Atomistic simulations of the interaction between transmutation-produced Re and grain boundaries in tungsten

Computational Materials Science xxx (xxxx) xxxx

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Atomistic simulations of the interaction between transmutation-produced Re and grain boundaries in tungsten Lixia Liua, Yangchun Chenb, Ning Gaoc, Wangyu Hua, , Shifang Xiaob, Fei Gaod, Huiqiu Dengb, ⁎

⁎

a

College of Materials Science and Engineering, Hunan University, Changsha 410082, China School of Physics and Electronics, Hunan University, Changsha 410082, China c Institute of Frontier and Interdisciplinarity Science, and Key Laboratory of Particle Physics and Particle Irradiation (MOE), Shandong University, Qingdao, Shandong 266237, China d Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA b

ARTICLE INFO

ABSTRACT

Keywords: Grain boundary Molecular dynamics simulation Transmutation element Tungsten

High energy neutron irradiation not only causes the transmutation of tungsten (W), but also induces transmutation elements to segregate and precipitate at grain boundaries. In this work, the segregation of transmutationproduced rhenium (Re) and their clusters at a Σ5(130) grain boundary (GB) in bulk W has been investigated by using both molecular dynamics and molecular statics methods. It is found that, for single interstitial, Re atom diffuses from the bulk to the GB region via a three-dimensional rotation and migration in the form of an interstitial 〈1 1 1〉 Re-W mixed dumbbell, because the rotation and migration barriers of a Re-W dumbbell are both small. When Re atoms are absorbed by the GB, it is noteworthy that Re atoms tend to occupy the substitutional sites near the GB. The segregation energy of interstitial Re is ranged from −7.67 to −6.12 eV, and the effective interaction distance between the GB and Re interstitials is about 7 Å, which increases with increasing the cluster size. By modeling the uniaxial tensile test of W-GB structure containing Re substitution clusters, the fracture strength and elongation of the GB are significantly reduced. The present results provide important insights into the detailed mechanisms of Re segregation at GBs and its possible effects on the mechanical properties of Wbased materials.

1. Introduction As the candidates of plasma-facing materials (PFMs) in future fusion reactors, tungsten (W) and W alloys have plenty of advantages, such as high melting point, good thermal properties, low sputtering erosion and so on. Different from the normal conditions, the understanding of the response under irradiation of W and its alloys has been confirmed to be necessary and required for their applications in fusion reactors [1]. In addition to the displacement damages [2–5], some transmutation elements (TEs), such as Re, Os and Ta are also produced in W under high energy neutron irradiation. Gilbert and Sublet [6] reported that after service in the environment of fusion reactor power plant for 3 or 5 years, one of the TEs of W-materials is Re whose content is 2.59 at% or 3.80 at%, respectively. In experiments, for W-based alloys, the irradiation-induced precipitation in neutron-irradiated W alloys can result in the transmutation of W atoms to Re atoms so that the composition of the alloys may exceed the solid solution limit of Re in W, and then, the σ-phases (WRe) can be formed [7]; Fukuda et al. also analyzed

⁎

the electron diffraction results and confirmed that the precipitates in neutron irradiated tungsten are σ and χ phases [8]. These precipitates would lead to the obvious hardening and embrittlement of W materials and seriously affected the safety of their applications in future fusion reactors [9]. Therefore, to understand the Re-related radiation damages is important and necessary for further application of W and W-alloys in fusion reactors. In order to enhance the radiation damage tolerance of materials, grain boundaries (GBs) have been suggested as effective sinks for radiation-induced defects, such as interstitials and vacancies [10–12], because the energy of defects is reduced nearby the GB, resulting in trapping or annihilation processes [12–14]. Meanwhile, the high dose irradiation is also expected to induce the segregation and precipitation of TEs at GBs, influencing the properties of W materials containing GB structures (W-GBs). For example, the experimental study of He et al. suggested that Re atoms tend to migrate to GBs during the neutron irradiation [15]. In addition to the GB precipitation of defects in neutron-irradiated W-Re alloys, Takaaki et al. observed the Re-rich and Os-

Corresponding authors. E-mail addresses: [email protected] (W. Hu), [email protected] (H. Deng).

https://doi.org/10.1016/j.commatsci.2019.109412 Received 11 August 2019; Received in revised form 8 November 2019; Accepted 12 November 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.

Please cite this article as: Lixia Liu, et al., Computational Materials Science, https://doi.org/10.1016/j.commatsci.2019.109412

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GB energy

(b)

(a)

Tilt axis<100>

31.60

2.8

27.71 23.83

Grain1 Grain boundary Grain2

Z<100>(Å)

2.1

19.94 16.05

1.4

12.16 8.275

0.7

4.388

0.0

0

2

4

6

X<031>(Å)

8

10

J/m2

0.5000

Fig. 1. (a) Schematic geometry of GBs; (b) Gamma surface is calculated to search the minimal energy configuration of a GB by plotting GB energy as a function of the X and Z displacements.

methods is molecular dynamics (MD) simulation, which has been applied to understand the interaction between the nanostructures and GBs for a long time [10–14]. The simulation results showed that interstitials and vacancies can be trapped into GBs under the irradiation [11,12,19–21]. In austenitic alloys, Cr, Ni, Mo, P and Si were observed to segregate at GB regions [22–26]. With computational simulations, similar phenomena have also been observed in other systems, like V impurity in Ni [27], Cu precipitation in Fe [28], W self-interstitial atoms (SIAs) and vacancies in W [29], and P impurity in W [30]. These simulation results suggest that the GB or phase boundary plays an important role in facilitating the radiation tolerance of materials. In order to explore the interaction between Re and GB, in this work, the energetics and kinetics of Re segregation near the GB in W are investigated by using MD and molecular statics (MS) methods, calculating the total energy difference, segregation energy, diffusion barriers of Re nearby the GB. The effects of Re-rich clusters/precipitation on the strength of the GB in bcc W are also investigated. The present results are expected to provide important information to understand the detailed mechanisms of Re segregation at GBs and the possible effects on mechanical properties of W-based materials. 2. Model and methods 2.1. Interatomic potentials and models

Fig. 2. (a) The GB structure before relaxed. The atoms in the GB region are colored black, while the atoms in the bulk region are colored blue; (b) Stable structure of GB after relaxed. Here atoms are colored with their potential energies and the corresponding colors are shown in the color bar.

The Finnis-Sinclair (F-S) type interatomic potentials of W-W, Re-Re and W-Re developed and optimized by Chen et al. [31,32] in our group, and the latter optimizing potentials [32] have been used for the present simulations. The fitted potentials reproduce precisely the energies of various defects and the physical properties of the extended database obtained from DFT calculations [31,32]. For example, the most stable interstitial configurations of W and Re atoms are 〈1 1 1〉 W-W dumbbell and Re-W mixed dumbbell with the formation energy of 9.57 and 8.95 eV, respectively, as predicted by these potentials. The binding energy of 〈1 1 1〉 Re-W mixed dumbbell is 0.80 eV and the formation energy of a substitutional Re atom is 0.18 eV [32], which are in good agreement with ab initio calculations. The GB model is constructed based on the coincidence site lattice (CSL) method as demonstrated in Ref. [33], e.g. Σ5 and Σ3 which have been studied both experimentally [34] and theoretically [12,35–38]. In the present work, the symmetric tilt GB Σ5(130)/[100] in bcc W is selected as a model GB. To avoid the interaction between GBs, the distance between two GBs in the system is set to be larger than 12 nm, which is determined by pre-calculations and also confirmed in the previous studies [33].

rich precipitates along GBs in pure W and some of the matrix precipitates are also formed nearby the GB after neutron irradiation [16]. In simulations, Li et al. studied the mechanical properties of bcc W–based alloys and the effects of transmutation of W, and they found that Re addition decreased the ideal tensile strength of W [17]. Very recently, the DFT results of Zhang et al. found that the multiple Re atoms will form the planar structures, which have a strengthening effect on W GB and can be further enhanced by the aggregation of Re atoms [18]. Although many efforts have been carried out to investigate the behaviors between the transmutation-produced Re atoms and GBs in experiments, the segregation mechanisms of Re-rich clusters/precipitates nearby the GB in W are still unclear. Considering the difficulties to answer above questions from experimental viewpoints, computer simulations are expected to provide more details at atomic scale about how Re atoms move and segregate into the GB. One of the broadly used 2

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0.5

(a)

sub_Re

GB plane 4.0 Å

0

(b)

0.01

4.0 Å

GB plane

-2

0.00

6.64

E (eV)

E (eV)

0.0

0.01

0.32 -0.32

-4

7.08

8.0 Å

8.0 Å

7.0 Å

7.0 Å

absorbed by GB

-0.5

-6

GB region

-6.63 -7.07

-8

-1.0 -15

-10

-5

0

5

distance from GB(Å)

10

W Re

15

GB region

-15

-10

-5 0 5 distance from GB (Å)

10

15

0.30

1.6

(a)

Total binding energy (eV)

Binding energy of sub_Re-Re n(eV)

Fig. 3. The total energy difference of (a) a Re substitution and (b) a Re/W interstitial atom as a function of the distance from the GB.

0.24 0.18 0.12 0.06 0.00

3

6

9

Number (n) of Re atom

1.2

0.8

0.4

0.0

12

(b)

3

6

9

Number (n) of Re atoms

12

Fig. 4. (a) Binding energy of a substitution of Re atom to Ren cluster near the GB; (b) The total binding energy of the substitution of Re clusters near the GB.

Two systems with different sizes are constructed for different purposes in this work. The small system containing 66,200 atoms with a size of about 6.01 × 27.63 × 6.33 nm3 is used for calculating the energetic and kinetic properties of point defects related with Re atoms, and periodic boundary conditions (PBCs) have been applied in all three directions. However, the large system consisting of 198,780 atoms with a size of about 12.07 × 27.63 × 9.49 nm3 is used for simulating the uniaxial tensile test for a W-GB system, and the PBCs are applied only along the two directions parallel to the GB plane, but a fixed boundary condition applied in the direction perpendicular to the GB plane as suggested by previous studies [12–15,21,29,39]. Based on the convergence test of yield stress, strain rate was set to 0.001/ps. As shown in Fig. 1(a), a bi-crystal containing two GBs (because of the PBC applied along normal direction of the GB) is created by rotating one grain relative to another with the tilt axis located in the GB plane, and the detail has been reported in the previous work [39]. In order to establish possible minimal energy state after its construction, the gamma surface calculation is then performed to search the minimal energy state configuration of a GB by plotting the GB energy as a function of the X and Z displacements. Then the GB energy, GB , is calculated by: GB

=

EGB

magnitude of the GB energy. It is easy to obtain the related displacing vector of the lowest GB energy configuration from the gamma surface contours. The GB structure before relaxation is shown in Fig. 2(a), and common neighbor analysis (CNA) has been performed to identify the GB region and bulk region. In this work, the atomic configuration is viewed with OVITO software [40]. After MS relaxation, the stable structure of Σ5(130) symmetric tilt GB in bcc is shown in Fig. 2(b) with a GB energy of 2.34 J/m2 for W. Here, atoms are colored according to their potential energies as shown by the color bar. There is an expansion of 0.46 Å perpendicular to the GB plane comparing to the unrelaxed GB structure. The calculated properties include the total energy difference, segregation energies, and migration and rotation barriers of Re-related defects nearby the GB. From these energetic parameters, the interaction between irradiation induced defects and the GB is explored. 2.2. Computational details for MD and MS calculations All simulations are performed with LAMMPS code [41]. The conjugate gradient method and MD simulation are used to relax the system. The time step is 1 fs, and all the simulations are performed at either 0 or 300 K. As mentioned before, the most stable structure of Re or W interstitial atom (IA) in bulk W is 〈1 1 1〉 Re-W mixed dumbbell or W-W dumbbell, respectively. To introduce an interstitial, we firstly search for a perfect bcc unit cell near the GB region; then an initial configuration of the dumbbell along a 〈1 1 1〉 direction is created inside of this unit cell. The distance from the inserted interstitial to GB plane is set from

Eperfect 2AGB

(1)

where EGB is the total energy of the bi-crystal with the GB structure containing N atoms, Eperfect is the total energy of a perfect single crystal containing N atoms, and AGB is the interfacial area of the GB. Fig. 1(b) shows the gamma surface of Σ5(130) GB, where the colors indicate the 3

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0.16

Re-W in bulk

0.09

(b)

Re-W in bulk

Migration energy (eV)

Rotation energy (eV)

(a)

0.12

0.06

0.08

0.03

0.04 0.00

0.00 0

10

15

N

(c)

20

0 0.20

Re-W dumbbell

Migration energy (eV)

Rotation energy (eV)

0.16

5

0.12 0.08 0.04 0.00

5

10

N

(d)

15

20

Re-W dumbbell

0.15 0.10 0.05 0.00

0

5

10

15

Distance from GB ( Å)

0

20

5

10

15

Distance from GB ( Å)

20

Fig. 5. Diffusion of Re-W mixed dumbbell near the pristine GB. The Re interstitials of (a) rotation barriers and (b) migration barriers in bulk region; (c) Rotation diffusion barriers and (d) migration diffusion barriers as a function of distance from a pristine GB.

7

-8.30

Difussion barriers (eV)

6 5 4 3 2

-8.95

1 0 -1

1.55

A

1.38

1.38

B

C

D

Reaction coordinate Fig. 6. One path for substitution of Re atom diffusion near the GB. After the substitution of Re atom migrated from A to D, the barrier was 1.38 eV.

Fig. 7. (a) An example of single cell: the form of Re/W dumbbell clusters added, and the number (1–7) is the ordering insertion; (b) The profile of GB structure after absorbed 7 Re-W dumbbell clusters. The atoms in the GB region are colored black, while the atoms in the bulk region are colored blue. The Re atom is colored red and with a large size. The W atom absorbed by GB is colored green.

0 Å to 12 Å. On the other hand, a substitutional Re is created by replacing one W atom near the GB region (from 0 Å to 12 Å) in the system. The total energy difference ( E ) between the W-GB structure containing an impurity and the reference energy can be obtained by:

where Eperfect is the total energy of a perfect bcc W simulation cell with an impurity atom at a particular site. For the GB structure, after an impurity atom placing at a site , the simulation cell is firstly relaxed by conjugate gradient energy minimization process. According to Eq. (3), the negative energy value means that impurity atom prefers to segregate to the GB [27,44], while the positive energy value indicates that it prefers in the bulk region. In some references [13,14], this energy is also called the binding or segregation energy, which suggests that the system energy reduces sharply when an impurity atom moves from the bulk region to a particular site nearby the GB.

(2)

E = Etot Ebulk

where Etot is the total energy of the W-GB structure containing one impurity atom; the reference energy Ebulk is the total energy of the WGB structure with one impurity atom, which occupies a particular site in bulk region and is far away (> 8 Å) from the GB in order to avoid any interaction with the GB. The segregation energy Eseg for an impurity atom at the interstitial/ substitution site is calculated as follows [42,43]:

Eseg = (Etot

EGB )

(Eperfect

Eperfect )

(3) 4

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(a)

W-W W-Re

Binding energy of IACn-GB (eV)

Binding energy of IA-IACn (eV)

6.0

4.5

3.0

1.5

Relative total energy (eV)

0.0

0

0

3

6

9

Size of cluster

12

50 (b)

W W-GB WRe-GB

40 30 20 10 0

0

2

4

6

8

10

Size of cluster

(c) Re-W dumbbells

-10 -20

2 3 4 5 6 7

-30 -40 0

4

8

12

Distance from GB ( Å)

16

Fig. 8. (a) Binding energy of an IA of Re/W with IACn in bulk W; (b) Binding energy of IACn with GB; (c) The relative total energy of the system with a different size of Re-W dumbbell cluster as a function of the distance from the GB plane.

The relative total energy of the system containing an interstitial atom cluster (IAC) with n interstitials (IACn) , namely dumbbell clusters, is defined as:

3. Results and discussion

rel tot E IAC = E IAC n n

To study the Re segregation, the total energy difference ( E ) of the W-GB structure containing one impurity atom, a substitution Re or a Re/W interstitial, nearby the Σ5(130) GB is calculated as a function of distance from impurity to the GB plane. As demonstrated in Fig. 3, it is clear that there are two different regions nearby the GB as shown by E . The first region is located with the distance more than 4.0 Å (or 7.0 Å) for substitution Re (or interstitial Re), in which the E is almost zero, indicating the driven force for segregation of an impurity to the GB can be neglected. In the second region, where the distance is within 4.0 Å (or 7.0 Å) from the GB, E decreases with a maximum value up to around 0.56 eV (or 7.83 eV), which indicates that impurities are expected to be absorbed by the GB. Thus, the effective GB region for substitution and interstitial Re is about 4.0 Å and 7.0 Å from the GB plane, respectively. All these results also indicate that Σ5(130) GB has a stronger absorption and a wider denuded zone for interstitial Re than to substitution Re atoms. Fig. 3 also indicates that the lowest energy point for a substitutional Re or interstitial Re is located at the GB region. Thus, it is energetically favorable for a substitution Re or interstitial Re to be absorbed by the GB region. Different from the case of substitutional Re (as shown in Fig. 3(a)), there are two energy states of an interstitial Re atom during its migration to the GB in Fig. 3(b). When the Re atom is located at the bulk region, it takes the form of 〈1 1 1〉 Re-W mixed dumbbell. When the ReW dumbbell diffuses to the region within 7.0 Å from the GB plane, the Re-W dumbbell would decompose and Re would take the form of substitutional site near the GB and the left W will take the form of a 〈1 1 1〉 W-W dumbbell, which would be trapped into the GB, resulting in the energy decrease of around 6.64 eV in average. The left Re-

3.1. Energetics of Re atom segregation near a GB

(4)

EGB

tot is the total energy of the W-GB structure system containing where E IAC n an IACn . The formation energy of an IACn in the bulk region is defined as [45]:

f Ebulk , IACn = Ebulk , IACn

EGB

nEc

(5)

where Ebulk, IACn is the total energy of the system with an IACn in bulk region, and Ec is the energy of per atom in a perfect bcc lattice, which is −8.90 and −8.03 eV for W and Re, correspondingly. The binding energy of one IA to an IACn in bulk region is defined as [45]: b Ebulk , IA

IACn

=

f f f Ebulk , IACn + 1 + Ebulk , IACn + Ebulk, IA

(6)

For the IACn nearby the GB, its formation energy is defined as: f EGB , IACn = EGB, IACn

EGB

nEc

(7)

where EGB, IACn is the total energy of the GB system with an IACn at a particular site nearby the GB. The binding energy of the IACn to the GB is defined as: b E IAC n

GB

f = Ebulk , IACn

f EGB , IACn

(8)

5

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b ) of a Ren near the GB region was The total binding energy (Etot defined as: b Etot = EGB, Ren

rand Ebulk , Ren

(10)

rand where Ebulk , Ren is the total energy of the GB system with n Re atoms randomly distributed in the bulk region without any interaction with the GB. Fig. 4(a) shows that the substitution Re atoms prefer to aggregate nearby the GB as indicated by a positive binding energy of a substitution Re to the Ren cluster. Also, the total binding energy of substitution Re clusters near the GB is positive and increases with increasing the number of Re atoms, as demonstrated Fig. 4(b). These results implicate that Re atoms near the GB have a trend to aggregate, and the lowest energy configuration of the cluster is a plate-like cube, which is in good agreement with the recent DFT calculations [18]. This result may be important to understand the following tensile tests with the configuration of substitution Re clusters at the GBs.

3.2. Kinetics of Re atom segregation near the GB The kinetic process for one Re atom to the GB has been studied by following its migration, through a vacancy assisted mechanism (ReVac) or a Re-W mixed dumbbell assisted mechanism, at different temperatures. The simulations have clearly demonstrated that the Re-Vac remains immobile at 1200, 1500 and 1800 K even if the simulation time is extended up to 2 ns; but at higher temperature, e.g., > 2000 K, it starts to move with a low speed. Different from vacancy assisted mechanism, it is of interest to find that the Re-W mixed dumbbell diffuses easily even at 300 K by 3-dimensional (3-D) rotation in the bulk towards the GB region until it is absorbed by the GB. Using the nudged-elastic-band (NEB) method [47], the energy barrier of Re migration through vacancy-mediated way in W bulk is determined to be 1.57 eV that agrees with the previous work in a perfect W [32]. However, the rotation and migration barriers of Re interstitials are also separately determined to be 0.09 eV and 0.15 eV, respectively, as shown in Fig. 5(a) and (b). For the migration barrier, two nearest-neighbor dumbbells are used as the initial and final configurations, respectively, in which 21 images are inserted. For the rotation barrier, the Re-W 〈1 1 1〉 dumbbells in the initial and final configurations locate at the same position but along with two different orientations. We calculated all possible final orientations of Re-W 〈1 1 1〉 dumbbell for the initial sates and the one with the lowest energy is considered as the rotation energy. The large energy difference between these two diffusion mechanisms indicates that the Re atom prefers to migrate 3-D via the rotation and migration motion in the form of an interstitial 〈1 1 1〉 Re-W dumbbell, rather than through the vacancy assisted mechanism. The behavior of W atom in present work is consistent with the previous study [32], which reported that W atom diffuses by one-dimensional (1-D) migration in the form of a 〈1 1 1〉 W-W dumbbell at low temperatures. Different from the diffusion in bulk material, the rotation and migration barriers of a Re interstitial towards the GB have also been calculated with NEB method, and the results are displayed in Fig. 5(c) and (d), which suggest that the rotation and migration barriers for a Re-W interstitial are significantly reduced when it is close to the GB plane. The rotation barrier is increased firstly by 0.04 eV, and then reduced to zero when it is absorbed by the GB. The migration barrier is also decreased in a similar way, from the initial value of about 0.15 eV to the final zero. These results clearly confirm that once the Re-W dumbbell is located within the effective interaction distance to the GB plane, the stress filed of the GB would affect its kinetic process by decreasing the diffusion energy barrier, resulting in its quick absorption by the GB, similar to the case of W interstitials [29]. The energy barrier of Re migration to the GB through vacancy assisted mechanism has also been calculated. One of the lowest energy paths is given in Fig. 6. From these results, it is clear that a substitution Re atom has to overcome a high barrier of 1.38 eV to move towards the

Fig. 9. The schematic geometry of tensile (a) model-I : Re clusters located nearby the GB; and (b) model-II: Re clusters located in GB region. The atoms structure is shown in the right. The atoms near the GB region are colored black, while the atoms in the bulk region are colored blue. The Re atom is colored red and with a large size.

vacancy cluster would also interact with GB once it diffuses into the distance of around 4.0 Å to the GB plane. Above segregation process is different from W interstitial/other impurity atoms and the absorption of Re atoms by the GBs has been confirmed by experimental observations [15,16], which is also in accordance with the recently DFT results [18]. It should also be noted that during the segregation of Re-W interstitials, the Re-W dumbbell would rotate into one of the 〈1 1 1〉 directions before its absorption, and the angle with the GB normal is 43.09°. The similar rotation has also been observed in W that an interstitial with certain orientation will only spontaneously segregate into the Σ5(130) GB [29]. The reason may be explained from the anisotropic stress field of the GB, which has been investigated by Samaras et al [46]. They found that, as GBs in nanocrystalline structures generally contain alternatively regions of high compression and high dilatation, the interstitial clusters in Ni GB system are first attracted by the GB with tensile pressure, but when it arrives within a few atomic layers near the GB, the cluster sees the strong pressure gradients in the GB and changes its direction in order to arrive at a region of tensile pressure. For fully exploring the segregation behavior of substitution Re atoms, we also calculated the binding energy of the substitution Re clusters (Ren ) nearby the GB. The binding energy of one substitution Re atom to a Ren near the GB was defined as: b ERe

Ren

=

bulk , Re EGB, Ren + 1 + EGB , Ren

(9)

where EGB, Ren +1 is the total energy of the GB system with n + 1 substitution Re atoms and n + 1 vacancies, at site near the GB; and bulk ,Re EGB , Ren is the total energy of the GB system with n Re substitutions near the GB region and one Re substitution in the bulk region without any interaction with the GB. 6

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(a)

clean GB Re 2Å Re 4Å Re 6Å σ phases

30

40 (b)

Nearby the GB at 0K

30

Stress (GPa)

Stress (GPa)

40

20

20

10

10

0

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0

strain (c)

clean GB Re 2 Re 4 Re 6

30

In GB region at 0K

40

Stress (GPa)

Stress (GPa)

40

clean GB Nearby the GB at 300K Re 2Å Re 4Å Re 6Å σ phases

20 10

(d)

30

0.1

0.2

strain

clean GB Re 2 Re 4 Re 6

0.3

0.4

0.5

In GB region at 300K

20 10 0

0 0.0

0.1

0.2

0.3

strain

0.4

0.5

0.0

0.6

0.1

0.2

strain

0.3

0.4

0.5

Fig. 10. Stress-strain curve corresponding to Σ5(130) GB: in model-I included Re clusters/precipitation nearby the GB with different size at (a) 0 K and (b) 300 K; in model-II included Re clusters in GB region with different size at (c) 0 K and (d) 300 K.

3.3. Re cluster segregation behavior near the GB

Table 1 Model-I : The fracture strength (GPa) and elongation of Σ5(130) of the Re clusters near the GB. GB Structure

Clean GB GB-Re_2 Å GB-Re_4 Å GB-Re_6 Å GB-σ phases

0K

To further understand the segregation behavior of the IACn near GB region, the energies of the system containing 〈1 1 1〉 dumbbell clusters are calculated. Considering IAC2 ~IAC7 (or IAC2 ~ IAC11), which are composed of parallel compact 〈1 1 1〉 dumbbells, as shown in Fig. 7(a). It is noted that the dislocation loop is observed after MS relaxation with more than seven parallel 〈1 1 1〉 Re-W mixed dumbbells in the cluster, which is similar to the results of W-W dumbbells in the bulk. During the MD simulations at 300 and 900 K, a single Re interstitial diffuses along the 〈1 1 1〉 direction in 3-D model, while a Re IA-cluster shows immobile as pinned by Re atoms even at high temperatures, as observed in the previous work [32]. When a Re IA-cluster is formed near the GB region (for example, IAC2 and IAC7 are 7 Å and 12 Å from the GB plane, respectively), it will be absorbed by the GB. The absorption process of Re IA-cluster by the GB is similar to that of a single Re interstitial, that is, the decomposition of Re-W mixed dumbbell occurs with the Re atoms occupying substitutional sites nearby the GB and the W atoms quickly being absorbed by GB, as demonstrated in Fig. 7(b). As shown in Fig. 8(a), the binding energy of IA to an IACn in the bulk, the large binding energy indicates that the Re/W IACn is tightly bound together in the bulk; Fig. 8(b) demonstrates similar results of the binding energy between one Re/W IACn and the GB. The relative total energy of the system with defect clusters is plotted in Fig. 8(c), which shows that the relative total energy decreases with increasing the size of clusters, and the distance effect from the GB increases with increasing the cluster size. For example, the interaction distance increases from 7 Å to 12 Å with the cluster size from 2 to 7 atoms, respectively. The above results indicate that Re interstitials have a tendency to aggregate, and the GB is a strong sink for large interstitial defects too. It has been found that the process of a single Re interstitial is similar when

300 K

Fracture strength

Elongation

Fracture strength

Elongation

40.34 38.16 22.01 18.93 22.02

0.42 0.41 0.33 0.12 0.10

36.73 34.74 19.29 16.86 20.96

0.40 0.39 0.31 0.13 0.10

Table 2 Model- II: The fracture strength (GPa) and elongation of Σ5(130) of the Re clusters in GB region. GB Structure

Clean GB GB-Re_2 Å GB-Re_4 Å GB-Re_6 Å

0K

300 K

Fracture strength

Elongation

Fracture strength

Elongation

40.34 27.79 18.77 18.81

0.42 0.36 0.06 0.06

36.73 25.28 17.63 17.66

0.40 0.34 0.06 0.06

GB, which may occur only at high temperatures (> 2000 K), as indicated by the above simulations. Based on these results, a conclusion can be made that GBs are the sinks for Re atoms through interstitial or vacancy assisted mechanisms, although the energy barriers and kinetic absorption processes are different, which also strongly depends on temperatures. 7

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Fig. 11. Snapshots for the structure of bi-crystal under the process of tensile tested at 0 K (for containing Re 2 Å near the GB): (a) model-I , Re clusters near the GB; (a1) the details of the GB and clusters structure in model-I ; (b) model-II, Re clusters in GB region; (c) the clean GB. The atoms in GB region are colored black, while the atoms in the grain (bcc) are colored blue, and the phase transformed fcc atoms are colored green, and the Re atom expressed in a large size.

a Re IA-cluster is absorbed by the GB, and the effective interaction distance between the Re-W interstitial cluster and the GB increases with the increase of clusters size.

but it is placed symmetrically at the GB plane. When the stress is applied, stress-strain curves of the above two models are determined and shown in Fig. 10. The stress-strain curves correspond to the deformation of Σ5(130) GB containing different sizes of substitutional Re clusters and σ phase. The results of model-I and II obtained at 0 K and 300 K are shown in Fig. 10(a–d), respectively. The tensile strength and elongation of Σ5(130) GB are listed in Table 1 for model-I and Table 2 for model-II, respectively. All these results indicate that Re clusters drastically decrease the fracture strength and elongation, as compared to the results of the clean GB. Bain strain path induced by the tensile along the y-direction (i.e., [1 3 0] direction) results in the phase transformation from bcc to fcc [50], which is similar that observed in Fe, as explained in [39,40]. As displayed in Fig. 11, which shows several snapshots of the tensile test at 0 K for two models, the induced phase transformation in two models occurs with = 0.06; as explained in Refs. [39,51], noticeable shear component of stress ( xy ) and slide of the substitutional Re cluster along the x-direction appear when transited from = 0.06 to = 0.40 in the model-I , while from = 0.06 to = 0.35 in model-II. The fracture occurs in the region of the defect clusters, while occurs in the GB region at the clean GB structure. In the model-I , Re clusters located nearby the GB (Fig. 11(a)), it fractures nearby the GB region (the location of the defect clusters) at = 0.41. Fig. 11(a-1) exhibits the details of the GB and cluster structure evolution in the model-I . In the same way, as shown in Fig. 11(b), the fracture occurs close to the location of the defect clusters in GB region at = 0.36 for the model-II. The result of the clean GB is shown in Fig. 11(c), which fractures in the GB region. The similar results are observed at room temperature (300 K). These results manifest that the fracture strength and percentage of elongation of the GB containing Re clusters reduce significantly, as compared to the clean GB. It seems that the Re clusters have strong

3.4. Effects of Re clusters on strength of the GB Based on these results, the effects of Re clusters/precipitation on the strength of the GB in bcc W have been further studied by modeling a uniaxial tensile test of W crystal containing different sizes of Re substitution clusters within the GB at different temperatures, comparing with the clean GB. The effects of different sizes of Re clusters located at two different positions (nearby the GB or inside GB region) are studied. The clusters located nearby or inside the GB region are defined as model-I or model-II, respectively, as shown by Fig. 9. According to previous results (Section 3.1), the substitutional Re clusters, formed by clustering of substitutional Re atoms or by decomposing the Re-W interstitial clusters, form the plate-like shape near or in the GB region. It has also been confirmed by the very recent DFT calculations [18] that the multiple Re atoms will form a planar structure, and the segregation of Re in W-GB system is in accordance with previous experimental results [48,49]. In this work, the plate-like defect clusters are located at the center of the box nearby or in the GB plane, and each defect cluster has the same length and width (20 Å) along two directions paralleled to the GB plane. The size of the selected cluster depends on the thickness of plate-like cluster, which comes from the effective interaction range (about 7 Å) between Re atoms and GB region. In model-I , as shown in Fig. 9(a), Re interstitials are more likely to located near the GB region, three different thickness sizes (2 Å, 4 Å and 6 Å) are considered here. For comparison, the effect of σ phase (containing 114 Re atoms) of about 11 Å from the GB plane has also studied. The size of the Re clusters in the model- II is the same as the model-I , 8

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effect on impeding the plastic deformation of GBs, and may cause brittleness of the material, which is in good agreement with the previous first-principles work of Li et al [17].

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4. Conclusions In this paper, the transmutation-produced Re segregation at Σ5(130) GB in bcc W was investigated by using MS and MD methods. The interaction between Re atoms and the GB in W was studied by calculating their energetic and kinetic properties nearby the GB. When Re segregated near the GB, we further investigated the effects of Re clusters/precipitation on the strength of the GB by modeling a uniaxial tensile test of the W-GB structure containing Re clusters. The results suggest that the GB served as a sink for Re/W interstitials by reducing their energies and diffusion energy barriers close the GB. Due to the low rotation (0.09 eV) and migration diffusion barriers (0.15 eV) of Re interstitials in bulk region, the Re atom from bulk region to the GB region tended to be a 3-D migration with rotation and diffusion in the form of 〈1 1 1〉 Re-W dumbbells. Within a range of 7 Å from the GB, the Re atom can be absorbed by the GB instantly, which is similar to W interstitials. It is noteworthy that Re atoms were more likely to locate at substitutional sites nearby the GB. The segregation energy of an interstitial Re is generally large (from −7.67 eV to −6.12 eV), and its effective interaction distance with the GB is about 7 Å, which increases with increasing the cluster size. The comparison of stress-strain curves between the GB containing Re clusters/precipitation and a clean GB structure implies that the fracture strength and the elongation of the GB containing Re clusters reduced significantly. These results suggest that the Re clusters have strong effect on impeding the plastic deformation of the GBs, and may cause the embrittlement of materials. The present results provide important knowledge to understand the detailed mechanisms of Re segregation at GBs and the possible effects on mechanical properties of W-based materials. CRediT authorship contribution statement Lixia Liu: Investigation, Methodology, Formal analysis, Validation, Writing - original draft, Writing - review & editing. Yangchun Chen: Formal analysis, Methodology, Validation. Ning Gao: Conceptualization, Writing - review & editing. Wangyu Hu: Funding acquisition, Conceptualization, Formal analysis, Methodology. Shifang Xiao: Formal analysis, Methodology, Validation. Fei Gao: Conceptualization, Formal analysis, Writing - review & editing. Huiqiu Deng: Conceptualization, Funding acquisition, Resources, Supervision, Writing - review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was financially supported by the National Key R&D Program of China (2018YFB0704001), the National MCF Energy R&D Program of China (2018YFE0308101) and the National Natural Science Foundation of China (51771073). References [1] R. Causey, K. Wilson, T. Venhaus, et al., Tritium retention in tungsten exposed to intense fluxes of 100 eV tritons, J. Nucl. Mater. 266 (1999) 467–471. [2] B.D. Wirth, How does radiation damage materials, Science 318 (2007) 923–924. [3] K. Nordlund, J. Keinonen, M. Ghaly, et al., Coherent displacement of atoms during ion irradiation, Nature 398 (1999) 49. [4] C.Y. Lu, K. Jin, L.K. Béland, et al., Direct observation of defect range and evolution

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