# 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.

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.

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.

# New paper on grain boundary texture stability in UO2

A paper was recently published in Journal of Nuclear Materials on the stability of certain grain boundary configurations in UO2 (Uranium Dioxide) during grain growth. The paper is titled Stability of grain boundary texture during isothermal grain growth in UO2 considering anisotropic grain boundary properties, and can be found here or at the at Science Direct.

# Nucleation of Recrystallization

As a metallic material is plastically deformed, energy will be accumulated in the material through generation and rearrangement of defects, mainly dislocations. As dislocation networks develop they will tend to form entanglements that subsequently form dislocation cells which, in turn, eventually leads to the formation of subgrains in the grain interiors. These subgrains will be the birthplaces of the recrystallization nuclei. In order for the subgrains to become viable nuclei, however, two criteria need to be met locally in the microstructure. The first is a kinematic criterion requiring mobile high-angle grain boundaries to be formed by the nucleation event. The second is a thermodynamic criterion requiring a stored energy gradient to be present across the interface that is sufficient to provide enough positive driving pressure for grain growth to occur. These criteria are usually found to be met at grain boundary triple junctions and along grain boundaries. Such sites provide enough lattice curvature and sufficient stored energy differences for providing possible nucleation sites. Nucleation may also, but less common, take place in the grain interiors, for example along shear bands and near particle inclusions. The latter is often referred to as particle-stimulated nucleation. As nucleation preferentially takes place along pre-existing grain boundaries, it is common to observe so-called necklace patterns of recrystallized material along the boundaries.

The process of subgrain formation is in mesoscale models usually captured by considering an incubation time $t_{\mathrm{c}}$ or a threshold dislocation density $\rho_{\mathrm{c}}$, i.e. a threshold stored energy, that needs to be reached before nucleation is initiated. In macroscale models, the same process is usually formulated in terms of a critical strain $\varepsilon_{\mathrm{c}}$ that has to be achieved in order for nucleation to take place. Also the notion of a critical stress $\sigma_{\mathrm{c}}$ has been proposed to identify the onset of recrystallization.

During the formation of the subgrains, recovery processes very rapidly reduce the dislocation content in the subgrain interiors by annihilation of dislocations and by accumulation of dislocations at the subgrain boundaries. This increases the subgrain misorientation with respect to neighboring regions and eventually provide high-angle, mobile, boundaries. The same processes also provide subgrains with very low internal stored energy.

The crystallographic orientation of the nuclei will influence both the progression of recrystallization and also the evolution of any recrystallization texture in the material. Recrystallization nuclei are products of the cold worked parent material but they do not necessarily inherit their orientation from the parent grains although it is not uncommon that clear evidence of the texture present in the initial, cold worked, microstructure can be observed also in the recrystallized material. High-angle boundaries with respect to the surrounding material are also required in order to permit growth of the nuclei, causing differences to develop between the initial and recrystallized textures. It can be noted that the development of recrystallization texture is usually attributed to either orientated nucleation or oriented growth or possibly a combination of these processes.

Having defined the initiation of recrystallization, the next question is at what frequency new nucleation events take place. This issue has been approached in a number of ways in different studies. The two main trends is to consider either of  site-saturated nucleation where all nuclei are assumed to be present at the start of the simulation and no new nuclei are added over time or, alternatively, continuous nucleation where new nuclei are continuously added according to some expression of the rate of nucleation. In the latter scenario, both constant and non-constant rates of nucleation have been considered.

Although constant nucleation rates are often assumed, there are experimental studies indicating nucleation to be non-constant and continuous process. However, it is inherently difficult to experimentally determine the rate of nucleation as it is not easy to define through experimental observations if a region in the microstructure is a subgrain, a nucleus or an expanding grain and the exact time at which one changes into the other.

# New paper on recrystallization modeling

A new paper, titled ", was recently published in Modelling and Simulation in Materials Science and Engineering and can be found at the publisher's site following this link.

# Production of very fine grained materials

Since recrystallization is driven by reduction of stored energy, the amount of cold work influences the grain size evolution. Materials with grain sizes down to the nanometer scale can be obtained by exposing the material to severe plastic deformation (SPD). Such SPD processes include Equal Channel Angular Pressing/Extrusion (ECAP/ECAE), Asymmetric Rolling (ASR), Multi-Directional Forging (MDF), High-Pressure Torsion (HPT) and Twist Extrusion (TE). Some of these processes are discussed further below.

During ECAP, studied in Hallberg et al. (2010), the material specimen is pressed through a channel die as illustrated below.

The amount of effective plastic deformation, denoted by $\displaystyle \varepsilon_{\mathrm{eff}}^{\mathrm{p}}$, that is imposed onto the specimen in each pass can be estimated from knowledge of the die geometry according to

$\displaystyle \varepsilon_{\mathrm{eff}}^{\mathrm{p}}=\frac{N_{\mathrm{pass}}}{\sqrt{3}}\left[2\mathrm{cot}\left(\frac{\Phi}{2}+\frac{\Psi}{2}\right)+\Psi\mathrm{cosec}\left(\frac{\Phi}{2}+\frac{\Psi}{2}\right)\right]$

Varying the channel geometry thus allows control of the amount of plastic deformation that is exerted onto the specimen in each pass through the die. If the specimen is rotated between each pass through the die, different standardized processing routes can be obtained.

Another SPD process is asymmetric rolling, discussed in Hallberg (2013), where a conventional rolling process can be made asymmetric by different methods.

The asymmetry of the process can be induced by having different radii $r$ of the rolls, by different roller velocities, i.e. $\omega_{1}\ne\omega_{2}$ or by different friction/lubrication conditions at each side of the sheet. The asymmetry increases the shear deformation of the rolled sheet and hence the total amount of effective plastic deformation.

# Dislocation and grain boundary interaction

Mesoscale models of microstructure evolution allow studying of heterogeneous dislocation density distributions and related gradient effects. This is closely connected to the well-renowned Hall-Petch effect, stating a proportionality between the yield stress $\sigma_{\mathrm{y}}$ and the inverse of the square-root of the average grain size $d$ according to

$\sigma_{\mathrm{y}}\propto\frac{1}{\sqrt{d}}$

The interaction between dislocation motion and grain boundaries can be modeled by different approaches. Some examples are given below.

In Hallberg and Ristinmaa (2013) (also discussed in this conference presentation), a hybrid finite difference/cellular automaton model is established where dislocation density gradient are modeled in a reaction-diffusion system. This approach results in the expected Hall-Petch behavior of the macroscopic flow stress – without including an explicit dependence on the grain size – in addition to providing a physically sound dislocation density distribution, indicated in the figure below.

By this modeling approach, the dislocation density will be concentrated at grain boundaries and particularly at triple junctions, directly providing the sites for nucleation of recrystallization. This is in contrast to the common modeling approach where nuclei are placed manually at appropriate sites in the microstructure.

Another approach to modeling of dislocation and grain boundary interaction is taken in Hallberg (2013) where a level set formulation is used to model polycrystal grain structures.

# Grain boundaries

Many important aspects of the macroscopic material behavior in metals are controlled by the microlevel grain structure. Strength and ductility depends largely on the size and distribution of the grains. Especially the presence of grain boundaries plays an important role, not least in the macroscopic deformation hardening of the material. Grain boundaries pose obstacles to slip deformation by preventing dislocation motion, resulting in localized dislocation storage and heterogeneous deformation fields within the grains. This is closely related to the classical Hall-Petch relation, where the macroscopic flow stress is porportional to the inverse of the square root of the average grain size. More fine-grained materials will thus exhibit a higher flow stress than more coarse-grained materials.

The Hall-Petch effect will, however, tend to reverse as very small grains are considered. At a grain sizes of approximately 10 nm or below, other mechanisms such as grain boundary sliding usually cause a softening effect instead of the common Hall-Petch hardening.

Grain boundaries also contribute to the material behavior in many other aspects. They are often sites for initiation of defects such as cracks, they influence the material's susceptibility to chemical attack such as through corrosion and they influence the electrical conductivity of the material. In addition, migrating grain boundaries tend to interact with second-phase particles and the presence and arrangement of grain boundaries influence both the electrical conductivity and the diffusion properties of the material.

# Martensitic phase transformation and fracture

As the relatively ductile austenite phase is transformed in to harder and more brittle martensite in the vicinity stress-concentrations, the material conditions change and also the conditions for crack formation and propagation. Fatigue fracture can be considerably influenced by this kind of diffusionless phase transformation due to the higher fracture strength of martensite, compared to that of austenite. In Hallberg et al. (2012) the influence of martensite formation on fracture behavior and crack tip conditions is investigated, as illustrated below.

# Continuum scale modeling of phase transformation

Taking a continuum-mechanical perspective, the isothermal model in Hallberg et al. (2007) introduces the volume fraction of martensite as an internal variable. Along with a transformation condition, dependent on the state of deformation and on temperature, this allows the evolution of the martensitic phase to be traced. The presence of a transformation condition allows establishment of a transformation potential surface, much like the yield condition and yield surface found in plasticity theory. The transformation surface is illustrated in deviatoric stress space and in the meridian plane below.

Depending on which one is active, the yield and transformation conditions determine the response of the material. The relative influence of austenite and martensite on mechanical material properties is considered through a homogenization procedure, based on the phase fractions.

The above isothermal model is further elaborated in Hallberg et. al (2010b), where full thermo-mechanical coupling is considered. These models are suitable for large-scale simulations of metal forming processes involving materials exposed to martensitic phase transformation. The application to sheet metal forming is illustrated below by images from simulations of a deep-drawing process.

# Phase transformations

Phase transformations in metallic materials have a major impact on vital engineering aspects of the material behavior such as ductility, strength and formability. Some phase transformations, such as the formation of pearlite and bainite, occur through diffusion-based processes where the constituents in the microstructure are redistributed. Being based on diffusion, these kinds of phase transformations tend to be relatively slow. On the other hand, phase transformations can also proceed by pure displacements in the crystal lattice structure. This is typical for the very rapid and diffusionless formation of martensite in austenitic steels.

Specifically, the latter kind of materials, undergoing microstructural changes in terms of austenite-martensite transformation, have in recent years gained increasing attention in relation to shape memory alloys (SMAs) and alloys exhibiting transformation-induced plasticity (TRIP steels).

Description of phase transformations is further involved due to the strong temperature-dependence of the process. Combined with significant differences in mechanical properties between the phases and the volumetric deformations accompanying e.g. martensitic phase transformations, strongly thermo-mechanically coupled phenomena arise.

The presence of martensite also changes the fracture behavior of a material since the martensite is considerably harder than the more ductile austenite parent phase. This influences e.g. initiation and propagation of crack and may become detrimental to metal forming and forging processes.