Revisiting thermodynamics and kinetic diffusivities of uranium–niobium with Bayesian uncertainty analysis

Abstract

In this work, thermodynamic and kinetic diffusivities of uranium-niobium (U-Nb) are reassessed by means of the CALPHAD (CALculation of PHAse Diagram) methodology. In order to improve the consistency and reliability of the assessments, first-principles calculations are coupled with CALPHAD. In particular, heats of formation of γ-U-Nb are estimated and verified using various density-functional theory (DFT) approaches. These thermochemistry data are then used as constraints to guide the thermodynamic optimization process in such a way that the mutualconsistency between first-principles calculations and CALPHAD assessment is satisfactory. In addition, long-term aging experiments are conducted in order to generate new phase equilibria data at the γ 2 /α + γ 2 boundary. These data are meant to verify the thermodynamic model. Assessment results are generally in good agreement with experiments and previous calculations, without showing the artifacts that were observed in previous modeling. The mutual-consistent thermodynamic description is then used to evaluate atomic mobility and diffusivity of γ-U-Nb. Finally, Bayesian analysis is conducted to evaluate the uncertainty of the thermodynamic model and its impact on the system's phase stability.

Multiscale Modeling of Discontinuous Precipitation in U‐Nb

“…• The sluggish evolution time (of orders 10 14 (s) for 450 • C and 10 11 (s) for 550 • C) results from the estimated CALPHAD's slow bulk diffusivity (in orders of 10 −23 [cm 2 /s] for 450 • C and 10 −21 [cm 2 /s] for 550 • C [12], consistent with the experimental values from Peterson and Ogilvie [32,33]). In reality, the reaction happens much faster due to the fast boundarydiffusion condition at the reaction front of DP [6] as evidenced by the measured interphase boundary diffusivity triple products in [7].…”

“…Within this figure, (set) A, B, and C indicate the by-nature thermodynamics of U-Nb, the ordinary thermodynamics of U-Nb approximated by CALPHAD, and Djuric's phenomenological hypothesis, respectively. Here, the exact boundary of A is unknown hence it is represented by a dashed line; B is however known explicitly and believed to belong to A since the CALPHAD assessment is in reasonable agreement with experiments [12] (note that B is believed to represent the core of A -the ordinary thermodynamics defining U-Nb's chemical equilibria); therefore, it is proposed to be an available thermodynamic reference to examine C. According to the set theory, C has three possible positions relative to B:…”

“…To estimate thermodynamic driving force (in form of energy density [J/cm 3 ]), the previously assessed CALPHAD energetic data are used [12]; for simplicity, the used molar volumes for this estimation are assumed to be constant and take the average value of those from the initial orth and bcc phases. The interfacial energies are taken as the averages of those evaluated in [7].…”

“…The interfacial mobilities are chosen according to our empirical formula: µ = s D i M i D i +M i , where s = 10 6 is a scaling factor, i indicates either alpha or γ, M is the value of atomic mobility, and D is the value of interdiffusivity. Here, the atomic mobility and interdiffusivity of γ are taken from the previously assessed DIC-TRA database [12]; basing on the atomic packing factors of orth and bcc, it was assumed that the atomic mobility and diffusivity of α are three times faster than those of γ; since these kinetic coefficients do not affect the thermodynamic LE between the two reacting phases, their precise values are of only peripheral interest within the scope of this work. Nevertheless, it is noted that kinetic factors can play an important role in determining the lamellar microstructure of DP, as demonstrated later in this work; therefore a comprehensive knowledge of these physical quantities is beneficial for future developments and applications of the nuclear material.…”

“…Insight into the thermodynamic driving force of DP as well as those of general precipitation (before DP) and DC (after DP) are then given by tracking the energy path across the reactions, towards equilibrium [9]. Although this information mediates the understanding why specific length scales, growth rates, and phase compositions are selected, it does not infer explanations for the occurrence and growth of γ during DP, especially with the phase's composition residing near the center of miscibility gap's unstable region (see [10][11][12] To shed some light towards the addressing of these questions, we attempt, in the current work, the investigations of DP's origin from the fundamental thermodynamic and kinetic points of view. We choose to take our views from the interesting perspective of phase-field theory that accounts both thermodynamic and kinetic factors towards microstructural evolutions [13][14][15][16][17].…”