- Email: [email protected]
Stability of small vacancy clusters in tungsten by molecular dynamics
Nuclear Inst. and Methods in Physics Research B 464 (2020) 56–59
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Stability of small vacancy clusters in tungsten by molecular dynamics a,⁎
b
Jan Fikar , Robin Schäublin a b
T
Central European Institute of Technology, Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, 616 00 Brno, Czech Republic Laboratory of Metal Physics and Technology, Department of Materials, ETH Zürich, 8093 Zürich, Switzerland
ARTICLE INFO
ABSTRACT
Keywords: Clusters Molecular dynamics Tungsten
The vacancies produced in collision cascades of irradiated metals form voids and vacancy clusters. The stability of vacancy clusters and voids is usually studied by kinetic Monte-Carlo methods. We investigated the stability of these vacancy defects at high temperatures using molecular dynamics and recent embedded-atom method potential. We confirm that the vacancy cluster dissociation is thermally activated. We have obtained dissociation energies and characteristic temperatures, both increasing with the number of vacancies and tending to saturate at 3.5 eV and 1200 K, respectively, for large vacancy clusters. Our results qualitatively agree with Monte-Carlo results, but predict somewhat smaller values for both the dissociation energy and characteristic temperature.
1. Introduction High energy irradiation introduces in lattice point defects in the form of Frenkel pairs. The majority of these defects mutually annihilates in the cascade cool-down phase. Usually it is assumed that surviving vacancies cluster together and form 3D voids in order to minimize their energy, while surviving interstitials tend to cluster into planar objects, which collapse into energetically favorable prismatic dislocation loops. Hereafter, we focus on tungsten, which has been chosen as a divertor material for ITER due to its high melting point, good thermal conductivity and good sputtering resistance [1]. In fact, the surviving vacancies have more possibilities than interstitials, as they can not only form voids, but also dislocation loops an planar clusters (platelets). It has been shown that compared to dislocation loops the voids in tungsten are energetically more favorable than 1/ 2〈111〉 and 〈100〉 loops up to critical size of 6 × 10 4 vacancies and 6 × 105 vacancies, respectively [2]. The critical sizes correspond to void diameters of 12 and 27 nm and loop diameters of 65 and 200 nm for 1/ 2〈111〉 and 〈100〉 orientations, respectively. The transmission electron microscopy (TEM) visibility limit for voids is about 1 nm. Small voids at the limit of TEM visibility have been observed in tungsten by El-Atwani et al. [3] formed in room temperature irradiation. Also Ferroni et al. [4] reports 1.5 nm voids in tungsten irradiated at 500 °C and then annealed at 800 °C for 1 h. Recently, firstprinciples investigation in combination with Monte-Carlo simulations [5] showed that nano-size voids play important role for understanding the origin of anomalous precipitation of rhenium in neutron-irradiated tungsten at high temperature (900 °C) [6]. Hu et al. [7] observed
⁎
tungsten irradiated at low temperature and low dose using positron annihilation spectroscopy (PAS) after one hour annealing at temperatures 400–1300 °C. They observed a significant decrease of vacancy cluster density from temperature 800 °C and higher. Similar decrease of void size and concentration was observed from 1123 K by PAS by M-F. Barthe et al. [8]. The high-temperature dissociation of di-vacancy in tungsten was studied by kinetic Monte-Carlo (KMC) by Heinola et al. [9]. KMC was also used in the study of dissociation of vacancy clusters by Mason et al. [10]. In both cases thermally activated dissociation is assumed and first-principles calculations were used as an input for KMC. The objective of this paper is to examine the dissociation of voids at high temperature by molecular dynamics (MD) in comparison to the results of KMC. 2. Computational details We consider a cubic simulation block with periodic boundary conditions in all directions. Because the voids, compared to the dislocation loops, create a short range elastic deformation field in the surroundings, the dimensions of the simulation block can be limited to 20 × 20 × 20 lattice units. Such small simulation block allows for extended MD simulation times. The first nearest neighbor (1NN) di-vacancy in tungsten is stable, although it has low binding energy, which is in agreement with first-principles calculations [10,9]. The most stable tri-vacancy is composed of vacancies located in two 1NN and one in 2NN positions e.g. (0, 0, 0), (1/2, 1/2, 1/2) and (−1/2, 1/2, 1/2) in the BCC lattice units. The most stable tetra-vacancy is created from the previously
Corresponding author. E-mail address: [email protected] (J. Fikar).
https://doi.org/10.1016/j.nimb.2019.11.044 Received 1 October 2019; Received in revised form 27 November 2019; Accepted 27 November 2019 0168-583X/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Inst. and Methods in Physics Research B 464 (2020) 56–59
J. Fikar and R. Schäublin
described tri-vacancy by adding a vacancy in twice 2NN and once in 1NN position e.g. (0, 1, 0) [10]. The larger voids are created by removing the atoms inside a sphere of a certain radius. A radius between 1NN and 2NN distance gives 9 vacancies, one between 2NN and 3NN gives 15 vacancies and so on. In total 11 vacancy clusters were simulated up to 113 vacancies, which correspond to a diameter of 1.51 nm. It should be noted that such simple constructed voids are not necessarily the configurations with lowest formation energies or largest stability, as their faceting, if any, is not optimized for that. The formation energy of spherical voids containing N vacancies can be easily fitted and predicted by simple formula as an average free surface energy multiplied by the inner void surface [2].
4 3.5
Ef /N [eV]
3
1
3
9
a02
1 (86 385
100
(2)
+ 128
110
+ 27
111
+ 144
211).
0e
Ed / kT .
150
Ed kln
1s
.
0
(5)
3. Results and discussion Formation energies divided by the number of vacancies Ef /N are reported in Fig. 1. The dashed line is given by Eq. (1) using an average surface energy of 0.230 eV/Å2 [2]. It appears that the small vacancy clusters with N < 9 are not correctly described by free surface energy Eq. (1), probably because they are closer to point defects. The scatter of formation energies is probably mainly caused by the random faceting introduced by intersecting the sphere with a BCC lattice. Indeed, in reality the voids surface will be faceted preferentially by 110 planes, where the surface energy exhibits a minimum [2]. Other factors explaining the scatter include the unconsidered edge between facets. Typical results of a single MD run are reported in Fig. 2 for 15 vacancy cluster at 1860 K and Fig. 3 for 27 vacancy cluster at 2063 K. In the case of the 27-vacancy cluster the formation energy cannot be reliably used as the dissociation criterion as it initially decreases and later oscillates. It is probably due to the simple way the void was created. It should be noted that we rarely observed the motion of the whole vacancy cluster. When it is observed, it seems that it is often related to rejoin events, which are also infrequent. A typical Arrhenius plot for a 4-vacancy cluster can be seen in Fig. 4. The dashed fitted line gives the dissociation energy Ed and the prefactor 0 . For all studied vacancy clusters we have found an Arrhenius behavior. The resulting dissociation energy Ed as a function of number of vacancies N is reported in Fig. 5a, it increases with N steeply at the beginning and then tends to saturate. The characteristic temperature T1s at which the cluster dissociates in 1 s as a function of the number of vacancies N is reported in Fig. 5b; it exhibits a similar behavior and is qualitatively comparable to the results obtained by Mason et al. [10]. Both the 65- and 89-vacancy clusters dissociate somehow more easily than expected. Lower stability of these clusters is probably due to our simple spherical construction of the voids, which are not necessarily the most stable voids. All the fitted parameters are summarized in Table 1. For di-vacancy Heinola et al. [9] reported a dissociation barrier of 1.70 eV while our value is lower at 1.19 eV. They also reported the prefactor, which gives a characteristic temperature of 660 K, while we found a lower value at 440 K. Mason et al. [10] reported for di-vacancy an even higher characteristic temperature of 780 K. They observe qualitatively the same behavior of the characteristic temperature as ours, with an increase with N and at higher N a tendency to saturate. They report characteristic temperature of 1300 K for 9 vacancy and 1600 K for 15 vacancy, while we have obtained 790 ± 250 K and 1050 ± 90 K, respectively. Mason et al. used KMC with simple Kang-Wienberg model [15] and vacancy migration barrier of 1.75 eV obtained by DFT. On the other
(3)
In our MD simulations we use the recent embedded-atom method (EAM) potentials of Mason, Nguyen-Manh and Becquart (MNB) [10], which is an improved Ackland–Thetford potential [12]. The simulation block is first relaxed using the conjugate gradient (CG) method in LAMMPS [13] and then it is isothermally evolving over time at a specific temperature in the range of 800–3000 K. The boundary conditions and temperature are kept constant by a dumped barostat and thermostat to accommodate the thermal expansion. We used MD with an integration time step t of 2 fs and integrate for 1000 t , i.e. 2 ps, which we call a step. After each step a copy of the sample is relaxed by CG to find the formation energy Ef and the positions of the vacancies. Their position is detected using common neighbor analysis (CNA). The average position of the atoms with a number of neighbors different from perfect BCC lattice gives the center of the void. The covariance matrix of positions of these atoms is also recorded, as its largest eigenvalue max is a good indication of the cluster dissociation. Then the MD is continued for another step [14]. If for 1000 successive steps (2 ns) the largest eigenvalue of the covariance matrix max is larger than a reasonably chosen value, the cluster is considered as dissociated, simulation is stopped and the dissociation time is recorded. Alternatively, for small vacancy clusters the cluster dissociation can be seen by a sudden increase in the formation energy, but this criterion is problematic for voids N > 15 , as their formation energy at high temperature initially decreases and then oscillates. If the max is larger than the chosen value for time shorter than 2 ns and then it becomes again lower, it means that the escaping vacancies returned to the void and we call it re-join effect. In such case the calculation is continued until the final dissociation is reached. In some cases, especially at low temperatures, times up to 2 ms were simulated. If the measured dissociation time was shorter than 100 ps it was discarded, as it is too close to the simulation start and the system is not yet in equilibrium. The dissociation time is supposed to be thermally activated and the dissociation energy Ed and time constant 0 are determined by fitting a line to the Arrhenius plot. Typically, many points are needed in the Arrhenius plot due to the stochastic nature of the dissociation process.
=
100
temperature at which the cluster is dissociated in 1 s.
The simple approximation of average free surface energy for cubic crystals using 100, 110, 111, and 211 was recently analytically calculated by Mason et al. [11]
=
50
Fig. 1. Formation energy Ef of the vacancy clusters in W divided by the number of vacancies N. The dashed line follows Eq. (1).
T1s =
.
0
N
The fitting parameter b1 can be calculated from the average free surface energy
b1 =
2 1.5
(1)
E void = b1 N 2/3.
2.5
(4)
Usually, the time prefactor 0 is difficult to interpret; often the characteristic temperature T1s is used instead, which is here the 57
Nuclear Inst. and Methods in Physics Research B 464 (2020) 56–59
J. Fikar and R. Schäublin
Fig. 2. The 15-vacancy cluster in W at 1860 K, its (a, c) formation energy and (b, d) the square root of the largest eigenvalue of covariance matrix. The formation energy and the square root of the largest eigenvalue of the covariance matrix can be reliably used to determine the final dissociation of the cluster. The rejoin events (some are highlighted) are all the spikes crossing the threshold and returning in less than 2 ns (typically they are 0.1 ns or shorter). The final dissociation event, detailed in (c) and (d), is over the threshold for longer than 2 ns.
hand, the MNB potential used here gives lower vacancy migration barrier of 1.52 eV. Consequently, we observe lower stability of vacancy clusters. Moreover, MD and KMC use completely different timescales and also different criterion to distinguish dissociation and re-join events. Mason et al. used as a dissociation criterion of 100 steps, but these KMC steps are variable and temperature dependent. For our simulated large voids we found that the dissociation energy saturates at 3.5 eV and the one second characteristic temperature saturates at 1200 K. Size of these voids corresponds to nanovoids at the limit of TEM resolution. From our calculations these nanovoids are stable at room temperature. If we extrapolate the obtained values to a time interval of usual annealing of one hour, then these nanovoids will lose a vacancy in one hour at 700 ± 200 °C. This corresponds well to a decrease in density of vacancy clusters after one hour anneal at 800 °C observed by PAS by Hu et al. [7]. Note that our randomly created voids are not the thermally most stable voids and that we observe the start of a void dissociation, not the complete void evaporation.
Fig. 4. The Arrhenius plot following Eq. (4) for a 4-vacancy cluster in W.
Fig. 3. The 27-vacancy cluster in W at 2063 K, its (a) formation energy and (b) the square root of the largest eigenvalue of covariance matrix. The formation energy cannot be reliably used to determine the final dissociation of the cluster, while the largest eigenvalue can. The re-join and final dissociation (longer than 2 ns) events are highlighted.
58
Nuclear Inst. and Methods in Physics Research B 464 (2020) 56–59
J. Fikar and R. Schäublin
Fig. 5. (a) Dissociation energy Ed of the vacancy cluster in W as a function of the number of its vacancies N and (b) the characteristic temperature T1s at which the cluster dissociates in 1 s as a function of number of vacancies N. The error bars correspond to standard deviations of fitted parameters. Note the slightly lower stability of 65- and 89-vacancy clusters.
Funding acquisition, Supervision, Validation, Writing - review & editing.
Table 1 Dissociation energy Ed , time prefactors 0 and characteristic temperatures T1s for vacancy clusters containing N vacancies and corresponding diameter. The standard deviations of the fitted Ed and T1s are indicated as well. N
diameter [Å]
Ed [eV]
2
3.9
1.19 ± 0.09
3 4 9
15
27 51 59 65
89
113
4.5 4.9 6.5
7.7
9.4
11.6
1.25 ± 0.13 1.98 ± 0.19 2.31 ± 0.65 3.18 ± 0.26 3.39 ± 0.88
12.1
3.24 ± 0.35 3.46 ± 0.74
13.9
1.12 ± 0.78
12.5
15.1
2.26 ± 0.48
2.95 ± 0.89
0
[s]
Acknowledgments
T1s [K]
2.04 × 10
14
1.18 × 10
14
3.43 × 10
16
1.49 × 10
15
5.52 × 10
16
1.65 × 10
16
2.73 × 10
15
1.45 × 10
15
2.62 × 10
13
5.67 × 10
11
2.66 × 10
14
This work was supported by the Grant Agency of the Czech Republic, Grant No. 16-24402S. This research was carried out under the project CEITEC 2020 (LQ1601) with financial support from the Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Program II. Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum, provided under the program “Projects of Large Research, Development, and Innovations Infrastructures” (CESNET LM2015042), is greatly appreciated. The Swiss National Science Foundation is thanked for financial support with Grant #200021-172934/1.
440 ± 40 450 ± 50 650 ± 70
790 ± 250 1050 ± 90
1080 ± 310 1120 ± 130 1180 ± 280 910 ± 210 550 ± 390
1100 ± 360
References 4. Conclusions
[1] M. Rieth, et al., J. Nucl. Mater. 417 (2011) 463–467. [2] J. Fikar, R. Schäublin, D.R. Mason, D. Nguen-Manh, Nucl. Mater. Energy 16 (2018) 60–65. [3] O. El-Atwani, E. Esquivel, M. Efe, E. Aydogan, Y. Wang, E. Martinez, S. Maloy, Acta Mater. 149 (2018) 206–219. [4] F. Ferroni, X. Yi, K. Arakawa, S.P. Fitzgerald, P.D. Edmondson, S.G. Roberts, Acta Mater. 90 (2015) 380–393. [5] J.S. Wrobel, D. Nguyen-Manh, K.J. Kurzydlowski, S.L. Dudarev, J. Phys.: Condens. Matter 29 (2017) 154403. [6] M. Klimenkov, U. Jantsch, M. Reith, H.C. Scheneider, D.J.H. Armstrong, J. Gibson, S.R. Roberts, Nucl. Mater. Energy 9 (2016) 480. [7] X. Hu, T. Koyanagi, M. Fukuda, Y. Katoh, L.L. Snead, B.D. Wirth, J. Nucl. Mater. 470 (2016) 278–289. [8] M.-F. Barthe, et al., J. Nucl. Mater. (submitted 2019). [9] K. Heinola, F. Djurabekova, T. Ahlgren, Nuc. Fusion 58 (2) (2017) 026004. [10] D.R. Mason, D. Nguyen-Manh, C.S. Becquart, J. Phys.: Condens. Matter 29 (2017) 505501. [11] D.R. Mason, C. Ngyuen-Manh, M. Marinica, R. Alexander, A.E. Sand, S.L. Dudarev, J. Appl. Phys. 126 (2019) 075112. [12] G.J. Ackland, R. Thetford, Philos. Mag. A 56 (1987) 15–30. [13] S. Plimpton, J. Comput. Phys. 117 (1995) 1–19. [14] J. Fikar, R. Schäublin, D.R. Mason, D. Nguen-Manh, Nucl. Instrum. Meth. B (submitted 2018). [15] H.C. Kang, W.H. Weinberg, J. Chem. Phys. 90 (1989) 2824–2830.
We have investigated the dissociation of vacancy clusters by MD and established a reliable dissociation criterion. We have found that the void dissociation at high temperatures is thermally activated. We have obtained dissociation energies and characteristic temperatures, which both increase with the number of vacancies and tend to saturate for larger vacancy clusters at 3.5 eV and 1200 K, respectively. The same qualitative behavior was found by others using KMC. Our values of dissociation energies and characteristic temperatures are somewhat smaller, as the single vacancy migration barrier is a little underestimated by the used EAM potential as compared to DFT value used by KMC. Thus, in reality we would expect the voids to have slightly higher stability. CRediT authorship contribution statement Jan Fikar: Conceptualization, Data curation, Funding acquisition, Methodology, Visualization, Writing - original draft. Robin Schäublin:
59
Contents lists available at ScienceDirect
Nuclear Inst. and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Stability of small vacancy clusters in tungsten by molecular dynamics a,⁎
b
Jan Fikar , Robin Schäublin a b
T
Central European Institute of Technology, Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, 616 00 Brno, Czech Republic Laboratory of Metal Physics and Technology, Department of Materials, ETH Zürich, 8093 Zürich, Switzerland
ARTICLE INFO
ABSTRACT
Keywords: Clusters Molecular dynamics Tungsten
The vacancies produced in collision cascades of irradiated metals form voids and vacancy clusters. The stability of vacancy clusters and voids is usually studied by kinetic Monte-Carlo methods. We investigated the stability of these vacancy defects at high temperatures using molecular dynamics and recent embedded-atom method potential. We confirm that the vacancy cluster dissociation is thermally activated. We have obtained dissociation energies and characteristic temperatures, both increasing with the number of vacancies and tending to saturate at 3.5 eV and 1200 K, respectively, for large vacancy clusters. Our results qualitatively agree with Monte-Carlo results, but predict somewhat smaller values for both the dissociation energy and characteristic temperature.
1. Introduction High energy irradiation introduces in lattice point defects in the form of Frenkel pairs. The majority of these defects mutually annihilates in the cascade cool-down phase. Usually it is assumed that surviving vacancies cluster together and form 3D voids in order to minimize their energy, while surviving interstitials tend to cluster into planar objects, which collapse into energetically favorable prismatic dislocation loops. Hereafter, we focus on tungsten, which has been chosen as a divertor material for ITER due to its high melting point, good thermal conductivity and good sputtering resistance [1]. In fact, the surviving vacancies have more possibilities than interstitials, as they can not only form voids, but also dislocation loops an planar clusters (platelets). It has been shown that compared to dislocation loops the voids in tungsten are energetically more favorable than 1/ 2〈111〉 and 〈100〉 loops up to critical size of 6 × 10 4 vacancies and 6 × 105 vacancies, respectively [2]. The critical sizes correspond to void diameters of 12 and 27 nm and loop diameters of 65 and 200 nm for 1/ 2〈111〉 and 〈100〉 orientations, respectively. The transmission electron microscopy (TEM) visibility limit for voids is about 1 nm. Small voids at the limit of TEM visibility have been observed in tungsten by El-Atwani et al. [3] formed in room temperature irradiation. Also Ferroni et al. [4] reports 1.5 nm voids in tungsten irradiated at 500 °C and then annealed at 800 °C for 1 h. Recently, firstprinciples investigation in combination with Monte-Carlo simulations [5] showed that nano-size voids play important role for understanding the origin of anomalous precipitation of rhenium in neutron-irradiated tungsten at high temperature (900 °C) [6]. Hu et al. [7] observed
⁎
tungsten irradiated at low temperature and low dose using positron annihilation spectroscopy (PAS) after one hour annealing at temperatures 400–1300 °C. They observed a significant decrease of vacancy cluster density from temperature 800 °C and higher. Similar decrease of void size and concentration was observed from 1123 K by PAS by M-F. Barthe et al. [8]. The high-temperature dissociation of di-vacancy in tungsten was studied by kinetic Monte-Carlo (KMC) by Heinola et al. [9]. KMC was also used in the study of dissociation of vacancy clusters by Mason et al. [10]. In both cases thermally activated dissociation is assumed and first-principles calculations were used as an input for KMC. The objective of this paper is to examine the dissociation of voids at high temperature by molecular dynamics (MD) in comparison to the results of KMC. 2. Computational details We consider a cubic simulation block with periodic boundary conditions in all directions. Because the voids, compared to the dislocation loops, create a short range elastic deformation field in the surroundings, the dimensions of the simulation block can be limited to 20 × 20 × 20 lattice units. Such small simulation block allows for extended MD simulation times. The first nearest neighbor (1NN) di-vacancy in tungsten is stable, although it has low binding energy, which is in agreement with first-principles calculations [10,9]. The most stable tri-vacancy is composed of vacancies located in two 1NN and one in 2NN positions e.g. (0, 0, 0), (1/2, 1/2, 1/2) and (−1/2, 1/2, 1/2) in the BCC lattice units. The most stable tetra-vacancy is created from the previously
Corresponding author. E-mail address: [email protected] (J. Fikar).
https://doi.org/10.1016/j.nimb.2019.11.044 Received 1 October 2019; Received in revised form 27 November 2019; Accepted 27 November 2019 0168-583X/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Inst. and Methods in Physics Research B 464 (2020) 56–59
J. Fikar and R. Schäublin
described tri-vacancy by adding a vacancy in twice 2NN and once in 1NN position e.g. (0, 1, 0) [10]. The larger voids are created by removing the atoms inside a sphere of a certain radius. A radius between 1NN and 2NN distance gives 9 vacancies, one between 2NN and 3NN gives 15 vacancies and so on. In total 11 vacancy clusters were simulated up to 113 vacancies, which correspond to a diameter of 1.51 nm. It should be noted that such simple constructed voids are not necessarily the configurations with lowest formation energies or largest stability, as their faceting, if any, is not optimized for that. The formation energy of spherical voids containing N vacancies can be easily fitted and predicted by simple formula as an average free surface energy multiplied by the inner void surface [2].
4 3.5
Ef /N [eV]
3
1
3
9
a02
1 (86 385
100
(2)
+ 128
110
+ 27
111
+ 144
211).
0e
Ed / kT .
150
Ed kln
1s
.
0
(5)
3. Results and discussion Formation energies divided by the number of vacancies Ef /N are reported in Fig. 1. The dashed line is given by Eq. (1) using an average surface energy of 0.230 eV/Å2 [2]. It appears that the small vacancy clusters with N < 9 are not correctly described by free surface energy Eq. (1), probably because they are closer to point defects. The scatter of formation energies is probably mainly caused by the random faceting introduced by intersecting the sphere with a BCC lattice. Indeed, in reality the voids surface will be faceted preferentially by 110 planes, where the surface energy exhibits a minimum [2]. Other factors explaining the scatter include the unconsidered edge between facets. Typical results of a single MD run are reported in Fig. 2 for 15 vacancy cluster at 1860 K and Fig. 3 for 27 vacancy cluster at 2063 K. In the case of the 27-vacancy cluster the formation energy cannot be reliably used as the dissociation criterion as it initially decreases and later oscillates. It is probably due to the simple way the void was created. It should be noted that we rarely observed the motion of the whole vacancy cluster. When it is observed, it seems that it is often related to rejoin events, which are also infrequent. A typical Arrhenius plot for a 4-vacancy cluster can be seen in Fig. 4. The dashed fitted line gives the dissociation energy Ed and the prefactor 0 . For all studied vacancy clusters we have found an Arrhenius behavior. The resulting dissociation energy Ed as a function of number of vacancies N is reported in Fig. 5a, it increases with N steeply at the beginning and then tends to saturate. The characteristic temperature T1s at which the cluster dissociates in 1 s as a function of the number of vacancies N is reported in Fig. 5b; it exhibits a similar behavior and is qualitatively comparable to the results obtained by Mason et al. [10]. Both the 65- and 89-vacancy clusters dissociate somehow more easily than expected. Lower stability of these clusters is probably due to our simple spherical construction of the voids, which are not necessarily the most stable voids. All the fitted parameters are summarized in Table 1. For di-vacancy Heinola et al. [9] reported a dissociation barrier of 1.70 eV while our value is lower at 1.19 eV. They also reported the prefactor, which gives a characteristic temperature of 660 K, while we found a lower value at 440 K. Mason et al. [10] reported for di-vacancy an even higher characteristic temperature of 780 K. They observe qualitatively the same behavior of the characteristic temperature as ours, with an increase with N and at higher N a tendency to saturate. They report characteristic temperature of 1300 K for 9 vacancy and 1600 K for 15 vacancy, while we have obtained 790 ± 250 K and 1050 ± 90 K, respectively. Mason et al. used KMC with simple Kang-Wienberg model [15] and vacancy migration barrier of 1.75 eV obtained by DFT. On the other
(3)
In our MD simulations we use the recent embedded-atom method (EAM) potentials of Mason, Nguyen-Manh and Becquart (MNB) [10], which is an improved Ackland–Thetford potential [12]. The simulation block is first relaxed using the conjugate gradient (CG) method in LAMMPS [13] and then it is isothermally evolving over time at a specific temperature in the range of 800–3000 K. The boundary conditions and temperature are kept constant by a dumped barostat and thermostat to accommodate the thermal expansion. We used MD with an integration time step t of 2 fs and integrate for 1000 t , i.e. 2 ps, which we call a step. After each step a copy of the sample is relaxed by CG to find the formation energy Ef and the positions of the vacancies. Their position is detected using common neighbor analysis (CNA). The average position of the atoms with a number of neighbors different from perfect BCC lattice gives the center of the void. The covariance matrix of positions of these atoms is also recorded, as its largest eigenvalue max is a good indication of the cluster dissociation. Then the MD is continued for another step [14]. If for 1000 successive steps (2 ns) the largest eigenvalue of the covariance matrix max is larger than a reasonably chosen value, the cluster is considered as dissociated, simulation is stopped and the dissociation time is recorded. Alternatively, for small vacancy clusters the cluster dissociation can be seen by a sudden increase in the formation energy, but this criterion is problematic for voids N > 15 , as their formation energy at high temperature initially decreases and then oscillates. If the max is larger than the chosen value for time shorter than 2 ns and then it becomes again lower, it means that the escaping vacancies returned to the void and we call it re-join effect. In such case the calculation is continued until the final dissociation is reached. In some cases, especially at low temperatures, times up to 2 ms were simulated. If the measured dissociation time was shorter than 100 ps it was discarded, as it is too close to the simulation start and the system is not yet in equilibrium. The dissociation time is supposed to be thermally activated and the dissociation energy Ed and time constant 0 are determined by fitting a line to the Arrhenius plot. Typically, many points are needed in the Arrhenius plot due to the stochastic nature of the dissociation process.
=
100
temperature at which the cluster is dissociated in 1 s.
The simple approximation of average free surface energy for cubic crystals using 100, 110, 111, and 211 was recently analytically calculated by Mason et al. [11]
=
50
Fig. 1. Formation energy Ef of the vacancy clusters in W divided by the number of vacancies N. The dashed line follows Eq. (1).
T1s =
.
0
N
The fitting parameter b1 can be calculated from the average free surface energy
b1 =
2 1.5
(1)
E void = b1 N 2/3.
2.5
(4)
Usually, the time prefactor 0 is difficult to interpret; often the characteristic temperature T1s is used instead, which is here the 57
Nuclear Inst. and Methods in Physics Research B 464 (2020) 56–59
J. Fikar and R. Schäublin
Fig. 2. The 15-vacancy cluster in W at 1860 K, its (a, c) formation energy and (b, d) the square root of the largest eigenvalue of covariance matrix. The formation energy and the square root of the largest eigenvalue of the covariance matrix can be reliably used to determine the final dissociation of the cluster. The rejoin events (some are highlighted) are all the spikes crossing the threshold and returning in less than 2 ns (typically they are 0.1 ns or shorter). The final dissociation event, detailed in (c) and (d), is over the threshold for longer than 2 ns.
hand, the MNB potential used here gives lower vacancy migration barrier of 1.52 eV. Consequently, we observe lower stability of vacancy clusters. Moreover, MD and KMC use completely different timescales and also different criterion to distinguish dissociation and re-join events. Mason et al. used as a dissociation criterion of 100 steps, but these KMC steps are variable and temperature dependent. For our simulated large voids we found that the dissociation energy saturates at 3.5 eV and the one second characteristic temperature saturates at 1200 K. Size of these voids corresponds to nanovoids at the limit of TEM resolution. From our calculations these nanovoids are stable at room temperature. If we extrapolate the obtained values to a time interval of usual annealing of one hour, then these nanovoids will lose a vacancy in one hour at 700 ± 200 °C. This corresponds well to a decrease in density of vacancy clusters after one hour anneal at 800 °C observed by PAS by Hu et al. [7]. Note that our randomly created voids are not the thermally most stable voids and that we observe the start of a void dissociation, not the complete void evaporation.
Fig. 4. The Arrhenius plot following Eq. (4) for a 4-vacancy cluster in W.
Fig. 3. The 27-vacancy cluster in W at 2063 K, its (a) formation energy and (b) the square root of the largest eigenvalue of covariance matrix. The formation energy cannot be reliably used to determine the final dissociation of the cluster, while the largest eigenvalue can. The re-join and final dissociation (longer than 2 ns) events are highlighted.
58
Nuclear Inst. and Methods in Physics Research B 464 (2020) 56–59
J. Fikar and R. Schäublin
Fig. 5. (a) Dissociation energy Ed of the vacancy cluster in W as a function of the number of its vacancies N and (b) the characteristic temperature T1s at which the cluster dissociates in 1 s as a function of number of vacancies N. The error bars correspond to standard deviations of fitted parameters. Note the slightly lower stability of 65- and 89-vacancy clusters.
Funding acquisition, Supervision, Validation, Writing - review & editing.
Table 1 Dissociation energy Ed , time prefactors 0 and characteristic temperatures T1s for vacancy clusters containing N vacancies and corresponding diameter. The standard deviations of the fitted Ed and T1s are indicated as well. N
diameter [Å]
Ed [eV]
2
3.9
1.19 ± 0.09
3 4 9
15
27 51 59 65
89
113
4.5 4.9 6.5
7.7
9.4
11.6
1.25 ± 0.13 1.98 ± 0.19 2.31 ± 0.65 3.18 ± 0.26 3.39 ± 0.88
12.1
3.24 ± 0.35 3.46 ± 0.74
13.9
1.12 ± 0.78
12.5
15.1
2.26 ± 0.48
2.95 ± 0.89
0
[s]
Acknowledgments
T1s [K]
2.04 × 10
14
1.18 × 10
14
3.43 × 10
16
1.49 × 10
15
5.52 × 10
16
1.65 × 10
16
2.73 × 10
15
1.45 × 10
15
2.62 × 10
13
5.67 × 10
11
2.66 × 10
14
This work was supported by the Grant Agency of the Czech Republic, Grant No. 16-24402S. This research was carried out under the project CEITEC 2020 (LQ1601) with financial support from the Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Program II. Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum, provided under the program “Projects of Large Research, Development, and Innovations Infrastructures” (CESNET LM2015042), is greatly appreciated. The Swiss National Science Foundation is thanked for financial support with Grant #200021-172934/1.
440 ± 40 450 ± 50 650 ± 70
790 ± 250 1050 ± 90
1080 ± 310 1120 ± 130 1180 ± 280 910 ± 210 550 ± 390
1100 ± 360
References 4. Conclusions
[1] M. Rieth, et al., J. Nucl. Mater. 417 (2011) 463–467. [2] J. Fikar, R. Schäublin, D.R. Mason, D. Nguen-Manh, Nucl. Mater. Energy 16 (2018) 60–65. [3] O. El-Atwani, E. Esquivel, M. Efe, E. Aydogan, Y. Wang, E. Martinez, S. Maloy, Acta Mater. 149 (2018) 206–219. [4] F. Ferroni, X. Yi, K. Arakawa, S.P. Fitzgerald, P.D. Edmondson, S.G. Roberts, Acta Mater. 90 (2015) 380–393. [5] J.S. Wrobel, D. Nguyen-Manh, K.J. Kurzydlowski, S.L. Dudarev, J. Phys.: Condens. Matter 29 (2017) 154403. [6] M. Klimenkov, U. Jantsch, M. Reith, H.C. Scheneider, D.J.H. Armstrong, J. Gibson, S.R. Roberts, Nucl. Mater. Energy 9 (2016) 480. [7] X. Hu, T. Koyanagi, M. Fukuda, Y. Katoh, L.L. Snead, B.D. Wirth, J. Nucl. Mater. 470 (2016) 278–289. [8] M.-F. Barthe, et al., J. Nucl. Mater. (submitted 2019). [9] K. Heinola, F. Djurabekova, T. Ahlgren, Nuc. Fusion 58 (2) (2017) 026004. [10] D.R. Mason, D. Nguyen-Manh, C.S. Becquart, J. Phys.: Condens. Matter 29 (2017) 505501. [11] D.R. Mason, C. Ngyuen-Manh, M. Marinica, R. Alexander, A.E. Sand, S.L. Dudarev, J. Appl. Phys. 126 (2019) 075112. [12] G.J. Ackland, R. Thetford, Philos. Mag. A 56 (1987) 15–30. [13] S. Plimpton, J. Comput. Phys. 117 (1995) 1–19. [14] J. Fikar, R. Schäublin, D.R. Mason, D. Nguen-Manh, Nucl. Instrum. Meth. B (submitted 2018). [15] H.C. Kang, W.H. Weinberg, J. Chem. Phys. 90 (1989) 2824–2830.
We have investigated the dissociation of vacancy clusters by MD and established a reliable dissociation criterion. We have found that the void dissociation at high temperatures is thermally activated. We have obtained dissociation energies and characteristic temperatures, which both increase with the number of vacancies and tend to saturate for larger vacancy clusters at 3.5 eV and 1200 K, respectively. The same qualitative behavior was found by others using KMC. Our values of dissociation energies and characteristic temperatures are somewhat smaller, as the single vacancy migration barrier is a little underestimated by the used EAM potential as compared to DFT value used by KMC. Thus, in reality we would expect the voids to have slightly higher stability. CRediT authorship contribution statement Jan Fikar: Conceptualization, Data curation, Funding acquisition, Methodology, Visualization, Writing - original draft. Robin Schäublin:
59