Tag 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.

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.

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Paper published on evolution of grain boundary character distribution during grain growth

A new paper, titled "Influence of anisotropic grain boundary properties on the evolution of grain boundary character distribution during grain growth - A 2D level set study", was recently published in Modelling and Simulation in Materials Science and Engineering and can be found at the publisher's site following this link or here.

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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 . The grain boundary blocks the first dislocation and the other dislocations pileup behind it. The first dislocation will experience a stress due to the pileup corresponding to . 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 of homogeneously distributed particles of size (in microns). The recystallized volume fraction is shown on the vertical axis and time 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.

 

Recrystallization and grain growth

Recrystallization (RX) is one of the main mechanism to control the evolution of grain microstructures. RX is generally accepted to be defined as the formation of a new grain structure in a cold-worked material and occurs through the formation and migration of high-angle boundaries. The grain boundary migrations are primarily driven by stored energy reduction and minimization of grain boundary surface energy.

Dynamic discontinuous recrystallization (DDRX) in pure Copper, modeled by a cellular automata-based representative volume element. Compression at different temperatures is shown. Note that at the higher temperatures of 975K and 1075K grain coarsening, rather than grain refinement, takes place.
Dynamic discontinuous recrystallization (DDRX) in pure Copper, modeled by a cellular automata-based representative volume element. Compression at different temperatures is shown. Note that at the higher temperatures of 975K and 1075K grain coarsening, rather than grain refinement, takes place.

As a metallic materials is deformed through plastic slip, energy will be accumulated in the material. This energy is to a large extent expended as heat while the remainder is stored in the material microstructure through the generation and redistribution of imperfections, mainly dislocations. By this process, the material becomes increasingly thermodynamically unstable. During subsequent annealing of the material, reduction of the stored energy can take place through relatively slow recovery or by more rapid static recrystallization (SRX). While the recovery proceeds as a continuous process, SRX is discontinuous. During thermomechanical processing of the material, i.e. when the material is exposed to plastic deformation at elevated temperatures, stored energy generation through dislocation accumulation and stored energy reduction through nucleation of new grains work in parallel. This process is commonly labeled dynamic recrystallization (DRX). The latter process of DRX may be further subdivided into a relatively slow continuous dynamic recrystallization (CDRX) or a more rapidly progressing discontinuous dynamic recrystallization (DDRX).

Schematic illustration of microstructure evolution due to discontinuous dynamic recrystallization (DDRX), proceeding by nucleation and growth of new grains. In contrast, no distinct nucleation stage is observable during continuous dynamic recrystallization (CDRX).
Schematic illustration of microstructure evolution due to discontinuous dynamic recrystallization (DDRX), proceeding by nucleation and growth of new grains. In contrast, no distinct nucleation stage is observable during continuous dynamic recrystallization (CDRX).

In materials of high stacking-fault energy, such as aluminum, dynamic recovery is significant and recrystallization occurs mainly by CDRX. In this case, subgrains with low-angle boundaries are formed from dislocation networks. With progressing plastic deformation, misorientation is increased until enough energy is achieved and the initially mobile subgrain walls have become immobilized, allowing new grains to be separated by subgrain growth. In materials of low stacking-fault energy, such as copper, dynamic recovery processes such as cross slip and climb are less influential and the recrystallization is dominated by DDRX during which new grains are nucleated as regions of low dislocation density grow to consume more dislocation-dense surroundings. RX nuclei are commonly accepted to form from subgrains and DDRX will be most significant in the microstructure regions having the highest dislocation density, primarily at grain boundary triple junctions, secondly along grain boundaries and at inclusions and with lesser probability in the grain interiors.

Processing conditions, such as temperature and strain rate, as well as material purity will influence the recrystallization process. This allows some control to be exerted over the resulting microstructure. Simulation models can provide the means for design and processing of materials through recrystallization.

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.

Animation showing a 2D level set simulation of Dynamic Discontinuous Recrystallization (DDRX)

A simple 2D simulation of dynamic discontinuous recrystallization (DDRX) in pure Cu at an elevated temperature. The simulation is based on level sets in a finite element setting. Adaptive remeshing is performed in each step. The animation speed is increased compared to actual time.