Phase-field modeling of gas bubbles and thermal conductivity evolution in nuclear fuels

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Abstract

A phase-field model was developed to simulate the accumulation and transport of fission products and the evolution of gas bubble microstructures in nuclear fuels. The model takes into account the generation of gas atoms and vacancies, and the elastic interaction between diffusive species and defects as well as the inhomogeneity of elasticity and diffusivity. The simulations show that gas bubble nucleation is much easier at grain boundaries than inside grains due to the trapping of gas atoms and the high mobility of vacancies and gas atoms in grain boundaries. Helium bubble formation at unstable vacancy clusters generated by irradiation depends on the mobilities of the vacancies and He, and the continuing supply of vacancies and He. The formation volume of the vacancy and He has a strong effect on the gas bubble nucleation at dislocations. The effective thermal conductivity strongly depends on the bubble volume fraction, but weakly on the morphology of the bubbles.

Introduction

Due to the creation of fission products and radiation damage, a complex microstructure evolution, such as the formation of gas bubbles, voids, precipitates and dislocation networks, occurs in nuclear fuels and cladding materials. These microstructural changes can result in fuel instability (such as fuel swelling, embrittlement, cracking and surface roughening) and thermodynamic property changes (such as phase instability, diffusivity and thermal conductivity). Therefore, understanding and predicting the microstructure evolution and its subsequent impact on material properties are crucial for scientific design of nuclear materials, optimizing fuel operation, and reducing uncertainty in operational and safety margins.

There has been extensive experimental investigation of the influence of fission products on properties and performance of nuclear fuels and cladding materials over the past 40 years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Great progress has also been made in developing computational models for the prediction of gas release and fuel performance [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. For example, the most advanced models include FASTGRASS [18], VICTORIA [19], MFPR [20], [21] and FRAPCON [22]. These codes take into account a number of observed phenomena such as the production of fission gases, bubble nucleation, migration and coalescence, re-solution, temperature, temperature gradients, interlinked porosity and thermal conductivity. The codes are mainly based on two assumptions: (1) the evolution of the average concentrations of fission products follows kinetic rate equations; and (2) the growth kinetics of a single spherical gas bubble is described by the capillarity relationship. This relationship is valid only for the description of equilibrium crystals and generally fails under irradiation conditions when the fuel matrix is saturated with point defects. In addition, these models ignore or do not sufficiently take into account the effects of defects and defect structures, the morphology of bubbles, i.e., spatial and size distribution of bubbles, the elastic interaction among defects, bubbles and diffusive fission products and diffusivity inhomogeneity. All of these factors are important in determining both new phase nucleation and microstructure evolution kinetics, and hence, the microstructure and properties.

The phase-field method based on the fundamental thermodynamic and kinetic information has been emerging as a powerful computational approach at the mesoscale for predicting phase stability and microstructural evolution kinetics during many materials processes such as solidification, precipitation in alloys, ferroelectric domain evolution in ferroelectric materials, martensitic transformation, dislocation dynamics and electrochemical process [23], [24], [25], [26], [27], [28], [29], [30], [31]. This method describes a microstructure using a set of conserved and nonconserved variables that are continuous across the interfacial regions. The temporal and spatial evolution of the variables, i.e., the microstructural evolution, is governed by the Cahn-Hilliard nonlinear diffusion equation and the Alan Cahn relaxation equation. It uses input from thermodynamic and kinetic data from atomistic simulation, thermal dynamic calculations and experiments, and outputs the kinetic information of microstructure evolution. Compared with atomistic simulation methods such as first principles methods and molecular dynamics [32], [33], the phase-field method can deal with much larger length and time scales. Some kinetic Monte Carlo methods [34] might have time and length scales similar to the phase-field method. However, the phase-field method has the advantage of taking into account long-range interactions such as elastic interactions. If the calculation of energy differences requires evaluating the elastic solution at each MC step, then it will be a very time consuming simulation. Rate theory methods [14], [21], [35] have been used for simulating gas bubble evolution in metals and nuclear fuels. Although rate theories can deal with large time and length scales, it is difficult to include the effects of the morphology of gas bubbles, and the inhomogeneity of defect distributions and material properties on microstructure evolution with rate theory. It is very easy to integrate long-range interactions and the inhomogeneity of material properties into phase-field formulisms. These advantages give phase-field modeling the power to model the evolution of the microstructure and properties in the material at the mesoscale and linking atomistic and macroscopic simulations.

Microstructural evolution in irradiated materials is a complex process which involves extreme time and length scales ranging from individual atomic events such as point defect generation during radiation cascades on the scale of femtoseconds and nano-meters, to long term diffusion and macroscopic material property changes on the scale of years and meters. To study microstructural evolution requires not only accurate thermodynamic and kinetic properties of individual defects such as the generation rate of point defects, defect mobilities and their reactions, but also accurate thermodynamic and kinetic properties of the system such as chemical free energy and mechanical properties of different phases, interfacial energies, and interface mobility. In general, such information does not exist in a form that is readily used by such models. Therefore, this work develops and presents a qualitative phase-field model in order to demonstrate the ability to simulate microstructural evolution such as the accumulation of fission products and gas bubbles as well as internal stresses, and to study the effect of microstructures on thermal conductivity. All the results presented in this paper are qualitative because of both lack of accurate thermodynamic and kinetic properties and the simplifying assumptions used in the models. However, the work demonstrates the capability of the phase-field approach to provide important information such as the accumulation of fission products, three dimensional gas bubble morphology, the evolution of internal stresses and the effective thermal conductivity. With the development of accurate thermodynamic and kinetic data, and improvement of the model, quantitative results can be obtained in the future, which can be used as inputs for macro-scale fuel performance simulations such as FRAPCON.

Section snippets

Description of the phase-field model

The formation of gas bubbles is a direct consequence of the extremely low gas atom solubility in nuclear fuels and metals. However, the gas bubble nucleation and growth kinetics are strongly affected by a number of factors including the trapping of gas atoms by preexisting and evolving defects such as vacancies, dislocations and grain boundaries, mobility inhomogeneity as well as the generation rate of fission products. Fission products, depending on materials and nuclear reactions, will

Formation and growth of gas bubbles around defects

Trapping of vacancies and gas atoms by defects is a possible formation mechanism of voids and gas bubbles. To examine this hypothesis, the segregation of vacancies and gas atoms around two defects was studied: a shear dislocation loop and a vacancy cluster, both of which are often observed in fuels and cladding materials. In the simulations, a square dislocation loop with b = [1 0 1¯], n = [1 1 1] and an edge length L0 = 40r0; and a spherical vacancy cluster with average vacancy concentration cvac = 0.2

Conclusions

This work presented a phase-field approach to modeling the gas bubble evolution and calculating the effective thermal conductivity in a polycrystalline material with the microstructure obtained from the simulations. The dependence of bubble formation and growth kinetics on small vacancy clusters, dislocations and grain boundaries, the generation rates and mobility of fission products, and the evolution of effective thermal conductivity have been simulated. The results show that the gas bubble

Acknowledgments

This work was supported at Pacific Northwest National Laboratory and Los Alamos National Laboratory by the US Department of Energy. PNNL is operated for the US Department of Energy by Battelle Memorial Institute under Contract DE-AC06-76RLO 1830. S.Y. Hu wants to thank Dr Ken Geelhood and Dr Rick Kurtz at PNNL and Dr Xiang-Yang Liu at LANL for their helpful discussions.

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