RT \sum_{i=1}^{13} c_{i0}^{p} \right) n_{2}^{g}, $$

$$ \sum_{i=1}^{3} \sum_{j=1}^{3} B_{i,j,3}^{g} \frac{\partial u_{i}^{g}}{\partial x_{j}} - K n_{3}^{g} \int_{T_{0}}^{T} \alpha \, dT = - \left( \gamma_{SV} \left( \frac{1}{R_{1}^{g}} + \frac{1}{R_{2}^{g}} \right) + RT \sum_{i=1}^{13} c_{i0}^{p} \right) n_{3}^{g} $$

where $n_1^g$, $n_2^g$ and $n_3^g$ are the components of the normal vector $\bar{n}^g$. The $B_{i,j,k}^g$ expression is not given here, it is a function of $c_{11}$, $c_{12}$, $c_{44}$, $\theta^g$, $\varphi^g$ and $\chi^g$. In the cubic case, $K = \frac{1}{3}(c_{11} + 2c_{12})$. $\alpha$ is the thermal expansion coefficient.

### 2.1.2.2. Case of a solid grain g/solid grain g' interface.
Two partial derivative equations are solved simultaneously

inside solid grains g and solid grain g':

$$ \forall i \in \{1, 2, 3\} \sum_{j=1}^{3} \frac{\partial T_{ij}^{g}}{\partial x_{j}} = 0, \quad \forall i \in \{1, 2, 3\} \sum_{j=1}^{3} \frac{\partial T_{ij}^{g'}}{\partial x_{j}} = 0 $$

The relationship between displacements ($u_n$ is the normal component of the displacement $u$, and $u_{t1}$ and $u_{t2}$ its tangential components) of grains g and g' along their common boundary is due to continuity:

$$ u_{1}^{g} = u_{1}^{g'}, \quad u_{2}^{g} = u_{2}^{g'}, \quad u_{3}^{g} = u_{3}^{g'} $$

We assume that once two grains meet, they remain in contact. Signorini contact conditions [9] are not used. One assumes that forces are opposite along the boundary of two grains.

$$ \bar{\bar{T}}^{g}(\bar{n}^{g}) = -\gamma_{GB} \left( \frac{1}{R_{1}^{g}} + \frac{1}{R_{2}^{g}} \right) \bar{n}^{g} + \bar{\bar{T}}^{g'}(\bar{n}^{g}) $$

or

$$ \sum_{i=1}^{3} \sum_{j=1}^{3} B_{i,j,1}^{g} \frac{\partial u_{i}^{g}}{\partial x_{j}} = \sum_{i=1}^{3} \sum_{j=1}^{3} B_{i,j,1}^{g'} \frac{\partial u_{i}^{g'}}{\partial x_{j}} - \gamma_{GB} \left( \frac{1}{R_{1}^{g}} + \frac{1}{R_{2}^{g}} \right) n_{1}^{g}, $$

$$ \sum_{i=1}^{3} \sum_{j=1}^{3} B_{i,j,2}^{g} \frac{\partial u_{i}^{g}}{\partial x_{j}} = \sum_{i=1}^{3} \sum_{j=1}^{3} B_{i,j,2}^{g'} \frac{\partial u_{i}^{g'}}{\partial x_{j}} - \gamma_{GB} \left( \frac{1}{R_{1}^{g}} + \frac{1}{R_{2}^{g}} \right) n_{2}^{g}, $$

$$ \sum_{i=1}^{3} \sum_{j=1}^{3} B_{i,j,3}^{g} \frac{\partial u_{i}^{g}}{\partial x_{j}} = \sum_{i=1}^{3} \sum_{j=1}^{3} B_{i,j,3}^{g'} \frac{\partial u_{i}^{g'}}{\partial x_{j}} - \gamma_{GB} \left( \frac{1}{R_{1}^{g}} + \frac{1}{R_{2}^{g}} \right) n_{3}^{g} $$

## 2.2. Transport phenomena

### 2.2.1. Solid description

The sintering atmosphere contains elements Ar and H, so these elements also exist in the solid phase. The solid solution is an ideal solid solution between Ar, whose atoms would be on the sites of a CFC lattice, and the (U, Pu)O₂ solid solution. For the (U, Pu)O₂ solid solution, a four ionic sub-lattice ideal solution model is used. The solid solution reads

$$ [\text{Ar}_x((U'_M, U'_M, U_M, U_M^{2\bullet}, Pu'_M, Pu'_M, V_M^{4\bullet})_1(O_{i1}^{\prime\prime}, V_{i1}^X)_{16} \\ (U_{i2}^{4\bullet}, Pu_{i2}^{4\bullet}, V_{i2}^X)_1(O_O^X, V_O^{2\bullet}, H_O^{2\bullet})_2)_{1-x}] $$

where Kröger-Vink notations are used [10]. The number of anionic interstitial sites is given by Olander [8]. Atlas et al. [11] indicates that in reducing conditions, between 1100 and 1600°C, interstitial cations have to be taken into consideration. We have supposed they were on fluorine structure octahedral sites, which leads to the value 1 of the coefficient for the interstitial cations sub-lattice. The structure element site fractions are denoted $y_i^j$, where j is the number of the
sub-lattice, and *i* the number of the species in the sub-lattice. No distinction is made between the two anionic interstitial sites.

The previous structural formula does not imply that there is no empty lattice cell. An empty lattice cell reads

$$ (V_M^{4'})_1(V_{i1}^X)_{16}(V_{i2}^X)_1(V_O^{2\bullet})_2 $$

The species $H_O^{2\bullet}$ is pointed out by Sherman and Olander [12] and is taken into account: atomic hydrogen diffuses in the empty anionic sites in $UO_2$ single crystals. Argon is located in empty cells [13].

The structure elements can be arranged in relative building units [14] whose electrochemical potential can be expressed as a function of the chemical potential of the involved elements. Here are some of the relative building units $U_i$:

$$ \begin{aligned} U_1 &\equiv \{U_M^\bullet + U_M' - 2U_M^X\}, \\ U_2 &\equiv \{U_M^{2\bullet} + 2U_M' - 3U_M^X\}, \\ U_3 &\equiv \{Pu_M^X - U_M^X\}, \\ U_4 &\equiv \{-U_M' + Pu_M'\}, \\ U_5 &\equiv \{-4U_M' + 3U_M^X + V_M^{4'}\}, \end{aligned} $$

$$ \begin{aligned} U_6 &\equiv \{-2U'_M + 2U_M^X + O''_{i1} - V_{i1}^X\}, \\ U_7 &\equiv \{4U'_M - 4U_M^X + U_{i2}^{4\bullet} - V_{i2}^X\}, \\ U_8 &\equiv \{4U'_M - 4U_M^X + Pu_{i2}^{4\bullet} - V_{i2}^X\}, \\ U_9 &\equiv \{2U'_M - 2U_M^X - O_O^X + V_O^{2\bullet}\}, \\ U_{10} &\equiv \{2U'_M - 2U_M^X - O_O^X + H_O^{2\bullet}\} \end{aligned} $$

In this case, the entropy volumic production is

$$ T\sigma = -\vec{J}_O^T \cdot \vec{\nabla}\mu_O - \vec{J}_U^T \cdot \vec{\nabla}\mu_U - \vec{J}_{Pu}^T \cdot \vec{\nabla}\mu_{Pu} - \vec{J}_H^T \cdot \vec{\nabla}\mu_H - \vec{J}_{Ar}^T \cdot \vec{\nabla}\mu_{Ar} $$

This makes the flux $\vec{J}_i^T$ and the forces appear. The forces are the potential gradients of chemical elements and not of structure elements. Irreversible process linear thermodynamics gives a linear relationship between flux and forces whose coefficients are Onsager coefficients. In practice, inter-diffusion coefficients $\tilde{D}$ are used, as well as proportionality coefficients between stresses and fluxes $E_{i\alpha}$. The advantage of these parameters is that they can be measured through experiment. Fluxes are given in the laboratory referential for a motionless grain. A further simplification is made for Ar and H fluxes:

$$ \begin{aligned} (\vec{J}_U^T)_{\text{lab}} &= -\tilde{D}_{UU} \vec{\nabla} c_U - \tilde{D}_{UPU} \vec{\nabla} c_{PU} - \tilde{D}_{UO} \vec{\nabla} c_{O} \\ &\quad - \sum_{\alpha=1}^{6} E_{U\alpha} \vec{\nabla} T_{\alpha}, \\ (\vec{J}_{PU}^T)_{\text{lab}} &= -\tilde{D}_{PuU} \vec{\nabla} c_U - \tilde{D}_{PuPU} \vec{\nabla} c_{PU} - \tilde{D}_{PuO} \vec{\nabla} c_{O} \\ &\quad - \sum_{\alpha=1}^{6} E_{Pu\alpha} \vec{\nabla} T_{\alpha}, \end{aligned} $$

$$ \begin{aligned} (\vec{J}_O^T)_{\text{lab}} &= -\tilde{D}_{OU} \vec{\nabla} c_U - \tilde{D}_{OPU} \vec{\nabla} c_{PU} - \tilde{D}_{OO} \vec{\nabla} c_{O} \\ &\quad - \sum_{\alpha=1}^{6} E_{O\alpha} \vec{\nabla} T_{\alpha}, \\ (\vec{J}_{Ar}^T)_{\text{lab}} &= -\tilde{D}_{ArAr} \vec{\nabla} c_{Ar} - \sum_{\alpha=1}^{6} E_{Ar\alpha} \vec{\nabla} T_{\alpha}, \\ (\vec{J}_H^T)_{\text{lab}} &= -\tilde{D}_{HH} \vec{\nabla} c_H - \sum_{\alpha=1}^{6} E_{H\alpha} \vec{\nabla} T_{\alpha} \end{aligned} $$

i.e. the influence of U, Pu and O on Ar or H diffusion only appears through $\tilde{D}_{HH}$ and $\tilde{D}_{ArAr}$ variation with U, Pu and O contents. $C_X$ is the concentration of X.

### 2.2.2. Bulk diffusion in grains

$\vec{J}_{V,i}^g$ denotes the bulk diffusion flux for species *i* in grain *g*. Species 1–5 (U, Pu, O, Ar and H) exist in the solid phase. Bulk diffusion leads to the following relationship between fluxes and forces:

$$ \vec{J}_{V,i}^g = - \sum_{k=1}^{5} \tilde{D}_{ik}^V \vec{\nabla} c_k^g - \sum_{\alpha=1}^{6} E_{i\alpha}^V \vec{\nabla} T_{\alpha} - c_i^g \vec{v}_G^g, \quad i \in \{1, \dots, 5\} $$

$\tilde{D}_{ik}^V$ are the bulk diffusion coefficients, and $T_\alpha$ the $\alpha$th component of the stress tensor with Voigt notations. $E_{i\alpha}^V$ is due to *i* bulk diffusion under the influence of stresses. $E_{i\alpha}^V$ is approximated by $E_{i\alpha}^V = \tilde{V}_i \tilde{D}_{ii}^V S_\alpha / (\partial \mu_i^c / \partial c_i + \frac{1}{2} \partial \tilde{V}_i / \partial c_i \sum_{\gamma=1}^6 S_\gamma T_\gamma)$, where $S_\alpha$ is the $\alpha$th component of the strain tensor in Voigt notations, $\mu_i^c$ the chemical part of the potential of *i* in the solid, and $\tilde{V}_i$ the molar partial volume of *i*. $\vec{v}_G^g$ is the velocity of the mass centre of grain *g* in the laboratory referential.

### 2.2.3. Surface and grain boundary diffusion

$\vec{J}_{S,i}^g$ denotes the surface flux of the grain *g* for the species *i*. The coefficients $\tilde{D}_{ik}^S$ are differ from the bulk diffusion coefficients due to binding variations of atoms from surface to bulk.

$$ \vec{J}_{S,i}^g = - \sum_{k=1}^{5} \tilde{D}_{ik}^S \vec{\nabla} c_k^g - \sum_{\alpha=1}^{6} E_{i\alpha}^V \vec{\nabla} T_{\alpha} - \delta c_i^g \vec{v}_G^g, \quad i \in \{1, \dots, 5\} $$

where $c_k^g$ is the concentration of species *k* taken at the surface, and $\vec{\nabla}_S$ denotes the surface gradient operator $(\partial/\partial s_1)\vec{u}_1 + (\partial/\partial s_2)\vec{u}_2$. $E_{i\alpha}^S$ is approximated by $E_{i\alpha}^S = \tilde{V}_i \tilde{D}_{ii}^S S_\alpha / (\partial \mu_i^c / \partial c_i + \frac{1}{2} (\partial \tilde{V}_i / \partial c_i) \sum_{\gamma=1}^6 S_\gamma T_\gamma)$. A grain boundary flux is defined accordingly by replacing subscript S by GB.

### 2.2.4. Gas diffusion

Let $\vec{J}_i^p$ be the bulk flux of species *i* in the pore *p* (all 13 species exist in gaseous phase).

$$ \vec{J}_i^p = - \sum_{k=1}^{13} D_{ik}^p \vec{\nabla} c_k^p - c_i^p \vec{v}_G^p $$
where $D_{ik}^p$ is a bulk diffusion coefficient for the gaseous phase.

The gas kinetics theory gives the inter-diffusion coefficients for species *i* and *k* in pores p: $D_{ik}^p$. We assume that the more probable shocks are the ones where two particles are involved, thus these coefficients can be used in the case where the gas possesses more than two species. The literature [15] provides

$$D_{ik} (\text{m}^2 \text{s}^{-1}) = 1.858 \times 10^{-7} \frac{(T^{3(M_i+M_k)}/(M_i+M_k))^{1/2}}{p \sigma_{ik}^2 \Omega_{ik}^{(1,1)*}}$$

with $\Omega_{ik}^{(1,1)*} = \Omega^{(1,1)}$ for $T^* = kT/\epsilon_{ik}$ with $(\epsilon_{ik}/k)^2 = (\epsilon_i/k)(\epsilon_k/k)$, where $\epsilon$ is the depth of the Lennard-Jones potential [16], M the molar weight in $10^{-3}$ kg mol$^{-1}$, T the temperature in K, $\sigma$ the diameter in Å and $p$ the pressure in N m$^{-2}$.

### 2.2.4.1. For a non-polar gas.

$$\begin{aligned} \Omega^{(1,1)*} = & 1.06036(T^*)^{-0.1561} + 0.19300 \exp\{-0.47635T^*\} \\ & + 1.03587 \exp\{-1.52996T^*\} \\ & + 1.76474 \exp\{-3.89411T^*\} \end{aligned}$$

### 2.2.4.2. For a polar gas.

$$\begin{aligned} \Omega^{(1,1)*} = & 1.06036(T^*)^{-0.1561} + 0.19300 \exp\{-0.47635T^*\} \\ & + 1.03587 \exp\{-1.52996T^*\} \\ & + 1.76474 \exp\{-3.89411T^*\} + 0.19 \frac{\delta^2}{T^*} \end{aligned}$$

where $\delta$ depends on the dipolar moment $\mu$, $\delta = \frac{1}{2}\mu^{*2}$ with $\mu^* = \mu(\sigma^3\epsilon)^{-1/2}$. On the other hand, $\delta = \sqrt{\delta_i\delta_k}$.

The values given by Monchick and Mason [16] and Trappeniers et al. [17] are summarized in Table 1. Pressure $p$ is linked to gas phase concentrations by the perfect gas law.

### 2.2.5. Continuity equations

#### 2.2.5.1. Inside grains g and g'.

$$\text{div}(\vec{j}_{V,i}^g) + \frac{\partial c_i^g}{\partial t} = 0, \quad \text{div}(\vec{j}_{V,i}^{g'}) + \frac{\partial c_i^{g'}}{\partial t} = 0, \quad i \in \{1, \dots, 5\}$$

with coupling conditions

* equality of potential on each side of the interface (local thermodynamic equilibrium):

<table><caption>Table 1
Gas diffusion parameters</caption><thead><tr><th>δ</th><th>M (g mol<sup>-1</sup>)</th><th>σ (Å)</th><th>ε/k (K)</th></tr></thead><tbody><tr><td>Ar</td><td>39.95</td><td>3.451</td><td>119.5</td></tr><tr><td>H<sub>2</sub></td><td>2.016</td><td>2.827</td><td>59.7</td></tr><tr><td>O<sub>2</sub></td><td>32</td><td>3.323</td><td>137.0</td></tr><tr><td>H<sub>2</sub>O</td><td>1, 0</td><td>18.1</td><td>775</td></tr></tbody></table>

$$\begin{aligned} \mu_1^{\text{solid}}(c_1^g, \dots, c_5^g, T_1^g, \dots, T_6^g) &= \mu_1^{\text{solid}}(c_1^{g'}, \dots, c_5^{g'}, T_1^{g'}, \dots, T_6^{g'}) \\ \mu_5^{\text{solid}}(c_1^g, \dots, c_5^g, T_1^g, \dots, T_6^g) &= \mu_5^{\text{solid}}(c_1^{g'}, \dots, c_5^{g'}, T_1^{g'}, \dots, T_6^{g'}) \end{aligned}$$

* nil matter volume flux through interface in the interface referential:

$$\begin{aligned} \bar{V}_1^g(\vec{j}_{V,1}^g - \vec{v}_{\text{inter}}^{gg'} c_1^g)\vec{n} + \bar{V}_1^{g'}(\vec{j}_{V,1}^{g'} - \vec{v}_{\text{inter}}^{gg'} c_1^{g'})\vec{n} \\ + \bar{V}_1^g \text{div}_S(\vec{j}_{GB,1}^g + ((\delta \vec{j}_{V,1}^g - \vec{v}_{\text{inter}}^{gg'} \delta c_1^g)\vec{t})\vec{t} - \vec{v}_{\text{inter}}^{gg'} \delta c_1^g) \\ - \bar{V}_1^{g'} \text{div}_S(\vec{j}_{GB,1}^{g'} + ((\delta \vec{j}_{V,1}^{g'} - \vec{v}_{\text{inter}}^{gg'} \delta c_1^g)\vec{t})\vec{t} \\ - \vec{v}_{\text{inter}}^{gg'} \delta c_1^g) = 0, \\ \vdots \\ \bar{V}_5^g(\vec{j}_{V,5}^g - \vec{v}_{\text{inter}}^{gg'} c_5^g)\vec{n} + \bar{V}_5^{g'}(\vec{j}_{V,5}^{g'} - \vec{v}_{\text{inter}}^{gg'} c_5^{g'})\vec{n} \\ + \bar{V}_5^g \text{div}_S(\vec{j}_{GB,5}^g + ((\delta \vec{j}_{V,5}^g - \vec{v}_{\text{inter}}^{gg'} \delta c_5^g)\vec{t})\vec{t} - \vec{v}_{\text{inter}}^{gg'} \delta c_5^g) \\ - \bar{V}_5^{g'} \text{div}_S(\vec{j}_{GB,5}^{g'} + ((\delta \vec{j}_{V,5}^{g'} - \vec{v}_{\text{inter}}^{gg'} \delta c_5^g)\vec{t})\vec{t} \\ - \vec{v}_{\text{inter}}^{gg'} \delta c_5^g) = 0 \end{aligned}$$

where $\delta$ denotes the thickness of the grain boundary.

The last five equations are differential equations which mean that the flux of matter volume is nil through the interface in the interface referential:

$$\begin{aligned} \iint_{\Sigma} \bar{V}_1^g(\vec{j}_{V,1}^g - c_1^g \vec{v}_{\text{inter}}^{gg'}) \vec{n} \, dS \\ + \iint_{\Sigma} \bar{V}_1^{g'}(\vec{j}_{V,1}^{g'} - c_1^{g'} \vec{v}_{\text{inter}}^{gg'}) \vec{n} \, dS \\ + \int_{\Gamma} \bar{V}_1^g(\vec{j}_{GB,1}^g - \delta \vec{v}_{\text{inter}}^{gg'} c_1^g) + (\delta(\vec{j}_{V,1}^g - \vec{v}_{\text{inter}}^{gg'} c_1^g)\vec{t})\vec{n} \, ds \\ - \int_{\Gamma} \bar{V}_1^{g'}(\vec{j}_{GB,1}^{g'} - \delta \vec{v}_{\text{inter}}^{gg'} c_1^{g'}) \\ + (\delta(\vec{j}_{V,1}^{g'} - \vec{v}_{\text{inter}}^{gg'} c_1^{g'})\vec{t})\vec{n} \, ds = 0 \end{aligned}$$

where $\Sigma$, $\Gamma$, $\vec{n}$ and $\vec{n}_\Gamma$ are defined in Fig. 1. It can be simplified to give

$$\begin{aligned} \iint_{\Sigma} \bar{V}_1^g(\vec{j}_{V,1}^g - c_1^g \vec{v}_{\text{inter}}^{gg'}) \vec{n} \, dS \\ + \iint_{\Sigma} \bar{V}_1^{g'}(\vec{j}_{V,1}^{g'} - c_1^{g'} \vec{v}_{\text{inter}}^{gg'}) \vec{n} \, dS \\ + \int_{\Gamma} \bar{V}_1^g(\vec{j}_{GB,1}^g + (\delta(\vec{j}_{V,1}^g)\vec{t})\vec{t}) \vec{n}_\Gamma \, ds \\ - \int_{\Gamma} \bar{V}_1^{g'}(\vec{j}_{GB,1}^{g'} + (\delta(\vec{j}_{V,1}^{g'})\vec{t})\vec{t}) \vec{n}_\Gamma \, ds = 0 \end{aligned}$$

<figure><img src="image_72.png" /><figcaption>Fig. 1. Surface of a grain.</figcaption></figure>
F.2 Étude de moyens expérimentaux de validation du modèle
# Micro-XANES and X-ray Microtomography Study of Oxidation State, Morphology, and Chemistry Evolution During Nuclear Fuel Sintering

J. Léchelle, P. Bleuet, P. Martin, E. Girard, F. Bruguier, M. A. Martinez, A. Somogyi, A. Simionovici, M. Ripert, F. Valdivieso, and P. Goeuriot

*Abstract*—Nuclear waste management has led to recycling $PuO_2$ into mixed oxide fuel nuclear fuel (MOX). Its manufacturing route involves a sintering process. A multiscale model is under construction in order to predict the evolution of the sample characteristics during sintering and to improve sintering conditions. The purpose of the small scale model is to simulate grain size and chemistry evolution of micrometer large samples. In the green state, crystallites are 0.3 µm in size and are made either of $UO_{2+x}$ or of $PuO_{2-y}$. After sintering (U, Pu)$O_{2-z}$ solid solutions are formed with crystalline grains close to 5 µm in size. At high temperature, sintering is governed by diffusion phenomena. Diffusion coefficients vary with point defect concentration. These defects are related to the oxidation state of the cations, the spatial distribution of which can be observed by means of micro X-ray absorption spectroscopy (XAS).

Micro X-ray absorption near edge structure spectroscopy (micro-XANES) and high resolution X-ray microtomography study experiments have been performed at the European Synchrotron Radiation Facility beamline ID22, Grenoble. The applicability of these techniques has been demonstrated for testing simulation validity of (U, Pu)$O_{2-z}$ samples along the MOX sintering route.

***Index Terms***—Microtomography, micro-XANES, plutonium dioxide, uranium dioxide.

## I. INTRODUCTION

THE recycling of $PuO_2$ into nuclear fuel is one method of reducing plutonium waste in nuclear management. Since $PuO_2$ has different chemical properties than those of $UO_2$, its incorporation into nuclear fuels has necessitated particular attention during the manufacturing route (powder agglomeration, pressing behavior, sintering behavior). The manufacturing of mixed oxide (MOX) fuel involves a sintering process. Sintering has drawn much attention and a numerical model is under development [1] to simulate grain size evolution, grain pore evolution, and chemistry changes (chemical diffusion), on a scale smaller than the crystalline grain size (from ~ 0.1 µm before sintering to 5 µm after sintering). These criteria are of industrial

<figure><img src="image_73.png" /><figcaption>Fig. 1. Fabrication conditions of samples versus time.</figcaption></figure>

interest in order to predict fuel behavior in reactors. Crystalline grain size affects the migration duration of gaseous fission products (e.g., Xe) to reach grain boundaries. Pore size has an effect on the volume of material in which these fission products can be trapped.

The model developed to describe the evolution of grain and pore size as well as chemical changes during sintering [1] should decrease the number of laboratory experiments needed to perfect efficient and inexpensive ways of producing nuclear fuel. This paper aims to investigate a way of validating this numerical model. Synchrotron radiation based micro X-ray absorption near edge structure spectroscopy (micro-XANES) and high resolution X-ray microtomography have been used to investigate the three-dimensional (3-D) information of the 50 µm large grain powder samples: their morphology, their chemical element nature (U or Pu), and the oxidation state of their cations. U and $Pu L_{III}$ absorption edges were used for U and Pu detection.

## II. SAMPLES AND BEAMLINE LAYOUT

The first category of samples used in this experiment involved the $UO_2$ powder grains, which had been taken at different points of the sintering route (Fig. 1). The sintering atmosphere was a mixture of argon and dry hydrogen gases, at 1700 °C. This atmosphere reduces uranium dioxide from $UO_{2.19}$, which, as shown by standard laboratory XRD spectra (Cu tube, $K\alpha$ radiation), is a mixing of $UO_{2.013}$ and $\alpha - U_4O_9$ into $UO_{2.00}$. Grains of powder (or granulates) are about 50 µm in size and are made up of $UO_2$ aggregates, which are themselves made up of one to three strongly bounded $UO_2$ crystals, the size of which is 0.1 µm.

This category was divided into two sets, which differed by their initial density (number of neighbors and surface of con-

Manuscript received October 22, 2003; revised April 17, 2004. This work was supported both by the Commissariat à l'Energie Atomique (CEA Centre d'Etudes de Cadarache) and by COGEMA.

J. Léchelle, P. Martin, M. Ripert, F. Bruguier, and M. A. Martinez are with the Commissariat à l'Energie Atomique, Centre d'Etudes de Cadarache, F-13108 St-Paul lez Durance, France (e-mail: jacques.lechelle@cea.fr).

P. Bleuet, A. Somogyi, and A. Simionovici are with ESRF ID22, F-38043 Grenoble, France (e-mail: pierre.bleuet@esrf.fr).

E. Girard, F. Valdivieso, and P. Goeuriot are with ENSM, PECM, F-42023 St-Etienne, France (e-mail: egirard@emse.fr).

Digital Object Identifier 10.1109/TNS.2004.832225
<figure><img src="image_74.png" /><figcaption>Fig. 2. ESRF ID22 beamline layout.</figcaption></figure>

<table><caption>TABLE I
FreLoN CAMERA FEATURES</caption><thead><tr><th>Feature</th><th>Value</th></tr></thead><tbody><tr><td>Image Depth</td><td>14-bit</td></tr><tr><td>Number of pixels</td><td>2048 x 2048</td></tr><tr><td>Physical pixel size</td><td>14 µm</td></tr><tr><td>Electrons per ADC unit</td><td>20</td></tr><tr><td>Dark current per pixel</td><td>1 e<sup>-</sup>s<sup>-1</sup></td></tr><tr><td>Pixel capacity</td><td>320000 e<sup>-</sup></td></tr><tr><td>Readout Noise</td><td>25 e<sup>-</sup></td></tr><tr><td>Readout rate</td><td>20 Mpixels s<sup>-1</sup></td></tr><tr><td>Readout time for full frame</td><td>Readout serial multiplex</td><td>Binned 450 ms 110 ms</td><td>Unbinned 900 ms 220ms</td></tr><tr><td>Binning</td><td colspan="3">2 x 2 possible</td></tr><tr><td>Image data size</td><td>unbinned</td><td>8 MB</td><td></td></tr><tr><td></td><td>binned</td><td>2 MB</td><td></td></tr><tr><td>Effective</td><td>Objective used</td><td>Effective pixel size</td><td>Field of view</td></tr><tr><td></td><td>X 20</td><td>0.35 µm</td><td>0.7 mm</td></tr><tr><td></td><td>X 10</td><td>0.7 µm</td><td>1.4 mm</td></tr><tr><td></td><td>X 5</td><td>1.4 µm</td><td>2.8 mm</td></tr><tr><td></td><td>X 2.5</td><td>2.8 µm</td><td>5.6 mm</td></tr></tbody></table>

tact are of interest for the study of sintering). The first set consisted of grains of powder with a natural density (samples labeled agc, ag1, ag2, and ag3); the second and denser one, the so-called “forced-sieved” sample set, was obtained by forcing pressed UO₂ powder through the meshes of a sieve (samples labeled amc, am1, am2, and am3).

The second category of samples was made up of only one green UO₂ (95 weight %) and PuO₂ (5 weight %) mixture sample, which was destined to become a U₀.₉₅Pu₀.₅O₂ solid solution after sintering. Here the sample was observed in its green state and had not been sintered so that the composition of a randomly chosen aggregate in the powder grain sample was either entirely UO₂+x or entirely PuO₂-y.

The ID22 beamline of the European Synchrotron Radiation Facility (ESRF) is made up of two hutches [2]. Micro-XANES, microdiffraction, and microtomography can be performed routinely at the beamline (Fig. 2).

A fast-readout, low-noise (FreLoN) 2000 charge-coupled device (CCD) camera has been used for microtomography [3]. It was developed by the Analog Transient Electronics Group (ATEG) in the ESRF Instrument Support Group. At the ID22 beamline, FreLoN cameras are used in combination with an optical microscope that projects light from a scintillator screen onto the CCD chip of the camera. The magnification of the optic can easily be changed. Various single-crystal scintillators that differ in spatial resolution and efficiency are available.

The scintillator used is a 4 µm thick lutetium aluminum garnet crystal (LAG : Lu₃Al₅O₁₂ : Eu³⁺) onto a 170 µm thick substrate made of yttrium aluminum garnet (YAG : Y₃Al₅O₁₂). The line spread function of such a scintillator is 0.5 µm. Manufacturing of the scintillator consists of YAG substrate surface polishing, followed by the deposition of LAG. Both surfaces are then coated with an antireflection coating.

A 90° reflecting prism coupled to an Olympus optical system composed of an objective of 20× magnification and a 2.5× eye piece were used for optical demagnification of the beam.

The energy of the beam was varied from 17.110 to 17.250 keV to record UO₂ XANES spectra. It was moved up to 18.5 keV in order to differentiate UO₂ from PuO₂ grains and set at 20 keV (above both edges) to enable subsequent 3-D reconstructions. Beam hardening is negligible when using a highly monochromatic beam, such as the one at beamline ID22 of the ESRF. The energy width (ΔE) of the Si (111) crystal planes is about 2.4 eV at 18 keV. This high monochromaticity makes any beam hardening corrections useless.

Integration time per view was set at 1 s for a set of 625 views, and the corresponding angular range covered was 180°.

The resolution of the microtomography, carried out in hutch 2, was about 0.7 µm, whereas in the case of the micro-XANES experiment, with a focused beam, the spot size was 4 µm (H) × 1.5 µm (V).

### III. RESULTS

#### A. Three-Dimensional Density of UO₂ Green and Sintered Samples

Reconstructions of UO₂ samples were obtained using the ESRF high speed tomography software suite based on a stan-
<figure><img src="image_75.png" /><figcaption>Fig. 3. Three-dimensional reconstruction of a powder grain from a green UO₂ sample. Attenuation is averaged on a 0.7 µm scale. Powder grain size (whole sample) is about 50 µm. White lines were drawn to emphasize the three-dimensional character of the sample.</figcaption></figure>

<figure><img src="image_76.png" /><figcaption>Fig. 4. A forced-sieved UO₂ sample sintered at 1700 °C for two hours. Attenuation is averaged on a 0.7 µm scale. Sample size is about 50 µm. The gray bar provides relative density information.</figcaption></figure>

dard filtered-back-projection algorithm [4]. Fig. 3 gives the re-
construction of a UO₂ powder grain green sample, whereas
Fig. 4 shows the one of the UO₂ forced-sieved sintered powder
grains.

The composition of the UO₂ powder used to make the sam-
ples was known. It had a very low impurity content. The grain
density was obtained by crossing mercury porosimetry mea-
surements performed on a powder made of these granulates
and alcohol density measurements for open and closed porosity.
These were performed on homogeneous half pellets made of a
material with characteristics identical to those of the powder
granulates. They had also undergone the same thermal treat-
ment.

Each two-dimensional (2-D) slice (after three-dimensional
reconstruction) had a constant thickness of one pixel width (0.7
µm). Thus, attenuation was solely dependent on density (ab-
sorption coefficient of a fully dense material times relative den-
sity). A correlation law was found to relate relative density to
attenuation.

<figure><img src="image_77.png" /><figcaption>Fig. 5. UO₂ sample sintered at 1600 °C. Relative density variations inside a 0.7 µm thick slice.</figcaption></figure>

<figure><img src="image_78.png" /><figcaption>Fig. 6. Density variations inside a UO₂ powder grain. Crosses indicate forced-sieved samples sintered at 1700 °C for two hours. Diamonds indicate green UO₂ powder grain information.</figcaption></figure>

Because of the high monochromaticity of the beamline, beam
hardening effects could be neglected, and the density variation
is the origin of the gradient appearing in Fig. 5.

The resolution of the technique was high enough to obtain
density variation curves of UO₂ according to the distance from
the outer surface of the grain, but not high enough to be able
to observe grains and pores. In the green sample, the crystalline
grain size ranges between 0.1 and 0.3 µm. In the sintered one,
the crystalline grain size can be close to 5 µm.

Fig. 5 gives a slice (one pixel thick) of the reconstructed UO₂
sample sintered at 1600 °C, in which an internal microstructure
can be partially observed. Here, it was decided not to correct
artifacts such as streaks on edges or ring-artifacts to carry out
data interpretation in a way as quantitative as possible.

Density turns out to be higher on the outer surface of the
sample for both the UO₂ green powder grain and for the forced-
sieved sintered UO₂ one (Fig. 6).

Density variations of the green sample are possibly linked to
the atomizing-drying route to obtain UO₂, while forced-sieved
sintered sample density is possibly due to the stress undergone
by the samples when forced through the sieve meshes.

B. Point Micro-XANES Spectra of Green and Sintered UO₂

Here “point micro-XANES” refers to a measurement per-
formed on a beamspot of a few square micrometer dimensions
using a focused beam. Micro-XANES spectra have been
recorded for varying oxidation states of U (standard samples).
U oxidation states in solid materials can be +III (5f³), +IV,
+V, or +VI (5f⁰). The Lᵢᵢᵢ edge absorption spectra are known

154
<figure><img src="image_79.png" /><figcaption>Fig. 7. U L<sub>III</sub> edge absorption spectrum of a green “UO<sub>2</sub>” sample.</figcaption></figure>

for these cations [5], [6]. The aim of the experiment was to
determine the capability of this technique as a validation tool
for the sintering numerical model now under development.
The spectrum obtained for our green “UO<sub>2</sub>” is given in Fig. 7.
Such a sample contains aggregates at several oxidation states.
The local oxidation state depends on oxygen partial pressure,
humidity, and temperature, and it evolves toward its thermo-
chemical equilibrium value in the course of time.

From the absorption edge value of such a spectrum, it is pos-
sible to obtain the local mean oxidation state (averaged in a cube
of 0.7 µm edge size). Following this principle, if tomography
had been carried out at several energies in the vicinity of U<sup>n+</sup>
absorption edges (one for each abscissa value on the spectrum
of Fig. 7), it would have been possible to produce a three-di-
mensional map of the iso-oxidation state of uranium.

C. *Two-Dimensional Micro-XANES Spectra of Green and Sintered UO<sub>2</sub>*

Transmission XANES spectra have been recorded for green
and sintered UO₂ in transmission mode. The sample was lo-
cated on the microtomography setup, but remained fixed while
the beam energy was varied (goniometric head stuck in a given
rotation position). Energy ranged from 17.110 to 17.250 keV in
1 eV steps. At each step, a flat-field image was recorded (in-
tensity I₀) as well as a sample image (transmitted intensity I)
by making the beam move one pixel at a time. For each pixel
of the sample image, a XANES spectrum was calculated and
normalized. Such spectra are averaged in the direction perpen-
dicular to the acquisition plane. The L<sub>III</sub> absorption edge was
determined using a piece of ad hoc software. This was based
on the smoothening with a fourth degree polynomial by least
squares of the increasing front of the absorption curve. It was
then followed by a second derivative zero value computation by
means of the fourth degree polynomial coefficient values. L<sub>III</sub>
values were determined for each curve (one per pixel in the ma-
terial on the 2-D image, thus 20 256 curves for the UO₂ green
sample versus 3455 for the UO₂ forced-sieved sintered sample).
Iso-values of L<sub>III</sub> edge energy were then plotted on the trans-
mission images. The iso-L<sub>III</sub> edge absorption energy maps were
drawn for both samples (Figs. 8 and 9).

Areas of oxidized uranium can be observed, although acqui-
sition time for the recording of projections is shorter than that

<figure><img src="image_80.png" /><figcaption>Fig. 8. Iso-L<sub>III</sub> edge absorption energy map of green UO<sub>2</sub>.</figcaption></figure>

<figure><img src="image_81.png" /><figcaption>Fig. 9. Iso-L<sub>III</sub> edge absorption energy map of sintered UO<sub>2</sub>.</figcaption></figure>

<figure><img src="image_82.png" /><figcaption>Fig. 10. Green (U, Pu)O₂ sample at 17.5 keV, i.e., between the U and Pu L<sub>III</sub> edge</figcaption></figure>
<figure><img src="image_83.png" /><figcaption>Fig. 11. Green (U, Pu)O₂ sample at 18.5 keV, i.e., above Pu L₁₁₁ absorption edge.</figcaption></figure>

<figure><img src="image_84.png" /><figcaption>Fig. 12. Thresholded difference between Figs. 11 and 10. Two-dimensional projection of a PuO₂ crystalline supposed grain.</figcaption></figure>

necessary for the acquisition of the micro-XANES spectrum in Fig. 7.

### D. U and Pu Element Observation by Microtomography

A green (U, Pu)O₂ sample (i.e., a mixture of UO₂ and PuO₂ powder grains) had been prepared for the microtomography experiment. It belonged to the second set of samples. It was placed on the tip of a metallic steel sample holder with a double (polyethylene and Teflon) barrier to isolate it from the atmosphere of the hutch. Clear observations of the sample through this barrier were not possible with visible light while aligning. Thus the sample turned out to be on the side of the sample holder (Figs. 10 or 11).

The range of observable projections was limited and could not lead to a three-dimensional reconstruction. However, some projections were observed at two energies, one between the U and Pu L<sub>III</sub> edges (here 17.5 keV) and the other above the Pu L<sub>III</sub> edge (18.5 keV). UO<sub>2</sub> and PuO<sub>2</sub> grains can be distinguished in Figs. 10 and 11. The PuO<sub>2</sub> crystalline grain image was produced in Fig. 12 by subtraction of Fig. 11 and Fig. 10 followed by a numerical thresholding operation.

The chemical nature of the (U, Pu)O₂ sample was known from the chemical composition of the initial fabrication pow-

ders. The presence of PuO₂ in the selected powder grain was investigated by the counting of the ²⁴¹Am 59.5 keV gamma-ray line. This counting had also been a tool used in evaluating PuO₂ quantity in the sample (here 60 ng) for safety reasons. This study shows all the capability of this technique as a tool for the observation of these two elements.

## IV. CONCLUSION

High resolution microtomography and micro-XANES turned out to be valuable tools for the study of mixed oxide fuel sintering. It enabled us to distinguish U from Pu, as well as the several oxidation states of these cations. Their advantage compared with scanning electron microscope analysis is the three-dimensional character of the data. Literature has shown that exponents for crystalline grain size evolution laws as a function of time differ in three [7] and two dimensions [8], [9].

In order to be useful on a scale smaller than that of sintered crystalline grains, and to determine the fine microstructure of our fuel samples, a higher resolution is required. This could be achieved through a decrease in the focused beam spot size by means of Fresnel zone plates, or Kirkpatrick-Baez double mirror systems of higher magnification.

## ACKNOWLEDGMENT

The authors thank all contributors of the Plutonium, Uranium and Minor Actinide Service, as well as the Radioprotection Group at the Cadarache C.E.A. Center, for their help, which made the experiment possible.

## REFERENCES

[1] J. Léchelle, R. Boyer, and M. Trotabas, "A mechanistic approach of the sintering of nuclear fuel ceramics," *Mater. Chem. Phys.*, vol. 67, pp. 120-132, 2001.

[2] T. Weitkamp, C. Raven, and A. Snigirev, "An imaging and microtomography facility at the ESRF beamline ID22," in *Proc. SPIE Developments in X-Ray Tomography II*, vol. 3772, U. Bonse, Ed., 1999, pp. 311-317.

[3] A. Koch, F. Peyrin, P. Heurtier, B. Chambaz, B. Ferrand, W. Ludwig, and M. Couchaud, "An x-ray camera for computed microtomography of biological samples with micrometer resolution using Lu3Al5O12 and Y3Al5O12 scintillators," in *SPIE*, vol. 3659, 1999, pp. 170-179.

[4] A. P. Hammersley. (1999-2001) HST: High speed tomography. ESRF. [Online]http://www.esrf.fr/computing/scientific/index.html

[5] P. Jollivet, C. Den Auwer, and E. Simoni, "Evolution of the uranium local environment during alteration of SON68 glass," *J. Nucl. Mater.*, vol. 301, pp. 142-152, 2002.

[6] A. L. Ankudinov and S. D. Conradson, "Relativistic XANES calculations of Pu hydrates," *Phys. Rev. B*, vol. 57, no. 13, pp. 7518-7525, 1998.

[7] M. P. Anderson, G. S. Grest, R. D. Doherty, K. Li, and D. J. Slorovitz, "Inhibition of grain growth by second phase particles: three dimensional Monte Carlo computer simulation," *Scripta Metallurgica*, vol. 23, pp. 753-758, 1989.

[8] E. Morhacova, "Relation between Monte Carlo simulations of grain growth and real structures," *Cryst. Res. Technol.*, vol. 30, no. 1, pp. K9-K12, 1995.

[9] M. P. Anderson, D. J. Slorovitz, G. S. Grest, and P. S. Sahni, "Computer simulation of grain growth—I. Kinetics," *Acta Metallurgica et Materialia*, vol. 32, no. 5, pp. 783-791, 1984.
F.3 Acquisition de données de base pour le modèle
# Volume interdiffusion coefficient and uncertainty assessment for polycrystalline materials

Léchelle, J.ª,*, Noyau S.ª, Aufore L.ª, Arredondo A.ª, Audubert F.ª

ªCEA/DEN DEC/SPUA, Cadarache, bat. 717 - 13115 Saint-Paul-Lez-Durance

## Abstract

A method has been developed in order to assess small volume interdiffusion coefficients from experimental Electron Probe MicroAnalysis concentration profiles of polycrystalline materials by means of Boltzmann-Matano or den Broeder methods and their complementary Hall method. These methods have been used as tools for the investigation of the quasi-binary $UO_2/U_{(1-y)}Pu_yO_{(2-z)}$ interdiffusion, for which obtaining a solid solution in the bulk of grains is of major interest. In this paper uncertainties on the interdiffusion coefficient as a function of concentration have been computed for each method. Small volume coefficient measurements were enhanced by means of a small angle acquisition profile line with respect to the interdiffusion interface.

**Keywords:** interdiffusion, volume diffusion, Boltzmann-Matano, Hall, den Broeder, uncertainty

## 1. Introduction

Experimental volume interdiffusion coefficients have to be assessed from binary interdiffusion experiments carried out on polycrystalline materials, as it is often the case in industrial research and development studies. They cannot be simply obtained by means of Boltzmann-Matano [1] [2] or den Broeder [3] methods completed in the plateau areas by Hall [4] method by

<footnote>*mailto:jacques.lechelle@cea.fr</footnote>
means of discrete integration and derivation techniques from discrete exper-
imental points due to the scattering effect of grain boundary diffusion. Such
a problem has often been solved by means of a manual smoothing of the
experimental curve [5], or using Oishi model [6]. In this work the problem
was sorted out by fitting the diffusion profile with an appropriate function
which can afterwards be handled in order to compute the volume diffusion
coefficient ($\tilde{D}$) as a function of concentration.

For the three methods uncertainty both upon $\tilde{D}$ and its variable either
the concentration $c$ or the molecular ratio $y$ of the diffusing species has been
assessed by means of an error propagation method. J.F.Cornet [5] had coped
with this problem by first computing $\tilde{D}(c)$ versus $c$, then reassessing the
concentration profile from the interdiffusion coefficient and comparing it with
the initial experimental profile.

## 2. Volume interdiffusion information

2.1. Acquiring concentration profiles

Interdiffusion profiles have been acquired by means of an Electron Probe
Micro Analyser (EPMA) equipped with an electron gun and 4 wavelength
dispersive spectrometers, which allow for simultaneous measurement of the
intensity of characteristic X-rays of 4 elements. Quantification of an un-
known composition is based on the measurement of standards. The analysed
sample line intensity is measured from a sample and from a standard. This
standard is a compound with known composition. The measured intensities
are corrected for dead time (detector dependent), background and line over-
lap. The relative intensity ratio of the analysed sample line intensities from
sample and standard is calculated. Finally this relative intensity ratio is cor-
rected for the atomic number (Z), absorption (A) and fluorescence (F) effect
to give the concentration of the analysed sample. For interdiffusion studies
the measurement mode used in EPMA are 1D-line scans. A linescan across
a sample is made by moving the sample stage by small 1µm-increments.

## 2.2. Fitting concentration profiles - Chosing a function

The raw measurement of an EPMA is the weight percent $w_e$ of an element $e$ (in our example U, Pu and O). In this work these raw measurements $w_e$ along the acquisition axis $x$ are first changed into concentration profiles $c_e(x)$
on the basis of the chemical formula of the material (e.g. $U_{(1-x)}Pu_yO_{2-z}$) and of the lattice cell parameter $a$. The weight percent profiles of several elements may be needed in order to compute the concentration profile of a single element. If we consider $U$ profile:

$$w_U = \frac{(1 - y) M_U}{(1 - y) M_U + y M_{Pu} + (2 - z) M_O} \quad (1)$$

where $M_U$, $M_{Pu}$ and $M_O$ are U, Pu and O molar atomic weight. However in our example we only used one weight percent profile at a time to compute concentration profiles. For this non stoichiometric material $z$ depends only on the plutonium atomic ratio $y$ on the cationic sublattice, on temperature during the interdiffusion experiment and on oxygen partial pressure $p_{O_2}$ (these latter are supposed constant in the volume of the material): $z = f(y, T_{exp.}, p_{O_2})$. $y$ can be deduced from (1) by solving a third degree equation and by means of thermodynamic models ([7],[8] or [9]) in order to compute $f(y, T, p_{O_2})$. Then $c_{Pu}$ can be estimated from the cell parameter $a$ and the thermal expansion coefficient $\alpha$. ($U_{1-y}, Pu_y) O_{2-z}$ is a Cubic Face Centered (CFC) solid solution the lattice parameter of which is for the 4th order holohedral lattice cell [10]: $a = [(547.0 + 30.1 z) + (11.0 z - 7.4) y]$ pm and the thermal expansion coefficient of which is $\alpha_{z,T} = \alpha_{z=0,T} (1 + 3.9 z)$ [10]. The molar volume $V_m$ reads:

$$V_m = \frac{\mathcal{N}}{4} [a^3 (1 + \alpha)^3] \quad (2)$$

and

$$c_{Pu} = \frac{y}{V_m} = \frac{4y}{(A_1 + B_1 y)^3 (1 + \alpha)^3 \mathcal{N}} \quad (3)$$

Simple fitting functions are mentioned in literature [11] in order to reduce interdiffusion profile noise. They can be tested in order to extract solely volume interdiffusion from profiles where intergranular diffusion also occurs. The proposed functions assume that concentration data are antisymmetric with respect to their inflexion point:

$$c(x) = 0.25 (\tanh(x) + 2) \quad (4)$$

$$c(x) = 0.75 - 0.5 \exp(-0.3x)^3 \quad (5)$$

$$c(x) = 0.5 + (3.49 \cdot 10^{-3} \tan^{-1}(x)) \quad (6)$$
$$c(x) = 0.75 - 0.25 \operatorname{erfc} \left( \frac{x}{4 \tilde{D} \Delta t} \right) \quad (7)$$

B. Messerschmidt [12] has carried out analytical fittings in order to facilitate the computation of $\tilde{D}$ vs. $c$, expanding $c$ as a polynomial in $x$:

$$c(x) = \sum_{i=0}^{N} c_i x^i \quad (8)$$

S. Mendez both to suppress noise and extract volume diffusion coefficient from polycrystalline materials in the case of small diffusion coefficients used [13]:

$$c(x) = \frac{1 - \exp(-\lambda_1 x)}{1 - \lambda_2 \exp(-\lambda_1 x)} \quad (9)$$

The first four of them show a major limitation because they are well suited only for an antisymmetric curve (relatively to its inflexion point). Messerschmidt method is well suited only for the study of the plateaus of the interdiffusion profiles. The last one (S. Mendez's work) does not make these assumptions, but implicitly makes the hypothesis that the origin has been taken far enough from the interface so that the concentration slope can be considered as equal to 0. An advantage of S. Mendez's fitting function is that it can handle interdiffusion profile data with very small diffusion coefficients using large values of $\lambda_1$ and $\lambda_3$.

In this work the following function has been used:

$$c(x) = \lambda_1 - \lambda_2 \frac{1 - \lambda_3 \exp(-\lambda_4 x)}{1 + \lambda_5 \exp(-\lambda_4 x)} \quad (10)$$

with a set of parameters $\lambda_i > 0$ $i \in \{1, \dots, 5\}$ which can be different for $x$ smaller or greater than the inflexion point abscissa $x_0$ and denoted by $l$ and $r$ subscripts for left and right hand side respectively. $\ln(\lambda_5)$ and $\ln(\lambda_3)$ are preferred to $\lambda_5$ and $\lambda_3$ for numerical handling in software when the diffusion coefficients are very small to avoid computational precision loss. The smaller the diffusion coefficient is, the higher the values of $\lambda_3$ and $\lambda_5$ are and intermediate values during computation may exceed machine storage representation.

Parameters for the left and right hand side of the function are linked by the
continuity of the function, which holds in the case of a solid solution but which may not be the case if a demixtion occurs at the interface, as well as by the continuity or not of the first derivative of the concentration. This derivative might not be continuous if the interdiffusion mechanism changes with composition (e.g. if the interdiffusion profile runs through two different phases). In that case a discontinuity of $\tilde{D}$ can be balanced by a discontinuity of the derivative of the concentration. Thus, there is an advantage expressing $c(x)$ in terms of $\lambda_1, \lambda_2, x_0$, $c(x_0)$ and $c'(x_0)$ for the continuous derivative case,

$$ \lambda_3 = - \frac{2 x_0 c'(x_0)}{c(x_0) + \lambda_2 - \lambda_1} \left( 1 - \frac{2(\lambda_1 - c(x_0))}{\lambda_2} \right) \quad (11) $$

$$ \lambda_4 = - \frac{2 c'(x_0)}{c(x_0) + \lambda_2 - \lambda_1} \quad (12) $$

$$ \lambda_5 = - \frac{2 x_0 c'(x_0)}{c(x_0) + \lambda_2 - \lambda_1} \quad (13) $$

whereas when the first derivative of the concentration is discontinuous at the inflexion point $c(x)$ can be expressed in terms of $\lambda_1, \lambda_2, \lambda_4, x_0$ and $c(x_0)$:

$$ \lambda_3 = \left[ 1 - 2 \frac{(\lambda_1 - c(x_0))}{\lambda_2} \right] e^{x_0 \lambda_4} \quad (14) $$

$$ \lambda_5 = e^{x_0 \lambda_4} \quad (15) $$

$\lambda_i$ are meaningful, their value is linked to the maximum and minimum values of the concentration $c$ along the profile (resp. $c^+$ and $c^-$):

$$ \lim_{x \to -\infty} c(x) = \lambda_{l1} + \frac{\lambda_{l2} \lambda_{l3}}{\lambda_{l5}} = c^{-} \quad (16) $$

$$ \lim_{x \to +\infty} c(x) = \lambda_{r1} - \lambda_{r2} = c^{+} \quad (17) $$

Expressing (10) in terms of $\lambda_{1l}, \lambda_{2l}, x_0, c(x_0), c'(x_0), \lambda_{1r}, \lambda_{2r}$ can be of interest in the case of continuity of the derivative at the inflexion point:

$$ c(x) = \left( \lambda_1 - \frac{\lambda_2 - (2f(x_0) + \lambda_2 - 2\lambda_1) e^{\frac{2f'(x_0)(x-x_0)}{f(x_0) + \lambda_2 - \lambda_1}}}{e^{\frac{2f'(x_0)(x-x_0)}{f(x_0) + \lambda_2 - \lambda_1}} + 1} \right) \quad (18) $$

in that case the parameters are denoted $m_i$ hereafter. Some features of these
<table><caption>Table 1: Meaning of $m_i$ parameters for the two fitting cases</caption><thead><tr><th>i</th><th>m<sub>i</sub> (continuous case)</th><th>m<sub>i</sub> (discontinuous case)</th></tr></thead><tbody><tr><td>1</td><td>&lambda;<sub>l1</sub></td><td>&lambda;<sub>l1</sub></td></tr><tr><td>2</td><td>&lambda;<sub>l2</sub></td><td>&lambda;<sub>l2</sub></td></tr><tr><td>3</td><td>x<sub>0</sub></td><td>&lambda;<sub>l4</sub></td></tr><tr><td>4</td><td>c(x<sub>0</sub>)</td><td>x<sub>0</sub></td></tr><tr><td>5</td><td>c'(x<sub>0</sub>)</td><td>c(x<sub>0</sub>)</td></tr><tr><td>6</td><td>&lambda;<sub>r1</sub></td><td>&lambda;<sub>r1</sub></td></tr><tr><td>7</td><td>&lambda;<sub>r2</sub></td><td>&lambda;<sub>r2</sub></td></tr><tr><td>8</td><td>not applicable</td><td>&lambda;<sub>r4</sub></td></tr></tbody></table>

functions make them convenient tools for diffusion studies:

- their derivatives are negative (for $\lambda_i > 0$) and tend towards 0 in $\pm\infty$:

$$ \frac{dc}{dx} = - \frac{\lambda_2 \lambda_4 (exp^{ln(\lambda_3)-\lambda_4 x} + exp^{ln(\lambda_5)-\lambda_4 x})}{(exp^{ln(\lambda_5)-\lambda_4 x} + 1)^2} \quad (19) $$

$c(x)$ has to be monotoneous ($\frac{dc}{dx} < 0$), otherwise the direction of the flux would change along the interdiffusion profile.

$$ \lim_{+\infty} \frac{dc}{dx} = \lim_{-\infty} \frac{dc}{dx} = 0 \quad (20) $$

This is required for interdiffusion so that when concentration is that of the initial materials (i.e. $c = c\text{st} = c^+$ or $c^-$ at $x \to \pm\infty$) $\frac{dc}{dx} = 0$ and the flux is null.

Two of their properties are of interest for the use of both Boltzmann-Matano and den Broeder methods as it will be described in sections 3 and 4 respectively:

- they possess a reciprocal function with an analytical expression:

$$ x(c) = \frac{\ln\left(\frac{\lambda_2 e^{\ln(\lambda_3) - \ln(\lambda_5)}}{c + \lambda_2 - \lambda_1} - \frac{c - \lambda_1}{c + \lambda_2 - \lambda_1}\right) + \ln(\lambda_5)}{\lambda_4} \quad (21) $$

- the function $g(x) = x \frac{dc}{dx}$ has got primitives, let denote $P$ the one which is $\frac{\lambda_2 \ln(\lambda_5) (e^{\ln(\lambda_3) - \ln(\lambda_5)} + 1)}{\lambda_4}$ at $-\infty$:

$$ P(x) = \frac{\lambda_2 (e^{\ln(\lambda_3) - \ln(\lambda_5)} + 1) (\ln(e^{\lambda_4 x} - \ln(\lambda_5) + 1) + \ln(\lambda_5))}{\lambda_2 (e^{\ln(\lambda_3) - \ln(\lambda_5)} + 1) x e^{\lambda_4 x}} \quad (22) $$
The existence of such analytical functions simplifies a lot the numerical estimation of $\tilde{D}$ vs. c.

A more mathematical approach consisting in expanding the solution of the second Fick law on a basis of erf functions has been developed by A.G. Nikitin [14], however this method also requires both to find the set of coefficients from the experimental data (deconvolution) and then to numerically rebuild $\tilde{D}$ vs. c. Handling a simple function as the one proposed in this work is easier.

### 2.3. Extracting volume diffusion profiles

Experiments provide profiles acquired by means of an Electron Probe MicroAnalyser (EPMA) which are due both to volume and faster grain-boundary diffusion phenomena. In the case where material grains are large enough, the contribution of grain-boundary diffusion may be subtracted by keeping along the profile the only points which pertain to the bulk of grains. Since the microstructure (grain boundaries) may not have been revealed by a chemical or thermal etching, a way of selecting "good" points would be to keep only local maxima for $c < c(x_0)$ and local minima for $c > c(x_0)$. A way of automatizing such a procedure is to give a weight $w_i$ to each point $(x_i, c_i)$, $i \in \{0, \dots, n\}$ smaller when the concentration is too low below the current smoothed curve (the one of the current iteration k) on the left side of the curve and smaller when the concentration is too high above this same curve on the right side of the inflexion point.

A Hooke and Jeeves procedure [15] has been used in order to minimize the test function:

$$ \chi_k^2 = \sum_{i=0}^{n} w_{k_i} (c_i - c(x_i))^2 \quad (23) $$

with

$$ w_{k_i} = 10^{-3} w_{0_i} \left| 1 - \frac{\text{contrib} - \chi_{k-1}^2}{\chi_{k-1}^2} \right|^p \quad (24) $$

where $w_{0_i}$ are the initial weights depending on the confidence associated with point #i (based on the discrepancy, for instance, between the sum over all elements of their weight percentage and 100%):

$$ w_{0_i} = \exp \left( -\frac{1}{2} \left( \frac{100 - \sum_{e=1}^{n_{el}} w_e}{\sigma_w} \right)^2 \right) \quad (25) $$
where $\sigma_w$ is the uncertainty upon the sum over the weight percent of the elements.

$contrib_{-\chi^2_{k-1}}$ is defined by:

$$contrib_{-\chi^2_{k-1}} = \frac{w_{k-1_i} (c_i - c(x_i))^2}{\chi^2_{k-1}} \qquad (26)$$

The parameters for the Hooke and Jeeves method are the $m_i$.

This method has been used on a $UO_2/U_{1-y}Pu_yO_{2-z}$ polycrystalline sample. Figure 1 gives the raw and fitted concentration curves for $\lambda_{l1} = 0.033 \text{ mol.cm}^{-3}$, $\lambda_{l2} = 0.0063 \text{ mol.cm}^{-3}$, $x_0 = 572.5 \text{ µm}$, $c(x_0) = 0.0336 \text{ mol.cm}^{-3}$, $c'(x_0) = -0.0003375 \text{ mol.cm}^{-3.\mu m^{-1}}$, $\lambda_{r1} = 0.0317 \text{ mol.cm}^{-3}$ and $\lambda_{r2} = 0.0093 \text{ mol.cm}^{-3}$, so that $\lambda_{3l} = 3.63785 \cdot 10^{22}$, $\lambda_{5g} = 2.78188 \cdot 10^{22}$, $\lambda_{3r} = 9.71611 \cdot 10^{15}$, $\lambda_{5r} = 7.59326 \cdot 10^{15}$.

A drawback of the method is the need to find a set of parameters close enough to the one which leads to a minimum in $\chi^2$ in order to converge. However it is not a strong constraint since equations (11), (13), (16) and (17) help assigning first values to $\lambda_i$.

An advantage of this method is to give the uncertainties upon the seeked parameters which are the values of the search steps in each direction (one direction per parameter) when $\chi^2$ stops decreasing.

## 2.4. Uncertainty upon c

### 2.4.1. Limits of EPMA for very small interdiffusion coefficients: a source of uncertainty

Uncertainties can be assigned to EPMA measurements using ISO/GUM recommendation [16]. Both, physical phenomena following Poisson law (X-Ray emission of the probed volume) and other key factors under study in the measurement process regarded as corrections (ZAF, surface roughness...) are taken into account:

$$w_{Spl} = \frac{\left( \frac{N_{Spl}}{t_{Spl}} - \frac{N_{bckgd_{Spl}}}{t_{bckgd_{Spl}}} \right)}{\left( \frac{N_{std}}{t_{std}} - \frac{N_{bckgd_{std}}}{t_{bckgd_{std}}} \right)} w_{std} C_{ZAF} C_{inhom.} C_{roughness} C_{n.m.el.} \qquad (27)$$

where $Spl$ stands for sample, $w_{Spl}$: weight percent of the measured element in the sample, $N_{Spl}$: count number due to the sample, $t_{Spl}$: time of acquisition for the sample, $bckgd$: background, $std$: standard, $w_{std}$: the weight
<figure><img src="image_85.png" /><figcaption>Figure 1: raw and fitted concentrations for $\lambda_1$ = 0.033 mol.cm⁻³, $\lambda_2$ = 0.0063 mol.cm⁻³, $x_0$ = 572.5 µm, $c(x_0)$ = 0.0336 mol.cm⁻³, $c'(x_0)$ = -0.0003375 mol.cm⁻³.µm⁻¹, $\lambda_1$ = 0.0317 mol.cm⁻³ and $\lambda_2$ = 0.0093 mol.cm⁻³ ($\lambda_{3l}$ = 3.63785 · 10⁺²², $\lambda_{5g}$ = 2.78188 · 10⁺²², $\lambda_{3r}$ = 9.71611 · 10⁺¹⁵, $\lambda_{5r}$ = 7.59326 · 10⁺¹⁵).</figcaption></figure>
percent of the measured element in the standard, $c_{ZAF}$: ZAF correction, $c_{inhom.}$: correction due to the inhomogeneity of the material, $c_{roughness}$: correction due to the roughness of the sample, $c_{n.m.el.}$: correction due to non measured elements. Main of the uncertainty causes aforementioned generally lead to very small corrections, except in the case of interdiffusion, the inhomogeneity inherent to the diffusion phenomenon itself. In the same manner as raw EPMA measurements can be corrected by a factor $c_{inhomog.}$ due to the inhomogeneity of the sample, the interdiffusion profile $c(x)$ expressed in terms of the concentration of an element (mol.cm$^{-3}$) can be corrected by an analog correction $c'_{inhomog.}$. In this work the diffusion contribution to $c'_{inhom.}$ has been analysed. Figure 2 represents the axis of the acquisition line along which measurements were carried out as well as the interface of the materials.

For simplicity the measurement volume of the microprobe is supposed to be a sphere of radius $R = 0.5\mu m$. If the acquisition line is orthogonal to the interface and $\bar{c}$ denotes the value of c measured by the microprobe,

$$ \bar{c}(x_i) = \frac{3}{4R^3} \int_{x_i-R}^{x_i+R} (R^2 - (x-x_i)^2) c(x) dx \quad (28) $$

where $c(x)$ is given by (18). As a matter of fact the measured value of concentration which is fitted is already an averaged one: c cannot be measured only $\bar{c}$ can. In order to evaluate the real point value of this function ($c(x)$) several acquisitions would be needed with a varying starting point. This cannot be the case with the EPMA sample holder the position of which can only be shifted by 1 $\mu m$ along two perpendicular directions and the process would however be too time consumming. Evaluation of the impact of this averaging effect is carried out on the measured concentration $\bar{c}$ instead, assuming that:

$$ c - \bar{c} \approx \bar{c} - \bar{\bar{c}} \quad (29) $$

with $\bar{c}$ the average of the measured $\bar{c}$ concentration. This should not be too rough an approximation as far as these differences remain small.

In order to be able to measure very small diffusion coefficients the acquisition line makes a very small angle $\alpha$ with respect to the interface. Let $Y_i$ be the orthogonal projection of the point of abscissa $x_i$ along the acquisition line onto the Y axis (perpendicular to the interface in the plane defined by the probed surface of the sample) and $c^{\perp}(Y_i)$ the concentration in the plane the
<figure><img src="image_86.png" /><figcaption>Figure 2: EPMA acquisition line on top of the sample (with iso-concentration planes)</figcaption></figure>

equation of which is $Y = cst = Y_i$.

$$ 
\begin{aligned}
c^{\perp}(Y) &= c_{\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5} \left( \frac{Y}{\sin(\alpha)} + x_i \right) \\
&= c_{\lambda_1, \lambda_2, \lambda_3} e^{-\lambda_4 x_i}, \frac{\lambda_4}{\sin(\alpha)}, \lambda_5 e^{-\lambda_4 x_i} (Y)
\end{aligned} 
\quad (30) $$

$\bar{c}(x_i)$ becomes:

$$ 
\begin{aligned}
\bar{c}(x_i) &= \bar{c}^{\perp}(Y_i) \\
&= \frac{3}{4R^3} \int_{Y_i-R}^{Y_i+R} (R^2 - (Y-Y_i)^2) c_{\lambda_1, \lambda_2, \lambda_3} e^{-\lambda_4 x_i}, \frac{\lambda_4}{\sin(\alpha)}, \lambda_5 e^{-\lambda_4 x_i} (Y) \, dY
\end{aligned} 
\quad (31) $$

The closer to the interface the iso-concentration planes are, the narrower they are one from each other for a constant concentration step. Thus, it can
<figure><img src="image_87.png" /><figcaption>Figure 3: Averaged EPMA concentration $\bar{c}$ and EPMA concentration $c$ versus $\bar{c}$</figcaption></figure>

be expected that the averaging effect of the measurement is greater near the interface where $||\vec{\nabla}c||$ is high except at the inflexion point if the profile is antisymmetric (both side contributions annihilate).

Figure 3 gives the relative difference between $\bar{c}$ and $c$. In the following $c$ notation is used instead of $\bar{c}$.

### 2.4.2. Uncertainty upon $x$

Uncertainty upon the $x$ coordinate along the profile has been evaluated taking into account several points:

- the surface of the sample may not be parallel to the sample-holder plane,

- the surface of the sample may not be a plane,

- the material may have a high thermal expansion coefficient so that dis-
tances between two points when the EPMA measurement is done may be
much shorter than the distance between these same points at the tempera-
ture of the experiment.

Despite efforts made to smoothen the sample surface prior to EPMA anal-
ysis, on a millimeter scale (i.e. interdiffusion profile length), the surface of
the sample may not be a plane. Or, even for a perfectly planar sample, its
plane may not be perfectly parallel to the sample-holder surface. The equa-
tion of the mean plane of the sample surface in the axis of the microprobe
is obtained as follow. The three coordinates relatively to the sample-holder
axis system of each acquisition point along a profile, as well as those of at
least two extra reference points, e.g. these denoted A and B in Figure 4 are
used for the determination of the mean least-square plane. In the case of

<figure><img src="image_88.png" /><figcaption>Figure 4: EPMA Pu X map with an acquisition line and the two points A and B at each end of the interface</figcaption></figure>
several acquisition profiles in the same area of the material points from all the profiles are taken into account. The initial points are projected onto this plane and then projected onto a least-square line in the projection plane. Least-square uncertainties upon these coordinates are then calculated. A subsequent correction is done along a profile on the basis of the Pu weight content and on the stoichiometry (deduced from Pu weight content, temperature and oxygen potential) and temperature in order to compute a thermal expansion coefficient and correct point spacing along an acquisition profile. In the case of our materials, and in the specific thermodynamic conditions used (composition, temperature, oxygen potential), the overall correction, obtained as the uncertainties associated to the least square fitting methods was close to 1/1000 which means $u(x) \approx 0.5\mu m$ near the inflexion point $x_0$ (the first measurement point was chosen as the origin and the profiles have approximately 1000 points).

In the case of Boltzmann-Matano method (see section 3), this uncertainty is taken into account as the uncertainty upon $x_0$ and $x_M$, in the case of den Broeder method (see section 4) as the uncertainty upon $x_0$, whereas in the case of Hall method as that upon $x - x_M$ (see section 4.3).

### 2.4.3. Uncertainty upon c function

Uncertainty upon concentration function has got several origins, - the acquisition technique (here the EPMA), its limitations have been described hereabove, its contribution on the uncertainty upon c is $u_{tech.}(c) = \delta c$, - the choice of the fitting function: small variations in the value of parameters $m_i$ can give acceptable values for the fitting curve, for the case of a concentration function $c(x)$ the derivative of which is continuous at the inflexion point the contribution to $u(c)$ reads:

$$ u_{fitt.}(c)^2 = \left(\frac{\partial c}{\partial \lambda_{l1}}\right)^2 u(\lambda_{l1})^2 + \left(\frac{\partial c}{\partial \lambda_{l2}}\right)^2 u(\lambda_{l2})^2 + \\ \left(\frac{\partial c}{\partial x_0}\right)^2 u(x_0)^2 + \left(\frac{\partial c}{\partial c(x_0)}\right)^2 u(c(x_0))^2 + \left(\frac{\partial c}{\partial c'(x_0)}\right)^2 u(c(x_0'))^2 + \\ \left(\frac{\partial c}{\partial \lambda_{r1}}\right)^2 u(\lambda_{r1})^2 + \left(\frac{\partial c}{\partial \lambda_{r2}}\right)^2 u(\lambda_{r2})^2 \qquad (32) $$

all these partial derivatives have been computed analytically from equation (18).

In order to express final $\tilde{D}$ values versus the atomic ratio $y$ of one cation of the interdiffusion species (here Pu), the relationship (34) between $y$, $c$ and
$z$ ($y = y(c, z)$) is used.

If U concentration is considered:

$$c_U = \frac{4(1-y)}{(A_1 + B_1 y)^3 (1+\alpha)^3 N} \qquad (33)$$

or Pu concentration:

$$c_{Pu} = \frac{4y}{(A_1 + B_1 y)^3 (1+\alpha)^3 N} \qquad (34)$$

in both cases it can be noticed that $y$ is a root of a third degree polynomial depending on $c$ and $z$ values, and its uncertainty, $u(y)$, depends both on $u(c)$ and $u(z)$. $\frac{\partial y}{\partial c}$ and $\frac{\partial y}{\partial z}$ have been computed analytically according to the propagation linked to the third degree polynomial resolution method.

$$u(y)^2 = \left(\frac{\partial y}{\partial c}\right)^2 u(c)^2 + \left(\frac{\partial y}{\partial z}\right)^2 u(z)^2 \qquad (35)$$

$u(c)$ is computed as aforementioned.

It could sound surprising that $y$ be computed from $c$ if we refer to (2.2). In section (2.2) values of $c$ and $y$ are those at the experimental points whereas in this section their value are considered at any point $x$ along the acquisition line whether it is at the same location of an experimental point or not and thus in this section $y$ is deduced from $c$.

Thus, the third origin of uncertainty is the partial lack of knowledge of the chemical composition of the particular sample we used: both initial parts of the sample had been finely characterized but as the heat treatment was stopped at the end of the interdiffusion experiment the oxygen content information from EPMA was not quantitative enough all along the profile to be used and the oxygen content had to be calculated by means of a thermodynamic model using temperature, oxygen potential and Pu EPMA concentration profile.

Table 2 sums up the different uncertainties used for the example of our study.

### 3. Boltzmann-Matano volume interdiffusion coefficient $\tilde{D}$

Although Boltzmann-Matano method is less used because of more recent advanced methods such as Sauer and Freise [17], Wagner [18], and den Broeder modifications, it remains a straightforward starting point to evaluate interdiffusion coefficients.
<table><caption>Table 2: Parameter uncertainties set by the user in the studied example</caption><thead><tr><th>parameter</th><th>value</th><th>uncertainty</th></tr></thead><tbody><tr><td>λ<sub>l1</sub></td><td>0.033 (mol.cm<sup>-3</sup>)</td><td>0.001</td></tr><tr><td>λ<sub>l2</sub></td><td>0.0063 (mol.cm<sup>-3</sup>)</td><td>0.0002</td></tr><tr><td>x<sub>0</sub></td><td>572.5 (μm)</td><td>0.5</td></tr><tr><td>c(x<sub>0</sub>)</td><td>0.0336 (mol.cm<sup>-3</sup>)</td><td>0.0006</td></tr><tr><td>c'(x<sub>0</sub>)</td><td>-0.0003375 (mol.cm<sup>-3</sup>μm<sup>-1</sup>)</td><td>0.0000025</td></tr><tr><td>λ<sub>r1</sub></td><td>0.0317 (mol.cm<sup>-3</sup>)</td><td>0.001</td></tr><tr><td>λ<sub>r2</sub></td><td>0.0093 (mol.cm<sup>-3</sup>)</td><td>0.0002</td></tr><tr><td>α</td><td>0.03176 (rd)</td><td>0.00046</td></tr><tr><td>z</td><td>computed</td><td>0.013</td></tr><tr><td>λ<sub>l3</sub></td><td>computed : 3.63785 · 10<sup>+22</sup></td><td>—</td></tr><tr><td>λ<sub>l5</sub></td><td>computed : 2.78188 · 10<sup>+22</sup></td><td>—</td></tr><tr><td>λ<sub>r3</sub></td><td>computed : 9.71611 · 10<sup>+15</sup></td><td>—</td></tr><tr><td>λ<sub>r5</sub></td><td>computed : 7.59326 · 10<sup>+15</sup></td><td>—</td></tr></tbody></table>

### 3.1. Boltzmann-Matano method

According to Boltzmann-Matano method $\tilde{D}$ expression is the following:

$$ \tilde{D}(c^*) = - \frac{\sin^2(\alpha)}{2 \Delta t} \frac{\int_{-\infty}^{x^*} (x - x_M) \frac{dc}{dx} dx}{\frac{dc}{dx}|_{x^*}} \quad (36) $$

where $\alpha$ is the angle between the acquisition line and the interface and $\Delta t$ the duration of the interdiffusion experiment (in isothermal conditions). In this work the abscissa $x_M$ of 'Matano plane', (the one crossed by an equal amount of matter one way and the other) can be expressed using $c(x)$ definition (18) and its properties:

$$ x_M = \frac{(c(x_0) + \lambda_{l2} - \lambda_{l1}) \left( x_0 + \frac{\log(2) (c(x_0) + \lambda_{l2} - \lambda_{l1})}{c'(x_0)} \right)}{+ \frac{c^--c^+}{(c(x_0) + \lambda_{r2} - \lambda_{r1}) \left( x_0 - \frac{\log(2) (c(x_0) + \lambda_{r2} - \lambda_{r1})}{c'(x_0)} \right)}} \quad (37) $$

For $x < x_0$:

$$ \int_{-\infty}^{x^*} (x - x_M) \frac{dc}{dx} dx = P(x^*, \lambda_{ri}) - x_M(c - c^+) \quad (38) $$
while for $x \ge x_0$:

$$ \int_{-\infty}^{x^*} (x-x_M) \frac{dc}{dx} dx = P(x_0, \lambda_{ri}) + P(x^*, \lambda_{li}) - P(x_0, \lambda_{li}) - x_M(c-c^+) \quad (39) $$

In any case:

$$ \left. \frac{dc}{dx} \right|_{x^*} = - \frac{\lambda_2 \lambda_4}{e^{\lambda_4 x^* - Ln(\lambda_5)} + e^{Ln(\lambda_5) - \lambda_4 x^* + 2}} + \frac{\lambda_2 \lambda_4}{(e^{\lambda_4 x^* - Ln(\lambda_3)} + e^{Ln(\lambda_5) - Ln(\lambda_3)}) (e^{\lambda_4 x^* - Ln(\lambda_5)} + 1)} - \frac{\lambda_2 \lambda_4 e^{Ln(\lambda_3) - Ln(\lambda_5)}}{e^{\lambda_4 x^* - Ln(\lambda_5)} + 1} \quad (40) $$

Figure 5 gives Boltzmann-Matano interdiffusion coefficient as a function of uranium concentration $c_U$. As concentration is close to its limits ($c^-, c^+$) the values of $\tilde{D}$ obtained by Boltzmann-Matano method are abnormally high as $c$ tends towards its limits $c^+$ and $c^-$. The uncertainty upon $\tilde{D}$ is discussed hereafter.

## 3.2. Uncertainty upon Boltzmann-Matano interdiffusion coefficient

Such uncertainty assessment with a fitting function have already been carried out: B. Messerschmidt [12] had calculated an uncertainty upon $\tilde{D}(c)$ on the basis of the polynomial expansion of $c(x)$ but with the aim of emphasizing the weakness of Boltzmann-Matano method in the plateau areas. J.F.Cornet used an iterative method [5].

The uncertainty upon $\tilde{D}$ from Boltzmann-Matano method with our fitting function has been studied hereafter. Several sources of uncertainties have been considered and are described hereafter.

### 3.2.1. EPMA averaging effect

The averaging effect of the EPMA has already been described for concentration $c$. The same assumption as that of section 2.4.3 is made here:

$$ |\delta \tilde{D}| = |\tilde{D}(c) - \tilde{D}(\bar{c})| \approx |\tilde{D}(\bar{c}) - \tilde{D}(\bar{c})| \quad (41) $$

which leads to a contribution to $u(\tilde{D})$:

$$ u_{tech.}(\tilde{D}) = \delta \tilde{D} \quad (42) $$

Although $\int (x - x_M) \frac{dc}{dx}$ has got an analytical expression on the basis of $c_{\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5}$, as well as $\frac{dc}{dx}$, it is not the case for analog expressions with $\bar{c}$.
<figure><img src="image_89.png" /><figcaption>Figure 5: Boltzmann-Matano interdiffusion coefficient ($\tilde{D}$) as a function of uranium concentration ($c_U$)</figcaption></figure>
For this reason analog expressions related to $\bar{c}$ such as the integral in (31) can only be computed numerically. They have been evaluated by a Gauss-Legendre method with 24 points $\xi_j$ in the reduced interval $[-1; 1]$.

$$ \begin{aligned} \left. \frac{d\bar{c}}{dx} \right|_{Y_i} &= \frac{\sin(\alpha)}{\frac{4}{3} R^3} \int_{Y_i-R}^{Y_i+R} (-2Y_i + 2Y) c_{\lambda_1, \lambda_2, \lambda_3} e^{-\lambda_4 x_i}, \frac{\lambda_4}{\sin(\alpha)}, \lambda_5 e^{-\lambda_4 x_i} (Y) \, dY \\ &= \frac{\sin(\alpha)}{\frac{4}{3} R^3} R \sum_{j=1}^{24} w_j (-2Y_i + 2(R\xi_j + Y_i)) c_{\lambda_1, \lambda_2, \lambda_3} e^{-\lambda_4 x_i}, \frac{\lambda_4}{\sin(\alpha)}, \lambda_5 e^{-\lambda_4 x_i} (R\xi_j + Y_i) \end{aligned} \quad (43) $$

where $\xi_i$ are Gauss-Legendre reduced abscissa and $w_i$ their associated weights. Since the analog computation of $\tilde{D}(\bar{c})$ is numeric its value may be slightly affected by the integration method, both $\tilde{D}(c)$ and $\tilde{D}(\bar{c})$ have been computed numerically as far as the evaluation of $|\tilde{D}(c) - \tilde{D}(\bar{c})|$ was concerned. For this purpose $\int (x - x_M) \frac{dc}{dx}$ is approximated by means of a simple trapezium rule scheme, and $\frac{dc}{dx}$ by a simple forward finite differences method.

Since the uncertainty due to the averaging effect of the measurement technique (the main cause of EPMA uncertainty in our case) is of prime interest, it is compared to the value of the interdiffusion coefficient. Figure 6 shows the uncertainty upon $\tilde{D}$ due to this averaging effect denoted $\delta D$ versus concentration for our example ($\lambda_{l1} = 0.033 \text{ mol.cm}^{-3}$, $\lambda_{l2} = 0.0063 \text{ mol.cm}^{-3}$, $x_0 = 572.5 \text{ µm}$, $c(x_0) = 0.0336 \text{ mol.cm}^{-3}$, $c'(x_0) = -0.0003375 \text{ mol.cm}^{-3.\mu m^{-1}}$, $\lambda_{r1} = 0.0317 \text{ mol.cm}^{-3}$ and $\lambda_{r2} = 0.0093 \text{ mol.cm}^{-3}$). Hence, in this specific case, it can be concluded that in the studied range of U concentration EPMA averaging measurement process can be regarded as negligible leading to reliable values of $\tilde{D}$ whereas for smaller interdiffusion coefficients other techniques may be required. For instance Rutherford Back Scattering (RBS) can be used if both parts of the interdiffusion couple are taken apart at the end of the experiment for analysis in the range $0.1 \text{ µm}$ up to $2 \text{ µm}$ [19] or, on a decreasing scale with tens nanometer resolution, Secondary Ion Mass Spectrometry (SIMS) can be applied [20] to analyse each separated part of the interdiffusion couple, or, with 1 nanometer resolution, coupled with Transmission Electron Microscopy (TEM) with high angle annular dark field (HAADF) imaging, Electron Dispersive Spectroscopy (EDS) and parallel electron energy loss spectroscopy (PEELS) [21].
### 3.2.2. Uncertainty upon $x_M$

Uncertainty upon $x_M$ value is taken into account, which is at the origin of the development of other methods such as den Broeder's method.

$$ u(x_M)^2 = \delta^2 x_M + \left(\frac{\partial x_M}{\partial \lambda_{l1}}\right)^2 u(\lambda_{l1})^2 + \left(\frac{\partial x_M}{\partial \lambda_{l2}}\right)^2 u(\lambda_{l2})^2 + \left(\frac{\partial x_M}{\partial x_0}\right)^2 u(x_0)^2 + \left(\frac{\partial x_M}{\partial c(x_0)}\right)^2 u(c(x_0))^2 + \left(\frac{\partial x_M}{\partial c'(x_0)}\right)^2 u(c(x'_0))^2 + \left(\frac{\partial x_M}{\partial \lambda_{r1}}\right)^2 u(\lambda_{r1})^2 + \left(\frac{\partial x_M}{\partial \lambda_{r2}}\right)^2 u(\lambda_{r2})^2 \quad (44) $$

Except for $\delta x_M$, analytical expressions have been used for the calculation of the partial derivatives $\frac{\partial x_M}{\partial m_i}$.

In the case of the studied profile $u(x_M) = 0.5$ and $x_M = 576.3 \pm 0.5 \mu m$.

This uncertainty leads to a contribution to the uncertainty upon $\tilde{D}$:

$$ u(\tilde{D}) = \frac{\partial \tilde{D}}{\partial x_M} u(x_M) \quad (45) $$

This effect is also given in Figure 6 and cannot be neglected as $c \to c^-$. A way of avoiding such a high uncertainty on this side of the concentration range would be for this half part of the domain ($c \ge c(x_M)$) to evaluate $\tilde{D}$ by means of:

$$ \tilde{D}(c^*) = \frac{\sin^2(\alpha) \int_{x^*}^{+\infty} (x_M - x) \frac{dc}{dx} dx}{2 \Delta t \frac{dc}{dx}|_{x^*}} \quad (46) $$

instead of (36).

### 3.2.3. Uncertainty upon $\alpha$

$\tilde{D}$ depends also on the angle $\alpha$ of the acquisition line with the interface: $\frac{\partial \tilde{D}}{\partial \alpha} = 2 \frac{\cos(\alpha)}{\sin(\alpha)} \tilde{D}$. Although the choice of a small $\alpha$ angle leads to a high dependency of $\tilde{D}$ on $\alpha$, it has to be noticed that it increases the number of points in the vicinity of the interface, i.e. where concentration varies a lot and thus increases the precision upon $\lambda_{l1},...,\lambda_{r2}$. The number of significant points $N_s$ (i.e. with a high weight because they influence greatly the shape of the curve $c = c(x)$) increases by a factor $\frac{1}{\sin(\alpha)}$ and $u(m_i)_\alpha \propto \frac{1}{\sqrt{N_s^\alpha}} = \frac{\sqrt{\sin(\alpha)}}{\sqrt{N_s^\alpha}}$. In our

example $\alpha = 0.033$, and $\frac{u(m_i)_{N_s^\alpha}}{u(m_i)_{N_s^\alpha}} \approx 5.5$ so that $m_i$ parameters ($\lambda_{l1},...,\lambda_{r2}$) are
at least 5 times more precise than they would have been if deduced from the
perpendicular profile.

This contributes to the uncertainty upon $\tilde{D}$:

$$u(\tilde{D}) = \frac{\partial \tilde{D}}{\partial \alpha} u(\alpha) = 2 \cdot \cot g(\alpha) \tilde{D} \cdot u(\alpha) \quad (47)$$

$u(\alpha)$ is assessed on the basis of the uncertainty upon the least-square profile line and the uncertainty upon A and B location with respect to the interface of the materials. This contribution to $u(\tilde{D})$ is minimum as $\tilde{D}$ reaches its minimum. This third effect is also represented in Figure 6 but remains smaller than 1/10 of the measured coefficient all along the profile.

### 3.2.4. Uncertainty upon $m_i$

$\tilde{D}$ depends also directly on the parameters $\lambda_{l1},...,x_0,...,\lambda_{r2}$. Its partial derivatives $\frac{\partial \tilde{D}}{\partial m_i}$ are calculated analytically once and are part of the software which has been developed.

Overall, $\tilde{D}$ uncertainty reads:

$$u(\tilde{D})^2 = \delta \tilde{D}_y^2 + \frac{\partial \tilde{D}}{\partial x_M}^2 u(x_M)^2 + \frac{\partial \tilde{D}}{\partial \alpha}^2 u(\alpha)^2 + \frac{\partial \tilde{D}}{\partial \lambda_{l1}}^2 u(\lambda_{l1})^2 + \frac{\partial \tilde{D}}{\partial \lambda_{l2}}^2 u(\lambda_{l2})^2 + \frac{\partial \tilde{D}}{\partial x_0}^2 u(x_0)^2 + \frac{\partial \tilde{D}}{\partial c(x_0)}^2 u(c(x_0))^2 + \frac{\partial \tilde{D}}{\partial c'(x_0)}^2 u(c'(x_0))^2 + \frac{\partial \tilde{D}}{\partial \lambda_{r1}}^2 u(\lambda_{r1})^2 + \frac{\partial \tilde{D}}{\partial \lambda_{r2}}^2 u(\lambda_{r2})^2 \quad (48)$$

Figure 7 gives the various relative contributions to $u(\tilde{D})$. For this particular case of study all parameter uncertainties have a nearly equal effect on $\tilde{D}$ in the uranium composition range [0.023; 0.0402]mol.cm$^{-3}$. The most sensitive ones are the position of the inflexion point $(x_0, c(x_0))$ as well as the slope at this point $c'(x_0)$. However the contribution to $\tilde{D}$ uncertainty $u(\tilde{D})$ turns out to be higher at each end of the diffusion profile. Contributions appear not to be symmetric with respect to the limits of the concentration interval simply reflecting the fact that the interdiffusion profile is not symmetric:

$$- c(x_0) = 0.0336 \text{ mol.cm}^{-3} \neq \frac{c_+ + c_-}{2} = 0.03145 \text{ mol.cm}^{-3},$$

- $\lambda_{li}$ and $\lambda_{ri}$ are not identical.
<figure><img src="image_90.png" /><figcaption>Figure 6: Contribution of the averaging effect of the Electron Probe MicroAnalyser ($\delta c$), of the uncertainty of the profile angle with the interface ($u(\alpha)$) and of Matano abscissa uncertainty ($u(x_M)$) to Boltzmann-Matano volume interdiffusion diffusion coefficient uncertainty for $\lambda_{l1} = 0.033 \text{ mol.cm}^{-3}$, $\lambda_{l2} = 0.0063 \text{ mol.cm}^{-3}$, $x_0 = 572.5 \mu m$, $c(x_0) = 0.0336 \text{ mol.cm}^{-3}$, $c'(x_0) = -0.0003375 \text{ mol.cm}^{-3}.\mu m^{-1}$, $\lambda_{r1} = 0.0317 \text{ mol.cm}^{-3}$ and $\lambda_{r2} = 0.0093 \text{ mol.cm}^{-3}$</figcaption></figure>
<figure><img src="image_91.png" /><figcaption>Figure 7: Contribution of uncertainties upon $\lambda_{1l}$, $\lambda_{2l}$, $x_0$, $c(x_0)$, $c'(x_0)$, $\lambda_{1r}$ and $\lambda_{2r}$ to Boltzmann-Matano interdiffusion coefficient</figcaption></figure>
## 4. Den Broeder volume interdiffusion coefficient $\tilde{D}$

Den Broeder method as been extensively used in the case of metals [22], [23] as well as ceramics [24]. The advantage of this method compared to Boltzmann-Matano method is mainly that it does not depend on the Matano abscissa and (to a lesser extent in our case) it can take into account atomic (or molecular) volume variations with concentration ([22],[23]).

### 4.1. Den Broeder method

In this method the interdiffusion coefficient is computed as a function of the atomic (or molecular in our case) ratio of one of the diffusing species $N_2$ and of its relative value $Y$. Let us say species #2 for instance. Then $\tilde{D}$ reads:

$$ \tilde{D}(N_2^*) = \frac{(N_2^+ - N_2^-) V_m (N_2^*)}{2 \Delta t \left(\frac{\partial N_2}{\partial x}\right)_{x=x^*}} \left[ (1 - Y^*) \int_{-\infty}^{x^*} \frac{Y}{V_m} dx + Y^* \int_{x^*}^{+\infty} \frac{(1 - Y)}{V_m} dx \right] \quad (49) $$

This equation can be expressed in terms of $c(x)$ for which the analytical form defined by equation (18) is used here too. In this section $c(x)$ is denoted by $c_1(x)$ where 1 labels the species 1:

$$ \tilde{D}(c_1^*) = \frac{-1}{c_1^- - c_1^+} \frac{1}{c_1 \left(\frac{dV_m}{dx}\right)_{x=x^*} + V_m \left(\frac{dc_1}{dx}\right)_{x=x^*}} \left[ (c_1^- - c_1^*) \int_{-\infty}^{x^*} (c_1 - c_1^+) dx + (c_1^* - c_1^-) \int_{x^*}^{+\infty} (c_1^- - c_1) dx \right] \quad (50) $$

where $c_1^-$ and $c_1^+$ are defined according to equations (16) and (17).

It is to be noticed that the discrete points used for the evaluation of all the terms are computed using the analytical expression of $c_1(x)$, which means that they can be chosen as close one to each other as needed.

The integrals in Den Broeder coefficient evaluation can be computed analytically. Since in the case of our material $\frac{dV_m}{dx}$ cannot be expressed analytically due to $V_m$ expression (2), it has been computed by means of a Lagrange polynomial based derivation matrix with three point stencils, and the whole expression of $\tilde{D}$ has been computed on a numerical basis. A simple trapezium rule was used for the determination of the integrals. Figure 8 gives den Broeder interdiffusion coefficient compared to that of Boltzmann-Matano. They are very close one to another.
<figure><img src="image_92.png" /><figcaption>Figure 8: Den Broeder interdiffusion coefficient ($\tilde{D}_{dB}$ blue) and Boltzmann-Matano interdiffusion coefficient ($\tilde{D}_{B-M}$ red) as a function of uranium concentration ($c_U$).</figcaption></figure>
## 4.2. Uncertainty upon Den Broeder interdiffusion coefficient

Since in our material applications the atomic ratio $y$ is a meaningful parameter, $\tilde{D}$ uncertainty has been estimated on the basis of the variables $y$ and $V_m$ (rather than $c$ and $V_m$) and their contribution to $u(\tilde{D})$: $\delta\tilde{D}_y$ which is the variation in $\tilde{D}$ due to the uncertainty upon $y$ (i.e. $u(y)$) and $\delta\tilde{D}_{V_m}$ which is the uncertainty in $\tilde{D}$ due to the uncertainty upon $V_m$.

$$u(\tilde{D})^2 = \delta\tilde{D}_y^2 + \delta\tilde{D}_{V_m}^2 + \left(\frac{\partial\tilde{D}}{\partial\alpha}\right)^2 u(\alpha)^2 \quad (51)$$

$\delta\tilde{D}_y$ (resp. $\delta\tilde{D}_{V_m}$) is simply obtained by computing $\tilde{D}$ using $y$ and $y \pm u(y)$ (resp. $V_m \pm u(V_m)$). $u(y)$ is computed as described in section 2.4.3, i.e. as a function of $u(z)$ and $u(c)$, so that the influence of each parameter $\lambda_1, \lambda_2, x_0, c(x_0)$ and $c'(x_0)$ can be studied. The partial derivatives of the expression of the molar volume given in section 2.4.3 lead to the uncertainties upon $V_m$:

$$u(V_m)^2 = \left[ \left( \frac{\partial V_m}{\partial z} \right)^2 u(z)^2 + \left( \frac{\partial V_m}{\partial y} \right)^2 u(y)^2 \right] \quad (52)$$

Thus, $\tilde{D}$ uncertainty in den Broeder method is built up of the same terms as for Boltzmann-Matano method, without $\frac{\partial\tilde{D}}{\partial x_M}u(x_M)$ term but with an extra $\frac{\partial\tilde{D}}{\partial V_m}u(V_m)$ which remains smaller.

Despite the higher uncertainty of Boltzmann-Matano method due to $x_M$, on our example both methods give identical results. This is mainly due to the choice of the fitting function which has lead to analytical expressions for the Boltzmann-Matano method. As concentration comes closer to its plateau values ($c^-$ and $c^+$) on each side of the interfaces, den Broeder or Boltzmann-Matano methods are not precise enough to be used, instead Hall method is preferred. The different sources of uncertainties are given in Figure 9. This method is commonly used in conjunction with both of these methods in order to estimate $\tilde{D}$ vs. $c$ on the whole concentration range (Boltzmann-Matano method in conjunction with Hall method: in the case of uranium and plutonium interdiffusion [25], or in the case of Ti and a Mo-alloy interdiffusion [26], den Broeder method in conjunction with Hall method: in the case of Ge in nickel alloys [27], Ti-Mo interdiffusion [28], or carbon diffusion in ferritic steels [29]).
<figure><img src="image_93.png" /><figcaption>Figure 9: Contribution of *y*, *Vm* and *α* uncertainties upon Den Broeder interdiffusion coefficient as a function of uranium concentration (*c*<sub>U</sub>)</figcaption></figure>

4.3. *Hall method*

In this section too, the discrete points used are generated as close as
needed one to each other by means of the analytical form of c(x) given by
equation (18).

The relative atomic ratio *Y* is defined by: *Y* = (*c*−*c*<sub>min</sub>) / (*c*<sub>max</sub>−*c*<sub>min</sub>) and *U* by
*U* = *erf*<sup>−1</sup>(2*Y* − 1). As *c* → *c*<sub>min</sub> (resp. *c*<sub>max</sub>), *U* versus x−x<sub>M</sub>/√Δ*t* becomes a
straight line: *U* → (x−x<sub>M</sub>)*h*−/√(Δ*t*) + *k*− (resp. *U* → (x−x<sub>M</sub>)*h*+/√(Δ*t*) + *k*+).
For the side of the curve left to the inflexion point (decreasing concentration
profile) $\tilde{D}$ reads:

$$ \tilde{D}_{-} = \frac{1}{4h_{-}^{2}} \left[ 1 - k_{-} \sqrt{\pi} \exp^{U^{2}} (1 - \operatorname{erf}(U)) \right] \quad (53) $$

with $\lim_{x \to -\infty} \tilde{D}_{-} = \frac{1}{4h_{-}^{2}}$ and on the right hand side:

$$ \tilde{D}_{+} = \frac{1}{4h_{+}^{2}} \left[ 1 + k_{+} \sqrt{\pi} \exp^{U^{2}} (1 + \operatorname{erf}(U)) \right] \quad (54) $$

with $\lim_{x \to +\infty} \tilde{D}_{+} = \frac{1}{4h_{+}^{2}}$.

$k_{-}$, $h_{-}$, $k_{+}$ and $h_{+}$ are obtained by fitting the asymptots of the interdiffusion profile $U$ vs. $\frac{x-x_M}{\sqrt{\Delta t}}$ (i.e. on each side, q% of the concentration interval $[c_{min}; c_{min} + \frac{q^{+}}{100} (c_{max} - c_{min})]$ and $[c_{max} - \frac{q^{-}}{100} (c_{max} - c_{min}); c_{max}]$), with a weighted linear least-square method with error in both coordinates [30]. $h_{-}$ (resp. $h_{+}$) is the root of a 'pseudo second degree polynomial' ([30]), the coefficients of which depend themselves slightly on $h_{-}$ ([30]):

$$ A h_{-}^{2} + B h_{-} + C = 0 \quad (55) $$

The solution $h_{-} = -21.6916s^{\frac{1}{2}} \mu m^{-1}$ is obtained, in our example, after 3 iterations from the non-weighted least-square slope $h_{-}^{0} = -16.8775s^{\frac{1}{2}} \mu m^{-1}$ as initial value.

$$ A = \sum_{i=1}^{N^{-}} \frac{w_{i} \left( \frac{x_{i}-x_{M}}{\sqrt{\Delta t}} - \frac{\langle x-x_{M} \rangle}{\sqrt{\Delta t}} \right) (U_{i} - \langle U \rangle)}{w\left(\frac{x_{i}-x_{M}}{\sqrt{\Delta t}}\right)} \quad (56) $$

$$ B = \sum_{i=1}^{N^{-}} w_{i}^{2} \left( \frac{\left( \frac{x_{i}-x_{M}}{\sqrt{\Delta t}} - \frac{\langle x-x_{M} \rangle}{\sqrt{\Delta t}} \right)^{2}}{w(U_{i})} - \frac{(U_{i} - \langle U \rangle)^{2}}{w\left(\frac{x_{i}-x_{M}}{\sqrt{\Delta t}}\right)} \right) \quad (57) $$

$$ C = - \sum_{i=1}^{N^{-}} \frac{w_{i} \left( \frac{x_{i}-x_{M}}{\sqrt{\Delta t}} - \frac{\langle x-x_{M} \rangle}{\sqrt{\Delta t}} \right) (U_{i} - \langle U \rangle)}{w(U_{i})} \quad (58) $$

(59)

where:

- $N^{-}$ is the number of points used to fit $h_{-}$,
<figure><img src="image_94.png" /><figcaption>Figure 10: Least square fitting of the example of the left hand side plateau (blue line) with non-weighted (LS-red) and weighted (WLS-black) with error in both coordinates least-square methods</figcaption></figure>

- global weight: $w_i = \frac{w\left(\frac{x_i-x_M}{\sqrt{\Delta t}}\right)w(U_i)}{h^2w(U_i)+w\left(\frac{x_i-x_M}{\sqrt{\Delta t}}\right)}$

- weight for $x$: $w\left(\frac{x_i-x_M}{\sqrt{\Delta t}}\right) \approx \frac{1}{u(x_i)^2+u(x_M)^2}$

- weight for $U$: $w(U_i) = \frac{\frac{dU}{dc}u(c_i)^2}{\frac{dU}{dc}u(c_i)^2}$

The uncertainty upon $\frac{(x_i-x_M)}{\sqrt{\Delta t}}$ is due to $x_M$, $u(x_M)$, and to the uncertainty upon $x_i$. This latter is obtained from that of $c(x_i)$: $u(x_i) = c^{-1}(c_i + u(c_i)) - (c_i - u(c_i))$, whereas $u(U_i) = \frac{dU}{dc}u(c_i)$ with $\frac{dU}{dc} = \frac{\sqrt{\pi}}{c_{max}-c_{min}}e^{(2Y-1)^2}$.

Figure 10 shows as an example the weighted and non-weighted least-square fits of the left hand side plateau on our material. Weights are simply the
inverses of the uncertainties on both coordinates. The weighted method is believed to be closer to real values, which is hardly the case in our example but above all enables computating uncertainties upon $h_+$ and $k_+$ (resp. on $h_-$ and $k_-$) which both depend on the abscissa uncertainty $u(\frac{x-x_M}{\sqrt{\Delta t}})$ and on the ordinate uncertainty $u(U)$ deduced from $u(c)$. Once $h_-$ is found $k_-$ is obtained from:

$$k_{-} = \langle U \rangle - h_{-} \left\langle \frac{x_{i} - x_{M}}{\sqrt{\Delta t}} \right\rangle \qquad (60)$$

Figure 11 gives Hall interdiffusion coefficient $\tilde{D}$ as well as Boltzmann-Matano value. They are very close one to each other, probably partly because of the choice of the fitting function.

### 4.4. Uncertainty upon Hall interdiffusion coefficient

The uncertainty upon $\tilde{D}$ is obtained by first expressing (53) and (54) as a function of concentration $c$ instead of $U$.

$$\tilde{D}_{-} = \frac{10^{-8}}{4 h_{-}^{2}} \left[ 1 - 2 k_{-} \sqrt{\pi} \exp \left[ erf^{-1} \left( \frac{2 c - (c_{min} + c_{max})}{c_{max} - c_{min}} \right) \right] \left( \frac{c_{max} - c}{c_{max} - c_{min}} \right) \right] \sin^2 \alpha \, cm^2 s^{-1} \qquad (61)$$

$$\tilde{D}_{+} = \frac{10^{-8}}{4 h_{+}^{2}} \left[ 1 + 2 k_{+} \sqrt{\pi} \exp \left[ erf^{-1} \left( \frac{2 c - (c_{min} + c_{max})}{c_{max} - c_{min}} \right) \right] \left( \frac{c - c_{min}}{c_{max} - c_{min}} \right) \right] \sin^2 \alpha \, cm^2 s^{-1} \qquad (62)$$

Hence, in the interval $[c_{max} - \frac{q}{100}, (c_{max} - c_{min}); c_{max}]$, $u(\tilde{D})$ reads:

$$u(\tilde{D})^2 = \left(\frac{\partial \tilde{D}}{\partial c}\right)^2 u(c)^2 + \left(\frac{\partial \tilde{D}}{\partial \alpha}\right)^2 u(\alpha)^2 + \left(\frac{\partial \tilde{D}}{\partial h_{-}}\right)^2 u(h_{-})^2 + \left(\frac{\partial \tilde{D}}{\partial k_{-}}\right)^2 u(k_{-})^2 \qquad (63)$$

with:

$$\begin{aligned} u(c)^2 = & \left(\frac{\partial c}{\partial \lambda_1}\right)^2 u(\lambda_1)^2 + \left(\frac{\partial c}{\partial \lambda_2}\right)^2 u(\lambda_2)^2 + \left(\frac{\partial c}{\partial x_0}\right)^2 u(x_0)^2 \\ & + \left(\frac{\partial c}{\partial c(x_0)}\right)^2 u(c(x_0))^2 + \left(\frac{\partial c}{\partial c'(x_0)}\right)^2 u(c'(x_0))^2 + (\delta c)^2 \end{aligned} \qquad (64)$$

$u(k_-)$, $u(h_-)$, $u(k_+)$ and $u(h_+)$ are obtained by means of Reed least-square fitting method for pairs of coordinates ([30]). Since their expressions are rather cumbersome [30], they are not given here but they depend on the
same variables as $h_-$ and $k_-$, i.e. A, B and C.

Tables 3 and 4 show that an approximate amount of 2000 points in the fitted area ($q_- = q_+ = 20$ for the width of the fitting intervals) is sufficient to determine $h_-$, $k_-$, $h_+$, $k_+$ and their uncertainties.

<table><caption>Table 3: Sensitivity of $h_-$, $k_-$ and their uncertainties to the number of points $N_-$</caption><thead><tr><th>N</th><th>h<sub>-</sub> (s<sup>1/2</sup>.µm<sup>-1</sup>)</th><th>u(h<sub>-</sub>) (s<sup>1/2</sup>.µm<sup>-1</sup>)</th><th>k<sub>-</sub></th><th>u(k<sub>-</sub>)</th></tr></thead><tbody><tr><td>300</td><td>-16.5665</td><td>0.0737749</td><td>0.361792</td><td>0.00569403</td></tr><tr><td>2000</td><td>-16.9767</td><td>0.0189383</td><td>0.332352</td><td>0.00146226</td></tr><tr><td>4000</td><td>-17.0093</td><td>0.0126968</td><td>0.330016</td><td>0.000980421</td></tr><tr><td>10000</td><td>-17.0272</td><td>0.00777959</td><td>0.328728</td><td>0.000600755</td></tr></tbody></table>

<table><caption>Table 4: Sensitivity of $h_+$, $k_+$ and their uncertainties to the number of points $N_+$</caption><thead><tr><th>N</th><th>h<sub>+</sub> (s<sup>1/2</sup>.µm<sup>-1</sup>)</th><th>u(h<sub>+</sub>) (s<sup>1/2</sup>.µm<sup>-1</sup>)</th><th>k<sub>+</sub></th><th>u(k<sub>+</sub>)</th></tr></thead><tbody><tr><td>300</td><td>-12.1487</td><td>0.0456292</td><td>-0.481375</td><td>0.0040949</td></tr><tr><td>2000</td><td>-12.3394</td><td>0.012405</td><td>-0.465739</td><td>0.00111574</td></tr><tr><td>4000</td><td>-12.3543</td><td>0.0084178</td><td>-0.464517</td><td>0.000757272</td></tr><tr><td>10000</td><td>-12.3625</td><td>0.0051977</td><td>-0.463844</td><td>0.00046764</td></tr></tbody></table>

The sensitivity of $\tilde{D}$ uncertainty to its different causes are given by the partial derivatives:

- $\frac{\partial \tilde{D}}{\partial c} \frac{\partial c}{\partial \lambda_1}$ for changes in $\lambda_1$,

...

- $\frac{\partial \tilde{D}}{\partial c} \frac{\partial c}{\partial c'(x_0)}$ for changes in $c'(x_0)$,

- $\frac{\partial \tilde{D}}{\partial c}$ for changes in $c$ due to EPMA averaging effect,

- $\frac{\partial \tilde{D}}{\partial \alpha}$ for changes in $\alpha$, due to $\alpha$ assessment uncertainty

- $\frac{\partial \tilde{D}}{\partial h_-}$ for changes in $h_-$, the uncertainty of which is given by the weighted least-square method with error in both coordinates,

- $\frac{\partial \tilde{D}}{\partial k_-}$ for changes in $k_-$.

From (53) it is noticeable that $\frac{\partial \tilde{D}}{\partial c}$ can be obtained in the following way:

$$ \frac{\partial \tilde{D}_{-}}{\partial c} = \left( \tilde{D}_{-} + \frac{k_{-} - 1}{2 h_{-}^{2}} \right) \frac{dU}{dc} = \left( \tilde{D}_{-} + \frac{k_{-} - 1}{2 h_{-}^{2}} \right) \frac{\sqrt{\pi}}{c_{max} - c_{min}} e^{(2Y-1)^2} \quad (65) $$
<figure><img src="image_95.png" /><figcaption>Figure 11: Comparison of Hall and Boltzmann-Matano interdiffusion coefficient ($\tilde{D}$) as a function of uranium concentration ($c_U$)</figcaption></figure>
<figure><img src="image_96.png" /><figcaption>Figure 12: Contribution of c, α, h_− and k_− to Hall $\tilde{D}_-$ coefficient</figcaption></figure>

The analog result also stands for $\frac{\partial \tilde{D}_+}{\partial c}$.

$\frac{\partial c}{\partial \lambda_1}, \dots, \frac{\partial c}{\partial \lambda_s(x_0)}$ are obtained like in the case of Boltzmann-Matano method. For the last three partial derivatives analytical expressions can be used:

$$ \frac{\partial \tilde{D}}{\partial \alpha} = 2 \cot g(\alpha) \tilde{D} \qquad (66) $$

$$ \frac{\partial \tilde{D}}{\partial h} = -\frac{2 \tilde{D}}{h} \qquad (67) $$

$$ \frac{\partial \tilde{D}}{\partial k} = \frac{\tilde{D}}{k} - \frac{1}{4k h^2} \qquad (68) $$

A noticeable property of these first two partial derivatives is that they vary
in the same way as $\tilde{D}$ does, so that their corresponding contribution to $u(\tilde{D})$ is minimum as $\tilde{D}$ reaches its minimum. It does not apply for the last one since $\forall c$, $\tilde{D}_-(c) \le \frac{1}{4h_-^2}$ and $\lim_{x \to -\infty} \tilde{D}_- = \frac{1}{4h_-^2}$ (resp. $\tilde{D}_+(c) \le \frac{1}{4h_+^2}$ and $\lim_{x \to +\infty} \tilde{D}_+ = \frac{1}{4h_+^2}$). Figure 12 gives the relative contribution of each cause to Hall estimation of the Hall interdiffusion coefficient versus concentration for the left hand side of the profile.

The main causes of error are the uncertainty upon $k_-$ (resp. $k_+$) and upon $c$ (due to those of the fitting parameters, this latter is overestimated, since uncertainties on the concentration function 18 parameters cannot all take a bad value at the same time otherwise the curve fitting would not be acceptable).

Although $\tilde{D}(c)$ obtained by Hall method has got rather high incertitude values (close do $10^{-14} cm^2 s^{-1}$ in our example) as $c$ varies, its limits which do not depend on $c$ are more precise. The assessed limits of $\tilde{D}$ obtained on our profile are the following:

$$ \lim_{x \to -\infty} \tilde{D}_- = 8.75 \pm 0.806 \cdot 10^{-15} cm^2 s^{-1} \qquad (69) $$

$$ \lim_{x \to +\infty} \tilde{D}_+ = 1.66 \pm 0.052 \cdot 10^{-14} cm^2 s^{-1} \qquad (70) $$

## 5. Discussion

The three coefficients are very close and in good agreement in the range of [0.0226; 0.0403] $mol \cdot cm^{-3}$. Below and over these values Hall diffusion coefficient has to be preferred.

The function 18 used in this work in order to fit interdiffusion profiles is very sensitive to the values chosen at the inflexion point $(x_0, c(x_0)$ and $c'(x_0))$. Its asymptots depend both on well known compositions at each end and on the values of the parameters sensitive to function variations at the inflexion points. Thus the parameters adjusted for the fitting function were precise enough to fit the asymptots so that den Broeder (or even Boltzmann-Matano) method could be very close to Hall method. The use of an analytical function for the three cases permitted the use of as many points as needed in function of the part of the curve (equally spaced as a function of $x$ position or of concentration $c$ value).
## 6. Conclusion

A way of computing uncertainty associated to Boltzmann-Matano, den Broeder and Hall methods has been proposed for assessing volume inter-diffusion coefficients changes with composition from Electron Probe Micro-analyser measurements on polycrystalline materials. This method has been used by S. Noyau [20] in order to study $UO_2/U_{1-y}Pu_yO_{2-z}$ interdiffusion in the temperature range 1500°C-1700°C. In the particular case of our material $U_{1-y}Pu_yO_{2-z}$ and experimental conditions (1700°C), the three methods have given agreeing results and the uncertainty study has given a concentration range for which the EPMA profile results could be reliably used.

In the central part of the concentration range, both den Broeder and Boltzmann-Matano methods give the same results with an uncertainty close to $8 \cdot 10^{-15}cm^2 \cdot s^{-1}$ while they both fail at the ends of this concentration range where Hall method give better results with an uncertainty close to $8 \cdot 10^{-16}cm^2 \cdot s^{-1}$. For these very small volume interdiffusion coefficients the choice of an acquisition line making a very small angle with the interdiffusion interface has lead to an enhanced precision on the fitted parameters of the concentration smoothing function. An optimized angle has to be chosen since interdiffusion profiles with a small angle with the interface otherwise decrease the precision upon interdiffusion coefficients.

In the central part of the interdiffusion profiles in our case of study values smaller than $10^{-14}cm^2 \cdot s^{-1}$ can hardly be measured. This value is also close to the lower limit of what can be obtained in terms of Hall interdiffusion coefficient on the plateaus of the interdiffusion profiles.

Single crystals could be used as benchmark materials in order to check the validity of the results as well as their precision.

A way of enhancing precision in the interdiffusion coefficient values versus concentration would be to use several interdiffusion couples with intermediate compositions.

The C/C++-language software developed in this work is available on request for whom is interested in.

## 7. Acknoledgements

Authors express their gratitude to N. Vigier at AREVA-NC for his support in this study as well as experimentalists in DEC/SPUA LEFCA facility chiefly M.A. Martinez and the Head of the Laboratory F. Adenot for his thorough proof readings.
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G

Schémas de décroissance radioactive
des isotopes de U, Pu et Am
<figure><img src="image_97.png" /><figcaption>FIGURE G.1 – Décroissance de $^{235}$U d’après [ENSDF2006]</figcaption></figure>
<figure><img src="image_98.png" /><figcaption>FIGURE G.2 – Décroissance de $^{238}$U d’après [ENSDF2006]</figcaption></figure>
<figure><img src="image_99.png" /><figcaption>FIGURE G.3 – Décroissance de $^{238}$Pu d’après [ENSDF2006]</figcaption></figure>
<figure><img src="image_100.png" /><figcaption>FIGURE G.4 – Décroissance de $^{239}$Pu d’après [ENSDF2006]</figcaption></figure>
<figure><img src="image_101.png" /><figcaption>FIGURE G.5 – Décroissance de $^{240}$Pu d’après [ENSDF2006]</figcaption></figure>
<figure><img src="image_102.png" /><figcaption>FIGURE G.6 – Décroissance de $^{241}$Pu d’après [ENSDF2006]</figcaption></figure>
<figure><img src="image_103.png" /><figcaption>FIGURE G.7 – Décroissance de $^{242}$Pu d’après [ENSDF2006]</figcaption></figure>
<figure><img src="image_104.png" /><figcaption>FIGURE G.8 – Décroissance de $^{241}$Am d’après [ENSDF2006]</figcaption></figure>
<figure><img src="image_105.png" /><figcaption>FIGURE G.9 – Décroissance de $^{242}$Am d’après [ENSDF2006]</figcaption></figure>
<figure><img src="image_106.png" /><figcaption>FIGURE G.10 – Décroissance de $^{243}$Am d’après [ENSDF2006]</figcaption></figure>
Index

Adjuvants de frittage, 23, 88, 124

Angle dièdre, 74–76, 112

Carte de microstructure, 23

Coefficients de diffusion, 43, 45, 46, 50, 101

Comportement élastique, 65, 66

Compression, 7, 9, 18, 82

Conditions algébriques, 63

Conditions aux limites, 63

Conservation de la matière, 70

Constantes élastiques, 46, 49, 99

Contrainte de frittage, 27, 113

Courbure, 1, 2, 7, 17, 27, 60, 62, 63, 73, 76, 77, 79, 81–83, 86, 113, 114, 123

Cous, 17–19, 24, 61, 112

Cycle thermique, 1, 9, 10, 61

Défauts ponctuels, 2, 14, 45, 53, 61, 124

Diffusion, 1, 18–21, 23, 28, 46, 47, 50, 53, 60–62, 70, 77, 83–85, 89, 90, 95, 101, 102

Diffusion en surface, 18

Diffusion en volume, 12, 18

DNPA, 113–118

Energie de surface, 7

Evaporation/condensation, 1, 9, 14, 15

Facettes, 50, 97

Flux, 1, 7, 10, 53, 61–63, 65, 70–72, 88

Formulation faible, 67

Frittage, 7, 63, 76, 83, 86–90, 97, 101, 107, 109–113, 118–120, 123, 124

Frittage différentiel, 27

Haute température, 9, 49, 52, 99, 111, 214

Interdiffusion, 2, 13, 28, 46, 47, 53, 87, 89, 90, 101, 102

Joint de grains, 7, 10–13, 17, 20, 21, 53, 60, 61, 68, 70, 73, 76, 77, 83–85, 87, 101, 123

Mécanique, 62

Maillage, 73, 76–78, 82, 83, 86

MEB environnemental, 111

Neumann, 61, 63, 65, 135

Non-rotation, 68

Non-translation, 68

Oxydes d'actinides, i, 1, 9, 35

Paramètres d'entrée, 93

Pas de temps, 62, 69, 73, 76, 77, 82, 86, 124

Potentiel chimique, 1, 7, 9, 10, 69, 70, 123

Puissance thermique, 38

Radioactif, 1, 37, 39–41, 85, 123

Raffinement, 73, 78, 83, 86

Retrait, 18, 85

Stœchiométrie, 1, 43–46, 61, 97, 101

Stades, 2, 17–21, 24, 25, 49, 60, 107, 113

Symétrie, 1, 9, 49, 50, 52, 66

Tension de surface, 18, 27, 49, 50, 62, 64, 97

Texture 3D, 118

Thermodynamique linéaire des processus irréversibles, 37

Tomographie, 107

Unité de structure, 53

Validation, 2, 83, 105, 107, 112, 113, 151

XANES, 107
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