Role of inclination dependence of grain boundary energy on the microstructure evolution during grain growth

Role of inclination dependence of grain boundary energy on the microstructure evolution during grain growth

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Role of inclination dependence of grain boundary energy on the microstructure evolution during grain growth Hesham Salama, Julia Kundin, Oleg Shchyglo, Volker Mohles, Katharina Marquardt, Ingo Steinbach PII: DOI: Reference:

S1359-6454(20)30156-7 https://doi.org/10.1016/j.actamat.2020.02.043 AM 15866

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Acta Materialia

Received date: Revised date: Accepted date:

3 July 2019 16 January 2020 17 February 2020

Please cite this article as: Hesham Salama, Julia Kundin, Oleg Shchyglo, Volker Mohles, Katharina Marquardt, Ingo Steinbach, Role of inclination dependence of grain boundary energy on the microstructure evolution during grain growth, Acta Materialia (2020), doi: https://doi.org/10.1016/j.actamat.2020.02.043

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Graphical Abstract

2D cross-section

Simulated microstructure

Grain boundary plane distribution

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Role of inclination dependence of grain boundary energy on the microstructure evolution during grain growth Hesham Salamaa , Julia Kundina , Oleg Shchygloa , Volker Mohlesa , Katharina Marquardtb , Ingo Steinbacha a Interdisciplinary

Centre for Advanced Materials Simulation (ICAMS), Ruhr-University Bochum, D-44801 Bochum, Germany b Faculty of Engineering, Department of Materials, Imperial College London

Abstract The role of inclination dependence of grain boundary energy on the microstructure evolution and the orientation distribution of grain boundary planes during grain growth in polycrystalline materials is investigated by three-dimensional phase-field simulations. The anisotropic grain boundary energy model uses the description of the faceted surface structure of the individual crystals and constructs an anisotropic energy of solid-solid interface. The energy minimization occurs by the faceting of the grain boundary due to inclination dependence of the grain boundary energy. The simulation results for a single grain show the development of equilibrium shapes (faceted grain morphologies) with different families of facets which agrees well with the theoretical predictions. The results of grain growth simulations with isotropic and anisotropic grain boundary energy for cubic symmetry show that inclination dependence of grain boundary energy has a significant influence on the grain boundary migration, grain growth kinetics and the grain boundary plane distribution. It has been shown that the model essentially reproduces the experimental studies reported for NaCl and MgO polycrystalline systems where the anisotropic distribution of grain boundary planes has a peak for the low-index {100} type boundaries. Keywords: Anisotropic grain boundary energy, Grain growth, Phase-field method

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1. introduction Microstructure evolution is an essential factor in studying the properties of any engineering material because it has a significant impact on the mechanical, thermal and electrical properties. Grain growth is a microstructure evolution phenomenon related to grain boundary migration. The driving force for this migration is the reduction of grain boundary energy between different grains. In a polycrystalline microstructure, due to its anisotropic nature, this driving force is orientation-dependent. This favours the growth of grains with orientations with lower grain boundary energy in order to minimize the total energy of the system. The process of grain growth was intensively investigated experimentally and theoretically for decades [1, 2], yet it is still challenging to model analytically. In recent years, many numerical approaches have been developed to study the grain growth phenomena using atomistic methods such as the Potts model [3] and mesoscale methods including front-tracking models [4], vertex models [5], cellular automata [6], the MonteCarlo model [7] and the phase-field model [8–10]. Among these models, the phase-field model has appeared as a promising method which can be used to simulate complex morphological evolution of polycrystalline microstructure [11, 12]. To date, most phase-field studies assumed an isotropic grain boundary energy and mobility during grain growth. However, the anisotropic character of the grain boundaries is a crucial factor in grain growth characteristics, including the grain size distribution [13], deviation of grain growth kinetics from the ideal parabolic law [14], and the distribution of grain orientation, or the texture development [15]. In general, grain boundaries are distinguished by five geometrical degrees of freedom [16], three parameters describing the misorientation between the crystals, e.g. three Euler angles, and two parameters describing the grain boundary plane, e.g. two inclination angles. However, in general the relation between the grain Preprint submitted to Elsevier

February 22, 2020

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boundary structure and the related properties is not fully understood even though it is clearly visible in polycrystalline ceramics [17, 18]. It has been reported that grain boundary plane distribution (GBPD) is a critical parameter controlling the properties of polycrystals [19] and is inversely correlated to the grain boundary energy [20–23]. Gruber et al. [24] concluded that grain boundary energy anisotropy has a significantly higher impact on the GBPD than grain boundary mobility anisotropy. Other studies on metals and polycrystalline ceramics showed that the GBPD is more significant for the grain boundary properties than the misorientation [25]. Therefore, in this study, we will concentrate on the investigation of the role of inclination dependency on the GBPD and their evolution during grain growth by means of the phase-field method and compare the isotropic and anisotropic systems. For polycrystalline materials, two types of phase-field models have been used to simulate grain growth, the multi-order-parameter models and the Kobayashi-Warren-Carter (KWC) model [26]. The multi-orderparameter models can be categorized into two models; the continuum-field (CF) model of Chen and Fan [27, 28]; and the multi-phase-field (MPF) model by Steinbach et al. [29]. To our knowledge, the use of the KWC model is restricted to systems with isotropic grain boundary energy and misorientation dependence of grain boundary energy due to the difficulty in introducing the grain boundary inclination dependence in this model. The MPF model is known to provide a consistent description of multi-junctions during grain growth. A recent study [30] shows that using the MPF model without adding a higher-order term can reliably simulate weakly anisotropic grain boundary energy without the presence of third phase contributions at interfaces. However, by including the higher-order term into the free energy functional, the model accuracy is further increased with strongly anisotropic grain boundary energy. Kamachali et al. [9] successfully used the MPF model to simulate large-scale 3D grain growth using tens of thousands of grains and studied statistics and mean-field characteristics of grain growth using isotropic grain boundary energy. Kim et al. [31] performed 3D grain growth simulation using the MPF model by including the effect of misorientation and inclination dependence on grain boundary energies via a grain boundary energy database for polycrystalline BCC-Fe. The authors observed that the grain growth kinetics are slowing down by using grain boundary energy anisotropy. However, the algorithm they used made the simulations very time-consuming in comparison to simulation methods based on a predefined analytical function describing the grain boundary energy dependence on the inclination angle. Despite the previous valuable efforts, to the best of the authors’ knowledge, there has been no largescale phase-field simulation of three-dimensional grain growth to study the inclination dependence of grain boundary energy. Therefore, in order to study whether and how the inclination dependence of grain boundaries can affect grain growth, we apply the MPF approach [29] to describe the grain growth with a specific anisotropic grain boundary energy. We use a function of the inclination angle first proposed in works of Caginalp [32] and McFadden et al. [33]. The grain boundary energy is defined as a weighted sum of the grain boundary energies of the neighboring grains of different orientation. In this study, we apply this method to polycrystalline ceramics due to their low sensitivity to the grain misorientation. The main objective of the present work is to study the role of inclination dependence of grain boundary energy on the microstructure evolution during grain growth via large-scale 3D phase-field simulations. We also examine the effect of anisotropic grain boundary energy on the morphological evolution of the microstructure compared with isotropic grain boundary energy. As a model material for the comparison with the experimental observations, we choose NaCl and MgO polycrystals because of their cubic symmetry and because these materials are widely investigated [20, 34] due to their importance for predicting of the stability of caverns and back-fill used for the storage of oil, gas, and nuclear waste [34]. The rest of the paper is organized as follows: Section 2 explains the methodology used to describe anisotropic grain boundary energy within the framework of the MPF model. The results of phase-field simulations are presented in Section 3, discussing the growth kinetics and morphological evolution. The conclusion is given in Section 4.

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2. Methods 2.1. Phase-field model In the present study, the MPF [11] has been applied using the open source library OpenPhase [35]. The model can treat a virtually unlimited number of grains using a set of phase-field variables φ(x, t) as functions of spatial coordinates and time, where each grain is distinguished from others by its phase or orientation. In polycrystalline materials, when many grains with different orientation but the same thermodynamic phase come in contact with each other, an interface is formed by the distortion of the atomic bonds which is associated with the grain boundary energy [36]. Individual orientations of the grains are defined by the Euler angles (ϕ1 , Φ, ϕ2 ) and every grain has a different phase field parameter φ(x, t) with a value between 0 and 1. The value of 0 indicates being outside the grain, and 1 indicates being inside the grain with a smooth transition between 0 and 1 indicating the diffuse interface. The sum over all phase-field variables in every point in space is constrained: N X

φα = 1.

(1)

α=1 80 81

The time evolution equation of the phase-field parameter is constructed as a sum over all dual interactions between the grains [29]:   N δF π 2 X µαβ δF ˙ − , (2) φα = − 8η N δφα δφβ β=1

82 83 84 85 86 87

where µαβ is the grain boundary mobility, defined separately for each pair of grains, N is the number of δF δF and δφ are the functional derivatives of the total grains in a given point, η is the interface width, and δφ α β free energy functional with respect to the phase-field variables φα and φβ . For the grain growth simulations, the total free energy density includes the grain boundary energy density which is the contribution of interest in the present work and the energy density responsible for the driving force. The grain boundary energy is given by   N N X X 8σαβγ − 4ησαβ ∇φα · ∇φβ + 4σαβ φα φβ + f GB = φα φβ φγ  , (3) π2 η η α6=β

γ6=α,β

97

where σαβ is the grain boundary energy between φα and φβ . The first and the second terms on the righthand side are the gradient term and the double obstacle potential which are responsible for stabilization of the interface profile. The third term in Eq. (3) represents the higher-order term which is added to the double-obstacle potential. The higher-order term were first suggested in [37] for a strongly anisotropic grain boundary energy of faceted type to prevent the presence of ghost phases. This term has later been used to increase the accuracy of phase-field models [38, 39]. In [37], the authors estimated the parameter σαβγ to be approximately 10 times the highest grain boundary energy in the system: σαβγ > π962 max{σαβ }. In the present work, we validated that the higher-order term improves the accuracy of the simulations. However, the efficiency of the simulations depends on the value of the third-order parameter σαβγ . By inserting Eq.( 3) into Eq. (2), the kinetic equation of the phase field reads:

98

    X N N N   2 2  X X p    π π µαβ π ∗ ∗ ∗ ∗ φ˙ α = σβγ − σαγ ∇2 φ γ + 2 φ γ + σ − σ φ φ + φ φ ∆G , (4) γ δ α β αβ βγδ  N  η η 2 αγδ η

88 89 90 91 92 93 94 95 96

β6=α

99 100 101

γ6=α,β

δ6=γ

where σ ∗ is the interface stiffness. The last term in Eq. (4) acts as a penalty force to keep the volume of the grain constant, and ∆Gαβ is the local driving force acting on the interface. This termï¿ 21 is usedï¿ 12 only to test the validity of the model when calculating the equilibrium shapesï¿ 12 and has been excluded in all grain growth simulations. 4

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2.2. Anisotropic grain boundary energy In polycrystalline materials, the shape of crystals is controlled by kinetic and energetic factors [40, 41] and the properties of grain boundaries are varying depending on the inclination along the same boundary. Therefore, the grain boundary energy anisotropy plays a vital role in determining the grain surface morphology and the kinetics of migration of grain boundaries in polycrystalline aggregates [1]. In principle, the classical description of the grain boundary energy anisotropy assumes that it can be a function of the crystallographic orientation of the interface, which results in equilibrium shapes of the individual crystals given by the Wulff construction [42], which minimizes the total surface energy of the system. The anisotropic model used in this work is based on the approach developed by Caginalp [32] and McFadden et al. [33]. In this approach, the grain boundary energy is a function of the inclination angle θ, which is defined as the angle between the grain boundary normal ~nα = ∇φα /|∇φα | and the nearest facet normal ~kiα for the grain α in the sample frame. The latter is defined as ~kiα = gα · ~ki , where gα is the orientation matrix of the grain α, ~ki is a facet normal in the crystal frame which is indicated by Miller indices. Using two unit vectors, the inclination angle is then calculated as n o θα = min arccos(~nα · ~kiα ) . (5) To define the grain boundary energy, we will use the description of the faceted surface structure of the crystal by introducing a virtual amorphous layer between two crystalline grains, as it can be observed in real minerals and ceramic systems with porosity [25, 43–45]. The amorphous layer serves as a disordered phase which allows to distinguish the interface of two neighboring grains α and β. Then, the energy of the solid-solid interface can be defined as a sum of two solid-amorphous interface enrgies σαβ = σα + σβ .

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In other words, the grain boundary is comprised of two solid-amorphous interfaces and the energy of the amorphous layer. Here, we assume that the thickness of the amorphous layer approaches zero and has no effect on the grain boundary energy. A similar anisotropic grain boundary model between neighboring grains was already suggested in the works of Moelans et al. [46] where the phase-field parameters were used as the weighting factors for the individual grain boundary energies.. The energy of a solid-amorphous interface is calculated following the approach of Caginalp [32] and McFadden et al. [33] as q (7) σα (θα ) = σ 0 sin2 (θα ) + a2 cos2 (θα ),

where σ 0 is a reference grain boundary energy which is equal for the same family of facets, a is the anisotropy strength that indicates the depth of the grain boundary energy cusp. The interface stiffness σ ∗ for the GibbsThomson-Herring equation [47] with respect to the inclination angle is given by σα∗ = σα (θα ) + σ 00 (θα ) =

130

(6)

σ 0 a2 sin2 (θα ) + a2 cos2 (θα )

(8)


23 .

Finally, the stiffness of the complete interface is defined as a sum of the two solid-amorphous interfaces: ∗ σαβ = σα∗ + σβ∗ .

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(9)

and the triple-junction term is given by ∗ σαβγ = σα∗ + σβ∗ + σγ∗ .

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(10)

Using Eqs. 7 and 8, the grain boundary energy σ and the grain boundary stiffness σ can be plotted as a function of the inclination angle θ as shown in Fig. 1. The grain boundary energy σ(θ) shows a minimum close to a cusp for small inclination angles. At the same time, the grain boundary stiffness exhibits a noticeable maximum at small inclination angles. Theoretically, if the stiffness is infinite at θ = 0, the curvature tends to zero and the crystal has a perfectly faceted structure. Then, the resulting orientation of the facet corresponds to the Wulff shape [42, 48]. ∗

5

00

Figure 1: Interface energy σ and the effective interface energy (stiffness) σ ∗ = (σ + σ ) as a function of the inclination angle θ

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2.3. Grain boundary plane distribution The GBPD is defined as the interface normal in the crystal frame ~n0α , which is calculated by ~n0α = gαT · ~nα ,

(11)

144

is the transposed matrix of gα . Calculating where ~nα is the interface normal in the sample frame and the frequency of plane normals for all observed grain boundaries over all misorientation yields the GPBD determined in terms of the relative grain boundary area. It can be plotted as a stereographic projection where the planes in the crystal reference frame represented as poles. The GBPD is constructed by means of MTEX software [49].

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3. Simulation Results

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146 147

gαT

For all simulations, the same model parameters were chosen as shown in Table 1 unless mentioned separately. The size of the simulation box will vary depending on the type of the study. Parameter Grid spacing Time steps Interface width Reference grain boundary energy Anisotropy strength Grain boundary mobility Third-order parameter

Symbol ∆x ∆t η σ0 a µ σαβγ

Value 1 × 10−7 0.1 6∆x 1.0 0.2 1 × 10−14 1

Units m s m Jm−2 m4 (Js)−1 -

Table 1: Model parameters. 148 149 150 151 152 153 154 155

3.1. Validation of the equilibrium shapes As a first application of the theoretical model introduced above, the case of equilibrium shape is investigated in 3D by performing a simulation of an initially spherical grain inserted in the melt in a simulation box of size 1283 grid points. The first test is done by using the grain boundary energy as a function of the inclination angle given in Eq. (7) which is bounded by {100} and {111} type facets. The development of the initially spherical grain into a faceted crystal is shown in Fig. 2. A local penalty driving force term is applied to keep the volume of the grain constant. Fig. 3 shows the equilibrium structures for different basic sets of facet normals corresponding to equal energy facets. 6

(a) 0 ms

(b) 5 ms

(c) 10 ms

(d) 20 ms

Figure 2: Evolution toward equilibrium shape of an initially spherical grain inserted in the melt with {100} and {111} facets

(a) Hexahedron

(b) Octahedron

(c) Cuboctahedron

(d) Rhombicuboctahedron

Figure 3: Equilibrium shapes using different basic sets of facet vectors: (a) only {100} facets; (b) only {111} facets; (c) {100} and {111} facets; (d) {100}, {110}, {111} facets. 156 157 158 159 160

For different grain boundary energies σ 0 and anisotropy values a, the equilibrium structure should change. A convenient test for the numerical solution is the case of a crystal bounded by {100} and {111} type facets. At the minimum of the total surface energy at constant volume of the crystal, the ratio of the surface area F100 and F111 of the respective facets, within certain limits, can be considered as a function of the ratio of the grain boundary energy σ 0 of {100} and {111} type facets 0 σ100 0 . σ111

e= 161

The ratio of the surface area is obtained (see Appendix A) F100 = F111

162 163 164 165 166 167 168 169 170 171 172 173 174 175

(12)

"

f2 f2 − e

2

#−1 

−3

3 f2



,

(13)

√ √ where f2 = tan(60◦ ) = 3 and 0.953 ≤ e ≤ 3. The details on the analytical solution are given in Appendix A. The selected geometric model loses its validity for e ≤ 0.953. Fig. 4 shows the equilibrium shapes and the ratios calculated from the simulated shapes in comparison to the analytical solution given by Eq. (13). For the value of e = 1, the result agrees sufficiently well with the analytical value. For e = 2, the {111} facets dominate the evolution of the crystal and the area is close to zero. Finally, for e = 0.5, an increase in the area of {100} facets is confirmed, however the analytical model loses its validity. The reason for this deviation from the analytical solution is the numerical resolution of the edges in the model which is limited by th diffuse interface width. Next, we analyse the relative grain boundary area and the grain boundary energy. We select the case of cubic symmetry with a crystal bounded only by {100} as shown in Fig. 3(a). Fig. 5 shows the stereographic projection of the grain boundary energy along with the corresponding GBPD. It can be observed that the distribution of the grain boundary planes is dominated by the {100} type facets, and the grain boundary energy has a minimum for {100} facets. This observation indicates the inverse correlation between the GBPD and the grain boundary energy distribution where the low-energy {100} facets are the slow-growing 7

(a)

(b)

(c)

(d) Figure 4: (a-c) Equilibrium shapes for different energy ratios of the facets:(a) e = 0.5; (b) e = 1.0; (c) e = 2.0; (d) The ratio of area of {100} over {111} type facets plotted versus the analytical solution in Eq. (13).

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and the high energy {111} facets are fast-growing while the system minimizes the overall grain boundary energy. As a result, the low-energy grain boundaries are favourable and more dominant in the distribution. We can relate these observations with MgO and NaCl systems, where the same behavior has been reported [25, 43].

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3.2. Validation of the triple-junction angles

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The triple junction angles were studied for isotropic and anisotropic systems in the 2D three-grain system shown in Fig. 6. The system has a size of 200∆x in the x-direction and 100∆x in the z-direction. The initial configuration contains three rectangular grains with different phase field parameters but the same thermodynamic properties. Boundary conditions are chosen such that the positions of the grain boundaries on the domain boundary are fixed. The interface width was varied between η = 6∆x and 8.5∆x. Other parameters are listed in Table 1. In the following study, we used the interface stiffness given in Eq. (8), and we examined the effect of the higher-order term on the accuracy of the simulations. The equilibrium P triple junction angle α was measured from the contour of the interfaces, which are calculated as αβ φα φβ . The grain boundary energy anisotropy was chosen such that the energy minimum occurs at either 0◦ or 45◦ . The obtained results are listed in Table 2. As expected, for the isotropic case, all angles are 120◦ . For the anisotropic case with the grains orientation of 45◦ , the angle α is close to 90◦ (see Fig. 6()a) and (c)). This is because the two symmetric planes between grains 1 and 3, or 2 and 3, try to reach an inclination of 0◦ with respect to the minimal energy. The boundary between grains 1 and 2 has the maximum energy 8

Figure 5: Stereographic projection in the crystal referece frame: (a) the surface energy and (b) the GBPD as the relative surface area expressed in multiples of random distribution (MRD). The <100>,<010> and <111> directions marked with black, blue and red solid circle, respectively

194 195 196 197 198 199 200 201 202 203 204 205

with an inclination angle of 45◦ , but it does not change its orientation due to the symmetric boundary conditions, but it can minimise its length. For the anisotropic case with the orientation of 0◦ , the angle α is close to 180◦ (see Fig. 6()b) and (d)). This is because in this configuration, all planes have an inclination angle close to 0◦ , which corresponding to  the minimum energy. The angles can be calculated by the Herring  P ∂σαβ nαβ = 0, where t and n are the unit vectors along the grain boundaries equation [47] αβ σαβ tαβ + ∂θ and normal to the grain boundaries, respectively. This gives the theoretical prediction of β = 176◦ for 0◦ orientation and β = 85◦ for 45◦ orientation (see Table 2). Our results show that the simulated angles approach the theoretical prediction, regardless of the chosen interface width (η ≥ 6∆x). However, by adding the higher-order term to the MPF model, the accuracy of the simulations is improved, and the obtained angles are significantly closer to the theoretical predictions. These results show that for a strongly anisotropic grain boundary energy, using a higher-order term in the phase-field equations improves the accuracy of the simulations. Therefore, the higher-order term has been used in the following grain growth simulations. isotropic Without higher-order term anisotropic 45◦ anisotropic 0◦ With higher-order term anisotropic 45◦ anisotropic 0◦

η=6 120◦

η = 8.5 120◦

theory 120◦

error 0%

93.8◦ 160◦

93.8◦ 160◦

85◦ 176◦

10.3% -9.0%

90.5◦ 167◦

90.5◦ 167◦

85◦ 176◦

6.5% -5.1%

Table 2: Triple junction angles α simulated for isotropic and anisotropic systems with and without the higher-order term.

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3.3. Grain Growth Simulation In the following step we investigate the microstructure evolution and the interaction between different grain boundaries in a multigrain structure. The MPF model with anisotropic grain boundary energy is used for studying the grain growth and coarsening in polycrystalline microstructures in 3D. The size of the simulation box in the present study is 4003 grid points with 18000 grains. The initial microstructure is obtained by Voronoi tessellation. In all simulations, random crystallographic orientations are assigned to all grains. Periodic boundary conditions are introduced along all directions. Two different grain boundary energies have been chosen to simulate grain growth. One is isotropic with a constant value of 1.0 Jm−2 fixed 9

(a)

(b)

(c)

(d)

Figure 6: Simulated triple junctions in the 2D domain of size (200X100)∆x2 with interface width η = 6∆x: (a,b) Anisotropic grain boundary energy with a minima at (a) 45◦ and (b) 0◦ without the higher-order term; (c,d) as(a,b) but with the higherorder term. The triple junction angle α and phase-field numbers are shown in white.

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for all grain boundaries, and the other one is anisotropic using Eq. (7) with minima in {100} planes. This allows for a comparison with the experimental observation in the polycrystalline materials MgO and NaCl, which have cubic symmetry. The grain boundary mobility is isotropic in all simulations. The other model parameters are listed in Table 1. Fig. 7(c) and (e) show the evolved microstructure at 900 [s] and 2300 [s] for isotropic and anisotropic grain boundary energy, respectively, with similar number of grains (≈ 4000). The simulations show that the anisotropic system exhibits retardation (slowdown) in grain growth compared to the isotropic system. It is noticeable in Fig. 7 that the anisotropic grain boundary energy affected the overall evolution of the microstructures (see Fig. 7(e)). It is anticipated that the presence of low-energy grain boundaries by using the anisotropic model would influences the microstructure evolution of the relevant grains. To examine this phenomena in more details, a 2D cross-section of the simulated microstructures in both the isotropic and the anisotropic model is presented in Fig. 8. The anisotropic system shows larger amount of nearly cubic grains structure compared to the isotropic system, which appeared equiaxed as can be seen in Fig. 8. Additionally, a magnifying view of the microstructure in both the isotropic and the anisotropic system (marked with green in Fig. 8) reveals the presence of more frequent triple-junction angles close to 90◦ and 180◦ (marked with yellow circles) as a result of the development of faceted grains. From that, it can be confirmed that the anisotropic grain boundary energy influences the morphological evolution of the microstructure. However, an analysis of the influence of the anisotropic grain boundary energy on grain growth is not trivial for the naked eye due to the complexity of the polycrystalline system. Therefore, a statistical study of the characteristics of the grain growth simulations under both conditions is necessary. The preferred orientations for grain boundary planes in both systems are presented in Fig. 7(b), (d) and (f). The GBPD are plotted as stereographic projections, and the colours represent the frequency of observed grain boundaries. The latter is calculated in multiples of a random distribution (MRD) so that values greater than unity indicate larger grain boundary plane areas compared to the isotropic GBPD. In Fig. 7(b), the initial microstructure of the polycrystalline material shows a random GBPD without any preferred orientations. As expected, the GBPD of the isotropic system is also homogeneous because the grain boundary energy is independent of the inclination (see Fig. 7(d)). When the grain boundary energy 10

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: The microstructure of the simulated grain growth: (a) Initial configuration; (c) isotropic grain boundary energy, at 900 [s]; (e) anisotropic grain boundary energy, at 2300 [s]; plotted side by side with the stereographic projection in the crystal reference frame that represents the GBPD as the relative surface area expressed in multiples of random distribution (MRD): (b), (d) and (f) respectively. The <100>,<010> and <111> directions marked with black, blue and red solid circles, respectively.

11

(a)

(b)

Figure 8: 2D cross section of the simulated microstructure shown in Fig.[7] for (a) isotropic grain boundary energy; (b) anisotropic grain boundary energy; with zoom onto triple junctions for both cross sections. The triple-junctions angle marked with yellow circles.

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is anisotropic and depends on the grain boundary inclination, the GBPD shows that the low-index grain boundary {100} planes occur more frequently in comparison to other grain boundary planes (see Fig. 7(f)) due to the existence of cusps in the grain boundary energy at the low-index planes. Therefore, the anisotropic system tend to develop faceted morphology of the grain boundary to minimize the surface energy of the system. As a result, the low-index planes are the most favoured type of surfaces because they have the lowest energies. The GBPD for the polycrystalline material agrees with the result obtained in section 3.1 for the single crystal with cubic symmetry. Comparing the GBPD obtained in our simulations to the experimental results for magnesia by Saylor et al. [43]. It is worth noting that the simulated distribution of boundary planes at all misorientations shows a similar trend as the experimental results, where the {100} planes were the most favoured type of boundary. The GBPD presented in Fig. 7(f) is also similar to an observation reported by Pennock et al. for synthetic NaCl (see Fig. 8 in [25]).

(a)

(b)

Figure 9: The time evolution during grain growth using isotropic and anisotropic grain boundary energy of the number of grains (a) and the squared average grain size (b). The black dashed lines represent the parabolic equations fitted to the curves. 252

The time evolution of the number of remaining grains is plotted in Fig. 9(a). The results show that the 12

253 254 255 256 257

number of remaining grains in the anisotropic system is much higher compared to the isotropic system at a given time step, which means that the grain growth kinetics is slower in the anisotropic system. In order to quantify the effect of anisotropic grain boundary energy on the kinetics of grain growth, the simulated results are compared to the theoretical mean-field theories for normal grain growth. The Mullins’ law shows that the square of the average grain size is a linear function of time [50]: < R >2 − < R0 >2 = Kt,

258 259 260 261 262 263 264 265 266 267 268 269 270

(14)

where < R > is the average grain size at time t, and K = Aµσ is the growth constant with A = 0.5 for isotropic 3D systems. The time evolution of the square of the average grain size is plotted in Fig. 9(b) for the isotropic and anisotropic systems. The initial stage of growth before a steady-state regime is reached is disregarded completely. The solid lines are the simulation results, and the dashed lines are Eq. (14) fitted to the curves over the time interval t > 400[s]. For the isotropic system, a parabolic equation is obtained with the growth kinetic constant K = 0.48 × 10−14 m2 s−1 . The parabolic equation is obtained for the anisotropic case with the growth kinetic constant K = 0.15 × 10−14 m2 s−1 . Since K = Aµσ, we find that A = 0.48 for the isotropic case and A = 0.15 for the anisotropic case. The constant A for the isotropic case is very close to the value predicted by Mullins [50]. Nevertheless, the constant A for the anisotropic case is ≈ 3 times smaller than that for the isotropic case. This means that the average grain size in the anisotropic system increases more slowly than in the isotropic system. The difference between the growth rates of the isotropic and the anisotropic systems is essentially the result of the surface energy average over the inclination angles which is smaller in the anisotropic case (see Fig. 1).

(a)

(b)

Figure 10: Simulated grain size distribution over time compared with Hillert’s 3D distribution for isotropic grain boundary energy (a) and anisotropic grain boundary energy (b). 271 272 273 274 275 276 277 278 279

The evolution of the grain size distribution during grain growth with isotropic and anisotropic grain boundary energy is shown in Fig. 10 and compared with Hillret 3D theory [51]. The x-axis in the plot displays the relative grain size with respect to the critical grain size given by < Rc >= 89 < R > [52]. The grain size distribution at the initial stage evolves very rapidly toward steady-state for the isotropic and anisotropic systems. In the case of using isotropic grain boundary energy, the peak of the distribution finishes close to Hillert’s 3D distribution, and is slightly shifted to smaller grains. It also exhibits the presence of a longer tail for larger grains. These results for the isotropic grain boundary energy are similar to those obtained by Kamachali et al. [9] and Kim [8]. However, for the anisotropic case, the grain size distribution exhibits a more systematic deviation from the Hillert’s distribution with a clear shift towards 13

280 281 282 283 284 285 286 287 288 289 290 291 292

smaller grain sizes. This deviation can be attributed to the fact that grain boundary energy anisotropy is orientation-dependent, which leads to grain boundaries rotation forming low energy grain boundaries close to the low index planes, and thus significantly reduces the motion of such grain boundaries. As a result, the population of grains surrounded by low energy grain boundaries increases (see Fig. 7(f)), leading to stronger deviation in the grain size distribution compared to the isotropic system. Furthermore, we investigate the impact of inclination dependence of grain boundary energy on the disorientation distribution among neighboring grains. The disorientation distribution for the isotropic and the anisotropic system during the microstructural evolution at a later stage of the simulation with the same number of grains is plotted in Fig. 11 compared to the Mackenzie distribution, which is a theoretical approach that determines the disorientation distribution for a cubic sample with random crystallographic orientations [53]. The disorientation distribution for both systems is almost perfectly matches the Mackenzie distribution (see Fig. 11) and remain unchanged during the simulation. Thus, the inclination dependence of grain boundary energy appears not to affect the disorientation distribution.

(a)

(b)

Figure 11: The disorientation distribution in grain growth simulations in comparison with the Mackenzie distribution for different grain boundary energies: (a) isotropic and (b) anisotropic

14

293

4. Conclusion

314

In order to study the role of inclination dependence of the grain boundary energy on the grain growth of polycrystalline materials in particular, three-dimensional MPF simulations were performed. The anisotropic grain boundary energy model used in this study follows the description of faceted surface structure of the crystal. As a result, the energy minimization occurs by faceting of the grain boundaries due to inclination dependence of the grain boundary energy. The obtained results can be summarized as follows: (1) Simulations of an initially spherical grain inserted in the melt successfully reproduced the expected equilibrium shapes. The obtained GBPD is inversely proportional to the grain boundary energy in accordance with the previous studies. (2) Fo polycrystalline materials, the inclination dependence of the grain boundary energy has a significant impact on the microstructure evolution and the grain boundary characteristics. This results in slower growth kinetics, and in a different morphological evolution expressed by the presence of more cubic grains structure and a higher number of triple-junction angles close to 90◦ and 180◦ . (3) The model closely reproduces the experimental studies reported for NaCl and MgO polycrystalline systems, where the anisotropic distribution of grain boundary planes has a peak for the low-index {100} type boundaries. (4) The grain size distribution matches Hillert’s prediction with a small shift toward smaller grains in the case of isotropic grain boundary energy. The simulated grain size distribution in the case of anisotropic grain boundary energy has a pronounced shift toward small grain size. No sign of abnormal grain growth was observed. (5) The inclination dependence of grain boundary energy appears not to affect the disorientation distribution.

315

Acknowledgements

294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313

319

The authors would like to acknowledge the support from the Fundamental Research Program of Korea Institute of Materials Science (PNK5570). JK thanks for the financial support from the German Research Foundation (DFG), grant KU 3122/3-1. VM gratefully acknowledges the financial support by the German Research Foundation (DFG), grant MO 974/6-1.

320

References

316 317 318

321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343

[1] G. Gottstein, L. S. Shvindlerman, Grain boundary migration in metals: thermodynamics, kinetics, applications, CRC press, 2009. [2] Maher, B., Taylor, R. Formation of ultrafine-grained magnetite in soils. Nature. 336, 368 (1988) [3] Grest, G., Anderson, M., Srolovitz, D. Domain-growth kinetics for the Q-state Potts model in two and three dimensions. Physical Review B. 38, 4752 (1988) [4] Becker, J., Bons, P., Jessell, M. A new front-tracking method to model anisotropic grain and phase boundary motion in rocks. Computers & Geosciences. 34, 201–212 (2008) [5] Weygand, D., Bréchet, Y., Lépinoux, J., Gust, W. Three-dimensional grain growth: a vertex dynamics simulation. Philosophical Magazine B. 79, 703–716 (1999) [6] Ding, H., He, Y., Liu, L., Ding, W. Cellular automata simulation of grain growth in three dimensions based on the lowest-energy principle. Journal Of Crystal Growth. 293, 489–497 (2006) [7] Anderson, M., Srolovitz, D., Grest, G., Sahni, P. Computer simulation of grain growth?I. Kinetics. Acta Metallurgica. 32, 783–791 (1984) [8] S. G. Kim, D. I. Kim, W. T. Kim, Y. B. Park, Computer simulations of two-dimensional and three-dimensional ideal grain growth, Physical Review E 74 (6) (2006) 061605. [9] R. D. Kamachali, I. Steinbach, 3-d phase-field simulation of grain growth: Topological analysis versus mean-field approximations, Acta Materialia 60 (6-7) (2012) 2719–2728. [10] R. D. Kamachali, A. Abbondandolo, K. Siburg, I. Steinbach, Geometrical grounds of mean field solutions for normal grain growth, Acta Materialia 90 (2015) 252–258. [11] I. Steinbach, Phase-field models in materials science, Modelling and simulation in materials science and engineering 17 (7) (2009) 073001. [12] Moelans, N., Blanpain, B., Wollants, P. An introduction to phase-field modeling of microstructure evolution. Calphad. 32, 268–294 (2008)

15

344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408

[13] Grest, G., Srolovitz, D., Anderson, M. Computer simulation of grain growth?IV. Anisotropic grain boundary energies. Acta Metallurgica. 33, 509–520 (1985) [14] E. A. Holm, G. N. Hassold, M. A. Miodownik, On misorientation distribution evolution during anisotropic grain growth, Acta Materialia 49 (15) (2001) 2981–2991. [15] G. S. Rohrer, Grain boundary energy anisotropy: a review, Journal of materials science 46 (18) (2011) 5881–5895. [16] Humphreys, F., Hatherly, M. Recrystallization and related annealing phenomena. (Elsevie,2012) [17] G. S. Rohrer, The role of grain boundary energy in grain boundary complexion transitions, Current Opinion in Solid State and Materials Science 20 (5) (2016) 231–239. [18] Sutton, A., Balluffi, R., Sutton, A. Interfaces in crystalline materials. (Clarendon Press Oxfor,1995) [19] Watanabe, T., Suzuki, Y., Tanii, S., Oikawa, H. The effects of magnetic annealing on recrystallization and grain-boundary character distribution (GBCD) in iron–cobalt alloy polycrystals. Philosophical Magazine Letters. 62, 9–17 (1990) [20] D. M. Saylor, A. Morawiec, G. S. Rohrer, Distribution of grain boundaries in magnesia as a function of five macroscopic parameters, Acta materialia 51 (13) (2003) 3663–3674. [21] Rohrer, G. Measuring and interpreting the structure of grain-boundary networks. Journal Of The American Ceramic Society. 94, 633–646 (2011) [22] K. Marquardt, G. S. Rohrer, L. Morales, E. Rybacki, H. Marquardt, B. Lin, The most frequent interfaces in olivine aggregates: the gbcd and its importance for grain boundary related processes, Contributions to Mineralogy and Petrology 170 (4) (2015) 40. [23] Holm, E., Rohrer, G., Foiles, S., Rollett, A., Miller, H., Olmsted, D. Validating computed grain boundary energies in fcc metals using the grain boundary character distribution. Acta Materialia. 59, 5250–5256 (2011) [24] Gruber, J., George, D., Kuprat, A., Rohrer, G., Rollett, A. Effect of anisotropic grain boundary properties on grain boundary plane distributions during grain growth. Scripta Materialia. 53, 351–355 (2005) [25] G. Pennock, M. Coleman, M. Drury, V. Randle, Grain boundary plane populations in minerals: the example of wet nacl after low strain deformation, Contributions to Mineralogy and Petrology 158 (1) (2009) 53–67. [26] Kobayashi, R., Warren, J., Carter, W. A continuum model of grain boundaries. Physica D: Nonlinear Phenomena. 140, 141–150 (2000) [27] D. Fan, L.-Q. Chen, Computer simulation of grain growth using a continuum field model, Acta Materialia 45 (2) (1997) 611–622. [28] L.-Q. Chen, W. Yang, Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: The grain-growth kinetics, Physical Review B 50 (21) (1994) 15752. [29] I. Steinbach, F. Pezzolla, A generalized field method for multiphase transformations using interface fields, Physica D: Nonlinear Phenomena 134 (4) (1999) 385–393. [30] Miyoshi, E., Takaki, T., Ohno, M., Shibuta, Y. Accuracy Evaluation of Phase-field Models for Grain Growth Simulation with Anisotropic Grain Boundary Properties. Isij International. pp. ISIJINT–2019 (2019) [31] H.-K. Kim, S. G. Kim, W. Dong, I. Steinbach, B.-J. Lee, Phase-field modeling for 3d grain growth based on a grain boundary energy database, Modelling and Simulation in Materials Science and Engineering 22 (3) (2014) 034004. [32] G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Annals of Physics 172 (1) (1986) 136–155. [33] G. McFadden, A. Wheeler, R. Braun, S. Coriell, R. Sekerka, Phase-field models for anisotropic interfaces, Physical Review E 48 (3) (1993) 2016. [34] G. Pennock, X. Zhang, C. Peach, C. Spiers, Microstructural study of reconsolidated salt, in: Proc. 6th Conf. Mech. Beh. Salt, SALTMECH6, Hannover, Germany, 2007. [35] Openphase. url, http://www.openphase.de. [36] H. J. Möller, Semiconductors for solar cells, Artech House Publishers, 1993. [37] Nestler, B., Garcke, H., Stinner, B. Multicomponent alloy solidification: phase-field modeling and simulations. Physical Review E. 71, 041609 (2005) [38] E. Miyoshi, T. Takaki, Validation of a novel higher-order multi-phase-field model for grain-growth simulations using anisotropic grain-boundary properties, Computational Materials Science 112 (2016) 44–51. [39] Moelans, N., Wendler, F., Nestler, B. Comparative study of two phase-field models for grain growth. Computational Materials Science. 46, 479–490 (2009) [40] W.-K. Burton, N. Cabrera, F. Frank, The growth of crystals and the equilibrium structure of their surfaces, Phil. Trans. R. Soc. Lond. A 243 (866) (1951) 299–358. [41] A. Chernov, The spiral growth of crystals, Physics-Uspekhi 4 (1) (1961) 116–148. [42] G. Wulff, On the question of speed of growth and dissolution of crystal surfaces, Z. Kristallogr 34 (1901) 449–530. [43] D. M. Saylor, A. Morawiec, G. S. Rohrer, The relative free energies of grain boundaries in magnesia as a function of five macroscopic parameters, Acta materialia 51 (13) (2003) 3675–3686. [44] P. Keblinski, S. Phillpot, D. Wolf, H. Gleiter, Amorphous structure of grain boundaries and grain junctions in nanocrystalline silicon by molecular-dynamics simulation, Acta materialia 45 (3) (1997) 987–998. [45] S. Ghanbarzadeh, M. Prodanovic, M. Hesse, Percolation and grain boundary wetting in anisotropic texturally equilibrated pore networks, Physical Review Letters 113 (2014) 048001. [46] N. Moelans, B. Blanpain, P. Wollants, Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems, Physical Review B 78 (2) (2008) 024113. [47] C. Herring, The use of classical macroscopic concepts in surface energy problems, in: Structure and properties of solid surfaces, 1953, p. 5. [48] F. Franc, On the kinematic theory of crystal growth and dissolution processes, Growth and Perfection of Crystal 411.

16

409 410 411 412 413 414 415

[49] [50] [51] [52]

Mtex toolbox. url, https://mtex-toolbox.github.io/. Mullins, W. Grain growth of uniform boundaries with scaling. Acta Materialia. 46, 6219–6226 (1998) Hillert, M. On the theory of normal and abnormal grain growth. Acta Metallurgica. 13, 227–238 (1965) C. Krill Iii, L.-Q. Chen, Computer simulation of 3-d grain growth using a phase-field model, Acta materialia 50 (12) (2002) 3059–3075. [53] Mackenzie, J. The distribution of rotation axes in a random aggregate of cubic crystals. Acta Metallurgica. 12, 223–225 (1964)

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416 417

418 419

Appendix A. Calculation of Equilibrium structure of a crystal, bounded by [111] and [100] facets The equilibrium shape of a crystal is determined by the minimum of the total surface energy σtotal at a constant volume of the crystal. σtotal = σ100 F100 + σ111 F111 ,

420 421

(A.1)

where σxyz is the surface energy of the facet per area and Fxyz is the associated surface area. The volume of the crystal is calculated as: V = Voctahedron − 6Vpyramid ,

422

The volume of octahedron is defined as

Voctahedron = A3 423

(A.2)

√

and the volume of the pyramid as

2 3

(A.3)

√ 1 3 2 L , (A.4) 2 3 where A is the edge length of the octahedron, formed by the 8 facets of [111]. L is the edge length of the pyramid edge whose base forms a [100] facet (see Fig. A.12). Insertion of Eq. (A.3) and A.4 into Eq. (A.2) yields: Vpyramid =

424 425 426

427

428

with f1 =

√

2 3 .

By normalizing a =

A √ 3 V

,l=

V = f1 (A3 − 3L3 ), L √ 3 V

, we obtain

1 = f1 (a3 − 3l3 )

and

a(l) = 429

(A.5)

The area of each type of facet is:

r 3

(A.6)

1 + 3l3 . f1

(A.7)

2

F111 = V 3 f2 (2a2 − 6l2 ),

(A.8)

2

430

with f2 =

√

F100 = V 3 6l2 .

(A.9)

3. The minimum of the grain boundary energy (A.1) can be found with e =

2 d dσtotal = σ111 V 3 (f2 (a2 − 3l2 ) + 3el2 ) dl dl Insertion of Eq. (A.7) and differentiation with respect to l yields "  − 31 # 1 e(l) = f2 1 − +3 f1 l 3

0=

431

432

σ100 σ111

(A.10)

(A.11)

and

l(e) =

(

f1

"

f2 f2 − e

18

3

#)

−3

(A.12)

433

within the validity limits 0
434 435 436

a 2

(A.13)

a (A.14) 0.953 = e( ) ≤ e ≤ f2 2 For e < 0.953, the geometric model shown here loses its validity. For e > f2 , the [100] facets disappear, i.e. the crystal is completely bounded by [111] Facets. Finally, the ratio of the area of [111] and [100] facets, as a function of the ratio e of the surface energies is F100 = F111

"

f2 f2 − e

2

#−1 

−3

3 f2



=

"

√

3 3−e

#−1


2 − 3

√

3.

A L

(a) Figure A.12: Octahedron where A is the octahedron edge length and L is the pyramid edge length.

19

(A.15)