Category Archives: Grain boundaries

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.

UO2_cross_section
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.

UO2_gas_bubbles
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.

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.

Simulated distribution of total dislocation density in an artificial polycrystal. Due to the reaction-diffusion modeling of dislocation density evolution and the presence of grain boundaries, dislocation pile-ups are formed along the grain boundaries and particularly at triple junctions.
Simulated distribution of total dislocation density in an artificial polycrystal. Due to the reaction-diffusion modeling of dislocation density evolution and the presence of grain boundaries, dislocation pile-ups are formed along the grain boundaries and particularly at triple junctions.

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.

Dislocation density distribution influenced by the presence of grain boundaries. The microstructure model is based on level sets, representing the individual grains. The extent of dislocation accumulation at grain boundaries can be controlled by the parameter w, as illustrated in images a-d.
Dislocation density distribution influenced by the presence of grain boundaries. The microstructure model is based on level sets, representing the individual grains. The extent of dislocation accumulation at grain boundaries can be controlled by the parameter w, as illustrated in images a-d.

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.

Dislocation pileup at a grain boundary
Schematic illustration of a dislocation pileup at a grain boundary. The individual dislocations glide along the same slip plane under an external resolved shear stress \tau_{rss}. The grain boundary blocks the first dislocation and the other (n-1) dislocations pileup behind it. The first dislocation will experience a stress due to the pileup corresponding to n\tau_{rss}. The pileup also gives rise to a backstress that interferes with the generation of additional dislocation loops. This is an important mechanism in work hardening.

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.

Microstructure of an ordinary construction steel (SS1312).
Microstructure of an ordinary construction steel (SS1312).

Impediment of grain boundary migration due to particle drag

Metallic materials used in engineering applications frequently contain some volume fraction of particles. These may be impurity particles or particles that are deliberately added to influence the microstructure behavior.

As grain boundaries migrate through the microstructure, driven by boundary curvature and/or stored energy gradients, any particles present will impede - or even prevent - the movement by exerting drag forces on the passing boundaries. Classically this effect is termed Zener drag or Zener pinning. This pinning pressure is usually written as

p=-z_{1}\gamma\frac{f_{\mathrm{V}}^{z_{2}}}{r_{\mathrm{p}}}

where the negative sign indicates a retarding pressure and where \gamma is the boundary energy while f_{\mathrm{V}} and r_{\mathrm{p}} are the volume fraction of particles and the average particle size, respectively. z_{1} and z_{2} are parameters.

In the original model by Zener (published by C.S. Smith in 1948), the formulation is based on the assumption of rigid grain boundaries between the particles, providing z_{i}=\left\{3/2,1\right\}. Later studies have tried to incorporate the tendency of the grain boundary to bow out between particles, arriving at other values of the parameters z_{1} and z_{2}.

A mesoscale RVE model of dynamic discontinuous recrystallization, considering particle drag, is established in Hallberg et al. (2014). The model is based on a 3D cellular automaton formulation. See illustration below.

Influence on recrystallization kinetics from a small volume fraction <img src='http://s0.wp.com/latex.php?latex=f_+%7B%5Cmathrm%7BV%7D%7D&bg=ffffff&fg=000000&s=0' alt='f_ {\mathrm{V}}' title='f_ {\mathrm{V}}' class='latex' /> of homogeneously distributed particles of size <img src='http://s0.wp.com/latex.php?latex=r_%7B%5Cmathrm%7Bp%7D%7D&bg=ffffff&fg=000000&s=0' alt='r_{\mathrm{p}}' title='r_{\mathrm{p}}' class='latex' /> (in microns). The recystallized volume fraction <img src='http://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=000000&s=0' alt='X' title='X' class='latex' /> is shown on the vertical axis and time <img src='http://s0.wp.com/latex.php?latex=t&bg=ffffff&fg=000000&s=0' alt='t' title='t' class='latex' /> on the horizontal axis. Results are shown at two different temperatures. It is obvious that an increased presence of particles retard the progression of recrystallization. Simulation results from a 3D cellular automata model.
Influence on recrystallization kinetics from a small volume fraction f_ {\mathrm{V}} of homogeneously distributed particles of size r_{\mathrm{p}} (in microns). The recystallized volume fraction X is shown on the vertical axis and time t on the horizontal axis. Results are shown at two different temperatures. It is obvious that an increased presence of particles retard the progression of recrystallization. Simulation results from a 3D cellular automata model.