Category Archives: Numerical modeling

Uranium Dioxide (UO2)

Different kinds of oxide fuels are used in nuclear power plants, most commonly used – and for the longest time – is Uranium Dioxide (UO2). A solid understanding of the fuel performance is central to control in-service performance, properties and degradation of the fuel as well as for safe handling of it.

As fuel material, UO2 is usually sintered into small cylindrical pellets, measuring about 10 mm in diameter and similar in length. These cylinders are stacked in fuel rods inside a Zircalloy cladding. The small radial gap between the pellets and the cladding is usually filled with a pressurized gas such as Helium. A number of such fuel rods are mounted in fuel assemblies together with control rods with a high capacity for neutron absorption. The fuel assemblies are then used as heat source in fission power plants.

As the fuel pellets are “burnt” in the reactor, fission processes take place and the degree of irradiation of the fuel pellets is usual measured in terms of the “burnup”, that is the fraction of the initial material that has undergone fission.

Under common in-service conditions, the core of the fuel pellets can be maintained at a temperature of 2000K while the pellet surface is at around 800K (the melting point of UO2 is approximately 3140K). The outside temperature is maintained by a constant flow of coolant through the fuel assemblies. Under such extreme thermal gradient conditions, the fuel material undergoes drastic changes. These changes have a strong influence on fuel performance and properties such as the thermal conductivity and structural rigidity. The grain structure will have different morphologies in different regions. This is schematically illustrated below.

Cross-section of a UO2 fuel pellet, showing characteristic microstructure variations.

The extreme thermal gradients will also cause so-called “hourglassing” of the fuel pellets along with cracking – both radially and circumferentially – due to thermally induced stresses and swelling due to solid fission products.

By the release of fission gasses (e.g., Xe, Kr, I and Cs), gas-filled pores or voids will form in the microstructure. The gas bubbles form in the grain interiors and migrate by diffusion to coalesce along the grain boundaries.

Formation of gas-filled bubbles, pores and voids in the grain structure of UO2 during irradiation.

The presence of gas bubbles can cause swelling and cracking of the fuel pellet and the gas can also be released inside the Zircalloy cladding, lowering the heat conduction capacity of the Helium that surrounds the pellets. In either case, the integrity of the Zircalloy cladding is compromised.

In Hallberg & Zhu (2015), the stability of grain boundary texture under grain growth in UO2 is studied through level set modeling, taking anisotropic grain boundary properties into account. The characteristic morphologies of faceted voids in UO2, due to heterogeneous interface energies, is studied in Zhu & Hallberg (2015) by 3D phase field simulations.

Mesoscale modeling of recrystallization

Several numerical algorithms have been employed in recrystallization modeling. Some techniques, suitable for mesoscale modeling of recrystallization, are summarized below. A more extensive discussion is given in Hallberg (2011).

Monte Carlo Potts models

Historically, Monte Carlo Potts (MCP) algorithms have been used to simulate recrystallization - mainly static recrystallization - in both 2D and 3D on fixed computational grids. The system energy is minimized through probabilistic changes to the state variables, defined at the grid points. Physical time is not available in MCP models where “Monte Carlo steps” are used as a measure of time. This makes comparison between MCP results and experimental results cumbersome.

 MCP models are relatively computationally efficient and can be used in both 2D and 3D. The method is also suited for computer parallelization

Cellular automata

An alternative method is given by cellular automata (CA) which have been employed frequently in studies on both static and dynamic recrystallization. As with MCP models, CA models are also usually defined on fixed grids, but use physical cell state switching conditions, based on recrystallization kinetics defined in physical time. The cell state switching can be performed as either deterministic or probabilistic. Cellular automata are attractive since high spatial and temporal resolution can be achieved at the grain-scale, cf. Hallberg et al. (2010c) and Hallberg et al. (2014).

Examples from 3D cellular automaton simulations of dynamic recrystallization in copper at different temperatures.
Examples from 3D cellular automaton simulations of dynamic recrystallization in copper at different temperatures.

The curvature of interfaces is an important aspect of grain boundary migration kinetics but being based on discrete grids with no direct representation of interface curvature, CA algorithms have shortcomings in this respect. The choice of grid type (square, hexagonal etc.) also influences the grain growth kinetics and may be detrimental to the simulation results. As with MCP, CA are computationally efficient and 3D implementations are straight-forward. CA methods also scale well when subject to computer parallelization.

Level set models

The level set formulation was introduced as a numerical tool to trace the spatial and temporal evolution of single interfaces. The method was later also extended to consider interfaces with multiple junctions. Standard level set formulations do not correctly capture the interaction between multiple grains, occurring for example at grain boundary triple junctions, and corrections have to be implemented to remedy this shortcoming. Level set modeling of recrystallization is discussed in Hallberg 2013 and of grain growth in Hallberg 2014.

A level set representation of a polycrystal with the finite element mesh discretization shown to the right.
A level set representation of a polycrystal with the finite element mesh discretization shown to the right.

Front tracking or vertex methods

Interface migration can also be described by front tracking or vertex models where the migration kinetics of grain boundary triple junctions are considered. The topology of the grain structure is represented by nodes placed at the triple junctions, interconnected by grain boundary line segments. The representation of curvature, however, comes with additional computational cost as intermediate nodes have to be introduced between the triple junctions. In addition, the extension of the method to 3D is not easily realized since it requires surface tesselations to be performed. Also, topological changes to the grain structure require dealing with different transformation conditions. It can also be noted that usual formulations of front tracking models do not hold information related to the grain interiors.

A combined crystal plasticity and vertex model is established in Mellbin et al. 2015, to allow concurrent modeling of large deformations, texture development and recrystallization.

Phase field models

The Phase-field method (PF) has received significant interest in recent years in simulations of a broad spectrum of physical processes, including recrystallization. In PF models of recrystallization, the grain microstructure is described by phase field variables. These are functions that are continuous in space and a distinction is made between conserved and non-conserved variables. A conserved variable is typically a measure of the local composition whereas a non-conserved variable contains information on the local structure and could represent for example the crystallographic orientation. Within a single grain, a phase field variable maintains a nearly constant value that corresponds to the properties of that grain. Grain boundaries are represented as interfaces where the value of the phase field variable gradually varies between the values in the neighboring grains on opposing sides of the grain boundary. Grain boundaries are hence described as diffuse transition regions of the phase field variables. The computational effort in treating the rapidly changing fields across diffuse interfaces can be considerable and the formulation of the energy densities to capture physical microstructure features is not trivial. In addition, topological changes such as nucleation of new grains are not easily handled.