A new equation of state for helium nanobubbles embedded in UO$_2$ matrix calculated via molecular dynamics simulations
Résumé
Molecular dynamics simulations have been carried out to determine the equation of state of helium inside nanobubbles embedded into UO$_2$ matrix.The parameters of this equation of state are fitted with the Brearley and MacInnes hard-sphere model based on the formalism of Carnahan-Starling used in fuel performance codes.This new equation of state takes into account the interactions between the surrounding UO$_2$ matrix and the helium atoms. Four nanobubble sizes (diameters 1, 2, 5, and 10 nm) have been investigated over four temperatures 300, 500, 700, and 900 K and for initial helium concentration inside the bubble ranging from 0.33 x 10$^5$ to 3.9 x 10$^5$ mol.m$^{-3}$ (corresponding to helium-to-vacancy ratio of 0.3-3.3, respectively). We observe that helium atoms are inhomogeneously distributed inside the bubble. A boundary layer of 1 nm thickness appears at the bubble surface in which helium atoms are more concentrated and diffuse into the UO$_2$ matrix. We also observe a saturation of the helium atoms that can be incorporated into the bubble.This concentration limit is equal to 1.6 helium atom per vacancy in UO$_2$. It corresponds to an atomic volume of 7.8 x 10$^{-30}$ m$_3$, which is almost half of the value proposed with the original Brearley and MacInnes model (13 x 10$^{-30}$ m$_3$). For this threshold concentration and for bubble of diameter higher than 5nm, micro-cracks and dislocations appear at the bubble surface.We calculated the critical pressures inside the bubble that yields to this onset of crack in UO$_2$. These critical pressures are in good agreement with those calculated with the Griffith criterion for brittle fracture.
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