Stability of grain boundary texture during isothermal grain growth in UO2 considering anisotropic grain boundary properties

Journal of Nuclear Materials 465 (2015) 664e673

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Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat

Stability of grain boundary texture during isothermal grain growth in UO2 considering anisotropic grain boundary properties Håkan Hallberg*, Yaochan Zhu Division of Solid Mechanics, Lund University, Box 118, S-221 00 Lund, Sweden

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 April 2015 Received in revised form 16 June 2015 Accepted 24 June 2015 Available online 2 July 2015

In the present study, mesoscale simulations of grain growth in UO2 are performed using a 2D level set representation of the polycrystal grain boundary network, employed in a finite element setting. Anisotropic grain boundary properties are considered by evaluating how grain boundary energy and mobility varies with local grain boundary character. This is achieved by considering different formulations of the anisotropy of grain boundary properties, for example in terms of coincidence site lattice (CSL) correspondence. Such modeling approaches allow tracing of the stability of a number of characteristic low-S boundaries in the material during grain growth. The present simulations indicate that anisotropic grain boundary properties have negligible influence on the grain growth rate. However, considering the evolution of grain boundary character distribution and the grain size distribution, it is found that neglecting anisotropic boundary properties will strongly bias predictions obtained from numerical simulations. © 2015 Elsevier B.V. All rights reserved.

Keywords: Uranium dioxide Grain growth Simulation Modeling Level set Grain boundary energy Coincidence site lattice

1. Introduction The microstructure of uranium dioxide (UO2) decisively determines the macroscopic thermo-mechanical properties of the material and also controls the processes whereby its microstructure evolves and eventually degrades. Grain growth occurs in the material under high-temperature conditions through migration of mobile, generally high-angle, grain boundaries. Grain boundary properties such as boundary energy and mobility tend to vary with the local grain boundary configuration, giving rise to anisotropic grain growth kinetics. In addition, grain boundary migration will be impeded by any presence of pores or particles. Further, as grain boundaries frequently constitute the origin of material damage and processes of creep, segregation of impurities and embrittlement it is evident that knowledge of the grain boundary structure is pivotal for tracing the state of the material. The present study focuses on the stability of grain boundary character distribution during grain growth in UO2 under sintering conditions when anisotropic grain boundary properties are considered. A full characterization of an individual grain boundary requires knowledge of three parameters in order to describe the

* Corresponding author. E-mail address: [email protected] (H. Hallberg). http://dx.doi.org/10.1016/j.jnucmat.2015.06.052 0022-3115/© 2015 Elsevier B.V. All rights reserved.

crystallographic misorientation across the interface and two additional parameters to define the inclination of the boundary plane [1e3]. Although both grain boundary energy and mobility are usually observed to vary with the full set of five parameters, the dependence of the parameters on grain boundary character is complex and no general theory exists to describe it. Experimental observations on grain boundary energy and mobility are usually made on limited ranges of misorientation and for chosen subsets of boundary configurations, such as symmetrical tilt and twist boundaries with well-defined boundary planes. Recognizing the complexity of these relations, the concept of coincidence site lattice (CSL) boundaries provides a simplifying approach. The CSL concept is based on the identification of some degree of geometrical agreement between the two adjacent crystal lattices, on opposite sides of the grain boundary. Although based on geometrical considerations, rather than on experimental observations, the CSL concept remains influential when it comes to describe grain boundary character and related grain boundary properties [3]. For example, CSL boundaries are often observed to have low energy and high mobility character [4]. The CSL concept also remains important in the field of grain boundary engineering [5]. Regarding UO2, it can also be noted that an increased diffusion rate for some CSL boundaries has been reported [6,7]. The CSL description of grain boundary character is adopted in the present work.

H. Hallberg, Y. Zhu / Journal of Nuclear Materials 465 (2015) 664e673

When it comes to mesoscale modeling of grain growth, different approaches can be taken as discussed in Ref. [8]. As examples, grain growth in UO2 is modeled by empirically-based closed-form expressions in Refs. [9e11]. Turning to numerical models, grain growth and pore migration in UO2 is studied using a 2D Monte Carlo Potts (MCP) model in Ref. [12] while a 2D phase field (PF) model is employed in Ref. [13] to study grain growth in porous UO2, assuming isotropic grain boundary properties. Zener-like pinning of grain boundaries by pores was found in Ref. [14] by using MCP simulations of grain growth in UO2 and grain growth in a porous material was also simulated by a 3D MCP model in Ref. [15]. The inhibition of grain boundary migration due to pores as a type of Zener pinning is also discussed in Ref. [16]. Further, 2D MCP models are used to trace the evolution of CSL boundaries during grain growth e although not in UO2 e in Refs. [17,18]. In passing it can also be noted that some model-based studies on UO2 focus on void growth while keeping the grain boundaries fixed. This is done analytically in Ref. [19] and by phase field modeling in Refs. [20,21]. The level set method was introduced in Ref. [22] and provides a method to trace the evolution of interfaces in different physical settings. Level sets have been used for mesoscale modeling of polycrystals in several studies. Recrystallization in isotropic systems is modeled by level sets in Refs. [23e26]. Grain growth with isotropic grain boundary properties is approached by the same method in Refs. [27,28]. Grain growth with a ReadeShockley description of grain boundary energy is considered in the level set model employed in Ref. [29]. Particle pinning of grain boundaries, modeled by level sets, is considered in Ref. [30]. The present level set formulation is adopted from Refs. [31], where the focus lies on dynamic recrystallization. A similar model was also adopted in Ref. [32] to trace the evolution of grain boundary character distribution during grain growth in polycrystals with cubic structure. Most mesoscale models of grain growth adopt the simplifying assumption of isotropic grain boundary properties in terms of boundary energy and mobility. Mesoscale models of grain growth where anisotropic grain boundary properties are taken into account are significantly more scarce, and particularly so related to UO2. If anisotropic energy is considered, it is usually done by adopting a ReadeShockley model for low-angle boundaries where the interface energy depends solely on the scalar misorientation across the interface. The common approach in these models is to assume a constant energy for all high-angle grain boundaries. Anisotropic grain boundary energy and mobility is an important aspect of the kinetics of microstructure evolution [3,33]. The grain boundary character will also influence the diffusion of fission products in the material [34]. It can be noted that grain structures in UO2 with a significant fraction of special boundaries e in terms of CSL boundaries e have been reported [35]. Such microstructure variations suggest anisotropic grain boundary properties to be an influential factor on the microstructure evolution in this material. This paper is structured into five sections. Beginning with a note on modeling of grain growth kinetics in Section 2, subsections provide details on the formulation of anisotropic grain boundary properties based on CSL correspondence. Section 3 contains the level set formulation of grain growth. Mesoscale simulations of grain growth in UO2 are performed in Section 4 where the results are also shown and discussed. Finally, some concluding remarks closes the paper in Section 5. 2. Modeling grain growth kinetics The local migration velocity of a grain boundary can be written as

v ¼ mp

665

(1)

where m and p is the grain boundary mobility and the driving pressure, acting on the grain boundary, respectively. Focusing the study on grain growth, the driving pressure due to a purely curvature-driven grain boundary motion can be formulated as

p ¼ kg

(2)

where the local grain boundary curvature k and the grain boundary energy g were introduced. In the present study, anisotropic grain boundary properties will be introduced in terms of the grain boundary energy g and the grain boundary mobility m. This is detailed in the following subsections.

2.1. CSL classification of grain boundary character In a polycrystalline aggregate it can be assumed that for some boundaries a certain degree of geometrical correspondence can be found between the atomic arrangements on opposite sides of the boundary. The notion of a Coincidence Site Lattice (CSL) is based on identification of the coinciding lattice positions in the two neighboring crystals. Boundaries where such geometrical agreement are found, have in several studies been discovered to posses “special” properties, such as low energy and high mobility. The degree of CSL correspondence is usually indicated by an integer number S, where 1/S is the fraction of coinciding lattice positions. Crystal orientations need to be defined by three parameters which in the present study are taken as the three EulereBunge angles (41, F, 42). Due to the many possible configurations of these parameter sets in adjacent crystals, it is unlikely to encounter ideal CSL boundaries. To address this, an acceptance criterion was introduced by Brandon in Refs. [36], defining a range of misorientation angles DqS related to an individual CSL configuration. The Brandon criterion is given by,

DqS ¼

qS S1=2

(3)

where qS is a parameter that determines the range of misorientation angles that are assumed to belong to each CSL. In the present work, the common choice of qS ¼ 15 is made. Based on the Euler angles, the orientation of each crystal can be defined by the orthogonal rotation matrix g(41, F, 42) that rotates the sample reference frame into the crystal reference frame. The misorientation between two crystals with rotations gi and gj can subsequently be evaluated as Dg ij ¼ g j g Ti . In a cubic system, there are 24 symmetrically equivalent variants of each orientation g. In evaluation of the misorientation, the common approach is to consider the set of variants that provide the minimum misorientation. By this approach, the scalar misorientation is found by performing the minimization.


 


1 q ¼ min

acos ftrðOs DgÞ  1g

2 O 2G s

(4)

c

where indices i and j are skipped for convenience and where Os are the operators in the cubic symmetry group G c . The trace of a tensorial quantity is denoted by tr($) and the absolute value is taken of the argument in Eq. (4) since a negative sign is merely an indication of the rotation axis pointing in the opposite direction. Due to the minimization in Eq. (4), the maximum misorientation that will be encountered in a cubic polycrystal is 62.8 , as shown in Ref. [37].

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To check whether a particular grain boundary is of a certain CSL type, all symmetry configurations related to the boundary need to be considered. This includes all symmetry variants of the two neighboring grains as well as the switching symmetry Dgij ¼ Dgji. Misorientations found by this approach, where it is also verified that the rotation axis reside in the fundamental zone, are usually termed the disorientation following [38]. The disorientation is used in the present study to identify CSL boundaries according to Eq. (3).

2.2. Variation of grain boundary properties with local grain boundary character If anisotropic grain boundary energy, denoted by g in the present study, is considered in mesoscale models of polycrystals, it is usually done by implementing the classical ReadeShockley relation.

gRS ðqÞ ¼

8 > :

   q q 1  ln m qm qm

gm

if

q < qm

if

q  qm

(5)

where qm is the angle that differentiates between low- and highangle boundaries. The usual choice of qm is adopted here. In addition, gm is the grain boundary energy for high-angle boundaries. According to Eq. (5), a gradually increasing energy is modeled for low-angle boundaries whereas a constant energy is assumed for all high-angle boundaries. The ReadeShockley relation is based on that low-angle boundaries can viewed as being constructed by dislocation planes, inserted between the adjoining crystals. The misorientation of low-angle boundaries will be proportional to a multiple of the Burgers vector b. By similar geometrical considerations also highangle CSL boundaries can be constructed by insertion of additional dislocation planes with a spacing proportional to b/S, cf [39]. Following this reasoning, the variation in boundary energy due to any CSL correspondence can be added to the ReadeShockley relation in Eq. (5). The variation of grain boundary energy with CSL type is provided by

8     ~m Dq Dq > < g 1 1  ln DqS DqS S gCSL ðqÞ ¼ > : 0

if

Dq < DqS

if

Dq  DqS (6)

In Eq. (6), the deviation from ideal CSL configurations is ~ m controls the magnitude described by Dq and the parameter g of the reduction in grain boundary energy for certain CSL type. For any disorientation, the variation in boundary energy with local grain boundary character is provided by Eq. (5) and Eq. (6) as

g ¼ gRS þ gCSL

   4  q fq ðqÞ ¼ 1  exp  5 qm

(8)

where qm, like in Eq. (5), is the angle that separate low- and highangle boundaries. In adopting this simplifying description of the anisotropy of grain boundary mobility, it can be noted that a number of studies indicate that anisotropy of the grain boundary energy can be expected to influence microstructure evolution to a larger extent than anisotropy of the grain boundary mobility [42e44]. One important aspect of the microstructure evolution in UO2 is the diffusion of oxygen and the creation of voids or pores. Grain boundaries are central to this process as the diffusion is up to five orders of magnitude faster along grain boundaries than in the bulk material [45]. Complex pore structures form along the grain boundaries and significantly alter the material properties, for example by reducing the thermal conductivity [46,47]. It has also been observed that the morphology of the voids will vary, depending on the local grain boundary character [48,49]. Very high pore fractions, up to 70%, at the grain boundaries have been reported [50]. Any presence of pores will impede grain boundary motion and retard the grain growth rate through a pinning effect [14,16]. In the most general case, the influence of pores on grain boundary mobility is described by a non-linear function fp that may depend on temperature, local grain boundary texture as well as size and morphology of the pores. Such complex dependencies are, however, beyond the scope of the present study where the stability grain boundary texture under sintering conditions, rather than full grain growth kinetics under in-service conditions, is in focus. Different mechanisms for the evolution of porosity are usually identified as being active in UO2 when different pore sizes are considered. For small pores e often referred to as “bubbles” e surface diffusion dominates whereas medium-sized and larger voids evolve by volume diffusion and processes of condensation and vaporization, respectively. As discussed in Refs. [51,52], pores with sizes below approximately 0.1 mm can be expected to move along with migrating grain boundaries and effectively result in a maintained retardation of the grain boundary velocity. In the present mesoscale model, grain sizes in the order of several microns are considered, and voids of sub-micron size are thus not resolved. The influence of porosity is instead included in the present model by a constant value of fp, assuming a homogeneously distributed and unchanged pore content. This results in a constant and uniform retardation of grain boundary migration. This approach is in line with the classical models of pore mitigated grain boundary migration in UO2, cf. [51e53]. The value of fp will be determined by considering experimental grain growth data later on. Combining the different contributions to the grain boundary mobility provides

mðqÞ ¼ mm fq ðqÞfp

(9)

(7)

also considered in Refs. [18,40]. The mobility of grain boundaries can also be expected to depend on the local grain boundary character. This dependence is, however, even more involved than for the grain boundary energy and no coherent model has been presented to describe it. Following [41], the essential characteristics of a low mobility for low-angle boundaries and a high mobility for high-angle boundaries is considered in the present formulation by adopting a disorientation dependence of the grain boundary mobility according to

where mm is the mobility of general high-angle grain boundaries in the material. Grain boundary mobility is commonly considered to be a diffusion-controlled process and following the Turnbull estimate [54,55], the mobility factor mm in eq. (9) can be written as

mm ¼ b

vm Db RTdb

(10)

where Db is the coefficient of grain boundary diffusion, db the grain boundary width and T the absolute temperature. Considering isothermal conditions, the temperature will be held constant in the

H. Hallberg, Y. Zhu / Journal of Nuclear Materials 465 (2015) 664e673

present study. The scaling parameter b 2 ]0, 1] is introduced since the Turnbull expression provides an upper limit estimate of the grain boundary mobility as diffusion along, rather than across, the boundary [31]. Through the model outlined above, anisotropic grain boundary properties are currently assumed to depend only on the local disorientation between neighboring grains. A full description of a grain boundary configuration should ideally consider not only the three orientation parameters for each crystal, but also two parameters to define the inclination of the boundary plane. However, according to [18], the influence of boundary plane inclination would be minor. In addition, several authors note that evolution of grain boundary texture can be expected also when adopting a disorientation-based anisotropy of grain boundary properties [17,18,42,56e58].

As a point of departure, the level set function f(x, t) is defined on a domain U, where x are the spatial coordinates and t the time. The zero-level contour where f ¼ 0 is taken to represent the spatial discontinuity G, i.e. the interface. Further, the level set function is defined as a distance function, for any point x representing the distance d(x, t, G) to the interface G at a certain time. In addition, a sign convention is also adopted whereby it holds that f > 0 inside G, f < 0 outside of the interface and f ¼ 0 at the interface. These preliminaries provide



fðx; tÞ ¼ dðx; t; GÞ; x2U G ¼ fx2U; fðxÞ ¼ 0g

In the previous section, a general formulation of grain boundary properties was established. Considering the particular case of uranium dioxide, material parameters are summarized in Table 1. In addition, the scaling parameter b in Eq. (10) is chosen as 0.1 which provides a grain boundary diffusivity which is in accord with the values considered in Ref. [13]. By considering grain growth data on UO2 in Refs. [59e61], the impediment of grain boundary mobility due to porosity is introduced by setting fp ¼ 0.012, cf. Eq. (9). The variation of grain boundary energy with boundary character is in the present work taken from Ref. [35] where atomistic modeling of UO2 was performed. It is noted in Ref. [35], that among the CSL boundaries, low-S boundaries are particularly frequent in UO2 and boundary energy data is provided for high-angle boundaries in general and for S3 and S5 boundaries in particular. These types represent frequently occurring boundary configurations together with general high-angle S1 boundaries. The current study is focused on these three boundary types. In Ref. [35] a grain boundary energy of 1.86 J/m2 is reported for general high-angle S1 grain boundaries while 1.04 and 1.58 J/m2 are found for S3 and S5 tilt boundaries, respectively. In relation to the value for general high-angle boundaries, this means that the energy drops by 44% for the S3 configuration and by 15% in the S5 case. It is assumed that these relative magnitudes continue to hold also for the isothermal case currently considered. From the above relations, and using Eqs. (7) and (9), grain boundary energy and mobility will vary with the local disorientation as shown in Fig. 1. 3. Level set modeling of grain growth The level set method was introduced in Ref. [22] and has found wide use to trace the evolution of interfaces in many different physical settings. The present level set implementation is based on that in Ref. [31] and similar formulations can also be found in Refs. [29,32,65]. The level set framework employed in the present study is briefly summarized in this section for completeness and in order to identify components that are of importance. Table 1 Material parameters entering the present model. The values are representative for UO2 at a temperature of 1700  C. Parameter

Value

Description

Source

Db

6.5  1013 m2/s 1 nm 0.45 J/m2 2.45  105 m3/mol

Grain boundary diffusivity Grain boundary width Grain boundary energy Molar volume

[62] [13] [63,64] [13]

db gm vm

(11)

Since f(x,t) is taken as a signed distance function, it also holds that

kVfðx; tÞk ¼ 1; 2.3. Grain boundary properties of UO2

667

x2U

(12)

The local interface normal n and the interface curvature k can be evaluated directly from the level set function according to

8 > > > <

Vf ≡Vf; kVfk   > Vf > > ≡V2 f : k ¼ VT n≡VT kVfk n¼

if

jjVfjj ¼ 1

(13)

Differentiation of Eq. (11) with respect to time provides the evolution law for the interface as

vf T vx þ V f ¼0 vt vt

(14)

Generalizing this result to a situation with an arbitrary number of N4 level sets provides

8 > < vfi þ vT Vfi ¼ 0; vt > : fi ðt ¼ 0; xÞ ¼ f0i ðxÞ;

 c i2 1:::Nf

(15)

where v denotes the local interface velocity vector and where f0i ðxÞ are the initial interface configurations at time t ¼ 0. When numerically evolving the interfaces, there is sometimes a tendency for a the level sets to separate or overlap at interfaces. This is mitigated by performing an interaction correction step with some regularity during the course of the solution. There is also a tendency of the level sets to drift from maintaining the property of being signed distance functions when continuously evaluating Eq. (15) which is remedied through reinitialization of the level set functions. Further details on the level set interaction correction and the reinitialization scheme are given in Ref. [31]. Considering level sets applied to modeling of grain growth, the interface velocity v appearing in Eq. (15) is with Eq. (1) given by

y ¼ vk kn;

vk ¼ mgb

(16)

where vk is usually referred to as the reduced boundary mobility. If advantage is taken of Eq. (13) and if it is assumed that kVfi k ¼ 1 holds due to level set reinitialization, then Eq. (15) can be reformulated as

8 > < vfi þ vk V2 fi ¼ 0; vt > : fi ðx; t ¼ 0Þ ¼ f0i ðxÞ;

 c i2 1:::Nf

(17)

In the present study, Eq. (17) is solved in a finite element setting. To consider the influence of anisotropic grain boundary properties in the numerical solution scheme, all elements containing

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Fig. 1. Misorientation dependence of the grain boundary properties. a) Variation of grain boundary energy g with misorientation according to Eqs. (5) and (6). b) Variation of the grain boundary mobility m with misorientation according to Eq. (9).

interfaces are identified when assembling the element matrices. Elements containing interfaces, being the zero-contour of any level set, are identified by evaluating si ¼ (max 4i)(min 4i) in each element for all i ¼ 1 ... N4 level sets. If si  0, the element contains the interface of the corresponding level set 4i. This approach also allows convenient identification of the level sets that meet in an individual element. Having identified the neighboring level sets, the interface properties can be evaluated according to Section 2 to provide the correct values of the grain boundary mobility m and energy g.

elements with linear interpolation, illustrated by the close-up picture in Fig. 2. The domain is continuously and adaptively remeshed by constrained Delaunay triangulation during the solution procedure to maintain appropriate resolution of the grain boundary network. The meshing is performed using the Triangle software [70]. Periodic boundary conditions are employed at all edges of the computational domain. The polycrystal is given an initial random texture by assigning to each grain values of the three EulereBunge angles (41, F, 42) according to

4. Numerical simulations

41 ¼ 2pr1 F ¼ acosð1  2r2 Þ 42 ¼ 2pr3

Isothermal conditions are assumed by holding the temperature constant at T ¼ 1700  C for a total annealing time of 100 h. The initial average grain size is taken as d0 ¼ 5 mm. These values are in accord with sintering conditions for the material and correspond to the temperatures and initial grain sizes considered in several experimental studies on UO2 [46,60,66e69]. Similar values for UO2 are also used in the analytic grain growth model in Ref. [9] and in the phase field model of grain growth in Ref. [13]. An initial grain microstructure is generated by Voronoï tesselation on a 100  100 mm square domain. The grain structure is then allowed to evolve for some time under purely curvature driven grain growth to equilibrate triple junction configurations and to provide a more realistic grain topology. By this procedure, an initial polycrystal with 576 grains is obtained, as shown in Fig. 2. The domain is discretized using approximately 150,000 triangular

(18)

where r1,2,3 are random numbers taken from a uniform distribution in the range [0, 1]. Considering the full set of 5760 grains, the corresponding misorientation distribution is shown in Fig. 3a together with the Mackenzie distribution for a randomly textured cubic polycrystalline aggregate [37]. Based on the individual grain orientations, the misorientation across each grain boundary can be evaluated according to Eq. (4). The resulting initial grain boundary texture map is shown in Fig. 2b. To get a statistically more relevant grain count, the 576-grain polycrystal is used repeatedly with new sets of crystal orientations assigned to the crystals. All simulations in the present study are performed on this total number of 576  10 ¼ 5760 initial grains. Considering this total set of grains, the corresponding

Fig. 2. Initial microstructure with 576 grains, represented in a 100  100 mm square domain. The close-up picture shows the finite element discretization in a small region of the domain.

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Fig. 3. a) Misorientation distribution in the total set of 5760 grains considered in the present study. The solid line represents the expected misorientation distribution for a randomly textured polycrystal cubic system as given by the Mackenzie distribution, cf. [37]. b) Initial grain boundary texture in the polycrystal. The coloring indicates the misorientation across each grain boundary, as obtained from Eq. (4).

misorientation distribution is shown in Fig. 3. Three different simulation scenarios are considered in the present work regarding the grain boundary properties: 1. Isotropic grain boundary properties, holding both g ¼ gm and m ¼ mm constant. 2. Anisotropic grain boundary properties, considering CSL configurations, with g ¼ gCSL þ gRS varying according to Eq. (7) and with m according to Eq. (9). 3. Anisotropic grain boundary properties with g ¼ gRS varying according to the ReadeShockley relation in Eq. (5), i.e. without considering CSL configurations, and with m according to Eq. (9). The polycrystal simulation model is employed to simulate isothermal grain growth at 1700  C during a total annealing time of 100 h, where after approximately 95% of the grains have been consumed by the growth process. Since no evolving grain growth stagnation mechanisms e such as particle pinning or thermal grooving e is considered in the model, the grain structure will develop towards a final equilibrium state containing a single grain. Such a configuration would provide little useful information to the present study and therefore the present annealing time of 100 h is chosen. The number of grains present in the microstructure as function of annealing time is shown in Fig. 4a while Fig. 4b shows the

corresponding increase of average grain diameter D, minus the initial average grain diameter D0, as grain growth progresses. In Fig. 4b, graphs for all three cases of isotropic or anisotropic grain boundary properties are shown together for comparison. It is evident that introduction of anisotropic grain boundary properties only has a very limited influence on the rate of grain growth in the system. Fig. 4b, also shows experimental data on grain growth in UO2 at 1700  C, taken from Refs. [59e61]. It is seen that the present setting of the porosity parameter fp in Eq. (9) provides grain growth rates that are comparable to what is observed in experiments. More refined adjustment of the grain boundary mobility due to porosity is from the results shown in Fig. 4b found to be beyond the scope of the present study. At present, the focus lies on the stability of grain boundary texture and it is evident from Fig. 4 that the influence of anisotropic grain boundary properties on grain growth rates is negligible. Fig. 5 shows the grain structure at four points in time during the process. The results in Fig. 5 are obtained using isotropic grain boundary properties, holding the grain boundary energy g and the grain boundary mobility m constant. The grain structure topology is, however, representative for all simulation scenarios under consideration. This observation is consistent with the graphs in Fig. 4a, showing almost identical evolution of the number of grains for all of the studied combinations of isotropic and anisotropic grain boundary properties.

Fig. 4. a) The number of grains in the considered polycrystal as function of the annealing time. b) The average grain size D, minus the initial average grain size D0, as function of annealing time. The three different cases of isotropic or anisotropic grain boundary properties employed in the present study are shown. For comparison, grain growth data on UO2 at 1700  C from Ref. [59] (,) [60], (B) and [61] (* and ), are also included.

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Fig. 5. The microstructure at four different stages during the annealing process when using isotropic grain boundary properties.

As seen in Fig. 5, the evolving grain structure is dominated by hexagonal grains connected at grain boundary triple junctions. As expected, grains with fewer than six sides tend to shrink and disappear while grains with more than six sides tend to prevail. Although the grain growth kinetics seem to be relatively unaffected by introducing anisotropic grain boundary properties, this is not the case for the evolution of grain boundary character distribution. Fig. 6 shows the length-weighted fraction of the S1, S3 and S5 types of boundaries in the polycrystal aggregate at the initial state compared to the final states obtained by the three different cases of isotropic or anisotropic grain boundary properties. The

Fig. 6. Distribution of grain boundary character in terms of length-weighted fractions of S1, S3 and S5 grain boundaries. The initial state (white bars) is compared to the distributions obtained at the end of the grain growth process in the three different cases of isotropic or anisotropic grain boundary properties (shaded bars).

length-weighted fractions are used as it has been observed that only using number fractions of grain boundaries of different CSLtypes tend to be misleading as the presence of S3 boundaries will be overestimated while the fraction of S5 boundaries is typically underestimated [35,71]. It can be noted in Fig. 6 that isotropic grain boundary properties seem to underestimate the presence of general S1 boundaries. In contrast, the fractions of S3 and S5 boundaries are overestimated. As a general observation, regardless of anisotropy or isotropy, the results in Fig. 6 show that grain growth tends to increase the fractions of S1 and S3 boundaries while the fractions of S5 boundaries are reduced. The latter is especially true as anisotropic grain boundary energy is modeled solely by the ReadeShockley formulation, with g ¼ gRS. In this case the S5 boundaries are missing entirely at the end of the grain growth process, as seen by the missing right-most bar in Fig. 6. The most pronounced influence of considering anisotropic grain boundary properties with g ¼ gCSL þ gRS, is in Fig. 6 found to be an increase of general highangle S1 boundaries. The evolution of S1, S3 and S5 boundaries, respectively, is also illustrated in Fig. 7. In these graphs it is again evident that by using isotropic grain boundary properties, the fraction of S1 boundaries is underestimated while the fraction of S3 boundaries is overestimated. In Fig. 7c, it can also be noted that the fraction of S5 boundaries drops to zero when g ¼ gRS. This corresponds to the missing right-most bar in Fig. 6. An influence of anisotropic grain boundary properties can also be found when considering the grain size distribution. Fig. 8a shows the initial grain size distribution in terms of the grain area A normalized by the averaged grain area 〈A〉. In Fig. 8b, the grain size

H. Hallberg, Y. Zhu / Journal of Nuclear Materials 465 (2015) 664e673

671

Fig. 7. Evolution of the length weighted fractions of three types of CSL boundaries under different assumptions of isotropic or anisotropic grain boundary properties: a) S1, b) S3 and c) S5.

distribution at the end of the grain growth process is illustrated for the three different cases of isotropic or anisotropic grain boundary properties currently under consideration. It can be seen from Fig. 8b that an anisotropic grain boundary energy where CSL configurations are considered (g ¼ gCSL þ gRS) shifts the distribution towards smaller grain sizes. In contrast, an anisotropic grain boundary energy, formulated only in terms of the ReadeShockley model (g ¼ gRS), closely resembles the result obtained in the isotropic case. While grain boundary texture is influenced by consideration of anisotropic grain boundary properties, as shown in Fig. 6, this is not necessarily the case for the macroscopic polycrystal texture. Fig. 9 shows the {111} pole figures for the full 5760-grain polycrystal

aggregate. In Fig. 9, RD is the horizontal direction and TD the vertical direction, respectively, compared to e.g. Fig. 2. As discussed in relation to Eq. (18) the initial polycrystal is given a random texture, shown in Fig. 9a. Fig. 9bed shows the pole figures at the end of the grain growth process for the three different cases of isotropic or anisotropic grain boundary properties. From these figures it is clear that no particular texture development can be distinguished for any of the three cases. This in line with the findings in Ref. [32] where anisotropic grain boundary properties were found to have little influence on the development of macroscopic texture even in the case of an initial cube texture. This is also discussed in Ref. [72] where it is observed that a strongly pronounced initial texture is required to influence the evolution of CSL boundaries.

Fig. 8. a) Initial grain size distribution in terms of the grain areas A normalized by the mean grain area 〈A〉 b) Grain size distribution at the end of the simulated annealing process for the three different cases of isotropic or anisotropic grain boundary properties under consideration.

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Fig. 9. Influence of isotropic or anisotropic grain boundary properties on the evolution of macroscopic texture in the polycrystal. The {111} poles are shown for: a) The initial polycrystal at t ¼ 0, b) The isotropic system, c) The anisotropic system, considering CSL configurations, cf. Eq. (7) and d) The anisotropic system, with grain boundary energy according to Eq. (5). Figures bed show the textures at the end of the annealing process.

5. Concluding remarks A numerical model of 2D grain growth is established, based on a level set formulation employed in a finite element framework. The model is used to perform numerical simulations of isothermal grain growth in uranium dioxide (UO2) to investigate the evolution of microstructure, especially in terms of grain boundary texture. The influence of anisotropic grain boundary energy and mobility is included in the model and different approaches to modeling of anisotropic grain boundary energy are considered. Three different formulations of isotropic or anisotropic grain boundary properties are compared. It is found that anisotropic grain boundary properties have negligible impact on grain growth kinetics and evolution of macroscopic texture. In contrast, both grain size distribution and the stability of grain boundary textures are significantly influenced when considering such anisotropy. In terms of grain size distribution, consideration of CSL correspondence shift the peak of the lognormal grain size distribution towards smaller grain sizes whereas a pure ReadeShockley type of energy anisotropy results in a grain size distribution that is similar to the isotropic case. Considering the grain boundary character distribution, it is observed that isotropic grain boundary properties underestimate the presence of general S1 grain boundaries while the fractions of S3 and S5 boundaries are overestimated. The fractions of the considered low-S boundaries clearly depend on how the anisotropy of grain boundary properties is modeled. Representation using length-weighted CSL fractions is preferable over number-weighted fractions which tend

to over- or underestimate the presence of different CSL types. The evaluation of length-weighted CSL fractions is further facilitated by the present level set modeling approach where clearly defined grain boundary segments and corresponding grain boundary texture is readily available. The present modeling approach e using level sets in a finite element setting e is indeed tractable when it comes to simulation of grain growth. However, suggestions for future work include consideration of the full 3D case, non-isothermal annealing conditions and inclusion of the influence of particles and evolving porosity, e.g. due to release and diffusion of fission gasses, on the grain growth process. Consideration of grain growth stagnation due to a presence of pores or particles would, for example, allow study of abnormal grain growth and reduced growth rates. Acknowledgment H. Hallberg gratefully acknowledges funding from the Swedish Research Council (Vetenskapsrådet, VR) under grant 2012-4231. References [1] D. Saylor, A. Morawiec, G. Rohrer, Distribution of grain boundaries in magnesia as a function of five macroscopic parameters, Acta Mater. 51 (2003) 3663e3674. [2] G. Gottstein, L. Shvindlerman, Grain Boundary Migration in Metals, CRC Press, 2010. [3] G. Rohrer, Grain boundary energy anisotropy: a review, J. Mater. Sci. 46 (2011) 5881e5895. [4] A. Brahme, J. Fridy, H. Weiland, A. Rollett, Modeling texture evolution during

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