Anisotropic diffusion of He in titanium: A molecular dynamics study

Solid State Communications 148 (2008) 178–181

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Anisotropic diffusion of He in titanium: A molecular dynamics study Min Chen a,b,∗ , Qing Hou a,b , Jun Wang a,b , Tieying Sun c , Xinggui Long c , Shunzhong Luo c a

Key Laboratory for Radiation Physics and Technology of Ministry of Education, China

b

Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610064, China

c

Institute of Nuclear Physics and Chemistry China, Academy of Engineering Physics, Mianyang 621900, China

article

info

Article history: Received 4 August 2008 Accepted 25 August 2008 by A.H. MacDonald Available online 29 August 2008 PACS: 61.72.-y 66.30.-h

a b s t r a c t We use the molecular dynamics to study the migration of He atom, dimer and trimer in Ti. The migration features of these three He species are shown in this paper. It is observed that the Arrhenius relation can well describe their diffusion. However, the diffusion is significant anisotropic. This anisotropy is represented by that the prefactor of the diffusion coefficients are quite different for these He species diffusing in different directions, but the activation energies are the same. Another counterintuitive observation is that He-dimer migrate more quickly than single He atom does. The results emphasize the importance of dynamics simulations in predicting diffusion behavior of He in metals. © 2008 Elsevier Ltd. All rights reserved.

Keywords: A. Ti C. Molecular dynamics simulation D. Helium diffusion in solids

1. Introduction The diffusion and aggregation of He in metals is known to result in the formation of He bubbles or platelets [1–4], and consequently to induce the change of physical properties of the metals [5]. Because it is of importance both in view of techniques [5–7] as well as fundamental researches [8], the behavior of He in metals has drawn considerable attentions [9–13]. Since diffusion is essential for the morphology evolution of He distribution, here we focus on the migration of He atom, He-dimer and He-trimer. For the diffusion of large He cluster (He bubble), a theory that can be deduced from Nernst–Einstein relation [14] has been demonstrated to be useful in explaining experimental observations [9,15]. Its applicability to small He cluster diffusion is questionable. For the diffusion of small He or He-vacancy cluster, the most relevant is the energy state of configurations [10–13]. However, investigations for adatom diffusion on surface have demonstrated that the entropic contribution to the diffusion can play important role. However, considering energetic contribution alone is not sufficient to determine the predominant mechanism in the diffusion [16–18]. This result, though it is found for adatom diffusion on surface, is instructive and motivates our dynamics investigation on the diffusion of Hen in bulk metals. The aim of the present paper is first to gain a picture description on the migration

∗ Corresponding author at: Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610064, China. Tel.: +86 28 85412104; fax: +86 28 85412374. E-mail addresses: [email protected] (M. Chen), [email protected] (Q. Hou). 0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.08.025

process of Hen (n = 1, 2, 3) in Ti and then to draw out the diffusion features as well as the roles of activation energy and prefactor, especially for the useful information that can be used for higher scale simulations such as kinetic Monte Carlo simulations [19]. Here, the choice of Ti as substrate is due to its important applications in nuclear materials. 2. Computational methods For our dynamics study, ab initio calculation is too demanding in computer capacity to be practical. Thus, we use the molecular dynamics (MD) for the study. Our implementation of the MD is outlined following. An empirical tight-binding potential, that was produced by Cleri and Rosato [20], is employed to describe the Ti–Ti interaction, and a Lennard-Jones potential is applied for the He–He interaction [21]. As for the He–Ti interaction, a pair-wise potential that takes the form of Lennard-Jones potential with its parameters determined by ab initial data is adopted. This set of potentials has been used in our recent studies [22,23], where the implementation is described in detail. The simulation box is 24 monolayers thick and each 1ayer contains 20 × 20 atoms arranged in hcp (001) planes. The x, y, zaxsis of the coordinate system are in the crystal directions [100], [120] and [001] respectively and the periodic boundary conditions are imposed along these three directions. Then, following what was described in our previous work [23], a He atom or He cluster is introduced around the center of the simulation box. Before simulating the diffusion process, the box is thermalized and

M. Chen et al. / Solid State Communications 148 (2008) 178–181

Fig. 1. The migration trajectory of a single He atom in Ti at temperature 333 K. The points denote the instance positions of the He atom. For clarity, the Ti atoms are hidden.

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Fig. 2. The migration trajectory of a He-dimer in Ti at temperature 333 K.

relaxed to reach thermal-equilibrium at given temperature. After this, the diffusion of Hen is simulated by relaxing the box further. Depending on the temperature, the number of helium atoms in the titanium, the simulation time is for 300 ps–2.5 ns. The time step of integral is 1fs at all times. In above processes, an electron-phonon coupling model [24], in which the electron gas is considered as a thermostat at constant temperature and the electron-phonon coupling acts as a friction force is applied to maintain the system to be canonical at the given temperature. This method is used in our resent research [22,23], and more detailed description is in our previous work [24]. By the approach described above, simulations are performed at a few temperatures between 333 K and 667 K separately. 3. Results and discussion 3.1. Diffusion of Single He atom Fig. 1 shows the moving trajectory of He atom in 1ns at the temperature 333 K. The position of the He atom at different times is denoted by points and the Ti atoms are hidden for clarity. The corresponding time step of two adjoining points is 0.2 ps. As expected, the He atom zigzags at interstitial sites for most of the time and jump from one site to a neighboring site in a much shorter time. This makes the trajectory show itself layer structured. The layers formed by the trajectory are in-betweens of the crystal monolayer of substrate. An interesting observation is that the inter-layer jump (along [001] direction) of the He atom happens much more frequently than the intra-layer jump (in (001) crystal plane). This indicates that the diffusion of He atom is anisotropic. With the increasing temperature, the frequency for He atom jumping from site to site increases and the layer structure of the trajectory tends to be ambiguous. However, one can also find that He atom migrates more quickly in the direction perpendicular to (001) plane than in (001) plane. This anisotropic diffusion will be discussed later according to the diffusion coefficient of the Hen in (001) plane and along [001] direction. 3.2. Diffusion of He- dimer For the migration of He2 , the first observation is that in the temperature range of the simulations, the two He atoms migrate without dissociation with the separation of the two He atoms around 0.70 lattice length of Ti. Thus we can denote the trajectory

Fig. 3. Typical snapshots of He dimer migration in Ti at temperature 333 K. Black spheres are the He atoms and the gray spheres are the Ti atoms. (a) A snapshot at 170 ps; (b) a snapshot at 184 ps; (c) a typical graph indicating the influence of He dimer on a (001) monolayer of the substrate, the atom pointed by the arrow is displaced from its original lattice position.

of He2 by its mass center. Fig. 2. displays the moving trajectory of He2 at the temperature 333 K. In contrast to the trajectory observed above for a single He atom at 333 K, the trajectory of He2 does not show obvious layer structure, however, it is similar to the trajectory of single He atom at 500 K (not shown here). This is probably because He2 can straddle two adjoining interstitial sites with its mass center settling on the crystal bridge in the migration process as shown by the randomly captured snapshots in Fig. 3(a)–(b). Thus, the migration of He2 looks relatively continuous. This feature of He2 migration also could be a reason for that He2 has large diffusion coefficient at room temperature as to be shown later. Moreover, it is also found that the molecular orientation of He2 changes in the migration process without obvious preference orientation. As to be shown below, this is different from that is observed for He3 . On the other hand, although He2 has no preference molecular orientation, what is similar to that observed for single He atom migration is that He2 migrates also more quickly in the direction perpendicular to (001)

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M. Chen et al. / Solid State Communications 148 (2008) 178–181

(Hen )

Fig. 5. Diffusion coefficients Dz

(Hen )

and Dxy

(He )

for Hen in Ti. The solid symbols and

(He )

the open symbols correspond to Dxy n and Dz n respectively. • and for single atom;  and  for dimer; M and N for trimer. Solid lines are the Arrhenius relation (He ) fitted to Dxy n data with activation energies (ev): EA(He1 ) = 0.18, EA(He2 ) = 0.06, (He1 )

EA(He3 ) = 0.13 and prefactors (cm2 /s): Axy (He3 )

(He2 )

= 2.1 × 10−4 , Axy

= 6.6 × 10−5 , (He )

Axy = 1.6 × 10 . Dot lines are the Arrhenius relation fitted to Dz n data with activation energies: EA(He1 ) = 0.18, EA(He2 ) = 0.06, EA(He3 ) = 0.13 and prefactors: (He1 )

Az

Fig. 4. Snapshots for He trimer migration in Ti at temperature 350 K. (a) At time of 1100 ps; (b) at 1450 ps; (c) 1464 ps; (d) 1477 ps and (e) 2250 ps. (f) A typical graph indicating the influence of He trimer on the substrate, The black line indicates the dislocation loop induced by He trimer.

plane than in (001) plane. Another observation for He2 migration is that He2 can induce rearrangement of its nearest Ti atoms. However, this effect is slight. Fig. 3(c) shows a snapshot of a (001) plane that He2 moves through. One can find that one Ti atom is displaced. The displaced Ti atom will restore it lattice position after He2 moves to other sites.

−5

(He2 )

= 2.6 × 10−3 , Az

He3 is also observed migrating without dissociation in the temperature range of the simulations and thus its trajectory can be denoted by the trajectory of its mass center. The bond length between two He atoms is around 0.62 lattice length of Ti. At room temperature, the trajectory of He3 migration shows itself layerstructured, a feature similar to the trajectory of single He atom. As single He atom acts, He3 also zigzags in a layer for most of the time and jumps from layer to layer. The inter-layer jump can span a few monolayers and the inter-layer migration is more frequent than intra-layer migration. However, in contrast to that in single He atom migration, the layers formed by the trajectory of He3 coincide with the (001) monolayer positions of the substrate. Fig. 4. exhibits a few snapshots of the migration process. Fig. 4(a) is a snapshot at 1100 ps. It is observed that He3 actually lie in a (001) monolayer of the substrate. The situation that the molecular plane of He3 tends to be coincident with (001) plane of the substrate remains for most of the time points. However, driven by temperature, He3 could change its orientation (defined by its molecular plane). At the points that the orientation of He3 greatly deviates from the

= 8.8 × 10−5 .

orientation of (001) plane, He3 migrates quickly in the direction that is perpendicular to (001) plane, as what demonstrated in Fig. 4(b)–(d). As the deviation of the orientation of He3 from (001) plane become small, He3 slows down its migration and tends to embed in a (001) monolayer as shown in Fig. 4(e). In the migration process, one can observe local reconstructions of the substrate in the neighborhood of the He3 . Fig. 4(f) shows the dislocation loop induced in a (001) monolayer at the point that He3 embeds in the monolayer. 3.4. Diffusion coefficients Diffusion coefficient gives more quantitative description on the diffusion features. Using the Einstein relation [16], we have (He ) calculated the diffusion coefficient Dxy n for Hen diffusing in (001) (He )

plane and Dz n for Hen diffusing along [001] direction, separately. (He ) Fig. 5 displays the dependence of the diffusion coefficients Dxy n (He )

3.3. Diffusion of He- trimer

(He3 )

= 1.2 × 10−4 , Az

(He )

(He )

and Dz n on the temperature. Dxy n and Dz n are obtained by the Einstein relation. The curves are the Arrhenius temperature dependence: D = A exp(−EA /kB T ), where the activation energy EA and the prefactor A is obtained by fitting this relation to (He ) (He ) corresponding Dxy n and Dz n data. By Fig. 5, a few remarks can be made. First, the Arrhenius relation can well describe the diffusion of single He atom as well as the diffusion of He2 and He3 , though the migration process of Hen for n > 1 is complicated by its internal motion and local reconstruction of the substrate. Second, the diffusion of these three He species is very anisotropic. This is consistent with above qualitative observation of Hen migration. (He ) (He ) Dz n is larger than Dxy n by a factor of 12, 1.8 and 5.5 for n = 1, 2, 3, respectively. Somewhat surprising is the fact that the activation energy for Hen diffusing in [001] direction and in (001) plane is the same. The significant anisotropy of diffusion is caused by the quite different prefactor for Hen diffusing in these two directions. Third, in contrast to what is intuitively assumed that a small cluster in a substrate would migrate faster than a large cluster would, an interesting observation is that at low temperature (333 K and 400 K) He dimer migrates more quickly than single He atom does. This is due to that the activation energy of He2 is much lower than that of single He atom. A possible reason

M. Chen et al. / Solid State Communications 148 (2008) 178–181

for the low activation energy of He2 is that a dimer can straddle two adjoining interstitial sites and only slightly disturbs its neighbor Ti atoms. However, this needs to be clarified in future. Moreover, the feature that He2 has no obvious preferred orientation in its migration makes its diffusion relatively less anisotropic in comparison with the diffusion of He atom and trimer, thus the (He ) (He ) difference between Dz 2 and Dxy 2 is not as large as that between (He1 )

Dz

(He1 )

and Dxy

(He3 )

and that between Dz

(He3 )

and Dxy

.

4. Conclusions We have simulated in detail the migration process of single He atom, dimer and trimer in Ti at the temperatures between room temperature and 667 K. The results demonstrate that He2 and He3 migrate without disassembling. This indirectly agrees with the statement that He–He has large bonding energy [9, 11]. The features of the migration of Hen have been presented and compared. The results demonstrate that the details of the migration process of Hen could be quite different for different n. The Arrhenius relation can well describe the diffusion of Hen in the temperature range of our simulation quantitatively. However, the diffusion is quite anisotropic as Hen migrates much more quickly in [001] direction than in (001) plane. Here, the prefactor plays predominant role. The result presents evidence that considering only the energy barrier in static lattice is insufficient in predicting the diffusion behavior of He in metal. Dynamics calculations are needed to obtain at least qualitatively correct prefactor. This conclusion is instructive to kinetics calculations, e.g., Monte Carlo simulations of long-term morphology evolution of He distribution in metal.

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Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant No 10775101 and 50671096. References [1] H. Rajainmäki, S. Linderoth, H.E. Hansen, R.M. Nieminen, M.D. Bentzon, Phys. Rev. B 38 (1988) 1087. [2] J.H. Evans, A.V. Veen, L.M. Caspers, Nature 291 (1981) 310. [3] A.J. Schwartz, M.A. Wall, T.G. Zocco, W.G. Wolfer, Phil. Mag. 85 (2005) 479. [4] D. Chen, W.Y. Hu, J.Y. Yang, L.X. Sun, J. Phys.: Condens. Matter 19 (2007) 446009. [5] S.E. Donnelly, Radiat. Eff. 90 (1985) 1. [6] D.F. Cowgill, Fus. Sci. Technol. 48 (2005) 539. [7] A. MÖslang, T. Wiss, Nature Mater. 5 (2006) 679. [8] R. Gomer, Rep. Prog. Phys. 53 (1990) 917. [9] C.D. Van Siclen, R.N. Wright, S.G. Usmar, Phys. Rev. Lett. 68 (1992) 3892. [10] C.S. Becquart, C. Domain, Phys. Rev. Lett. 97 (2006) 196402. [11] H. Trinkaus, B.N. Singh, J. Nucl. Mater. 323 (2003) 229. [12] K.O.E. Henriksson, K. Nordlund, A. Krasheninnikov, J. Keinonen, Appl. Phys. Lett. 87 (2005) 163113. [13] C.J. Ortiz, M.J. Caturla, C.C. Fu, F. Willaime, Phys. Rev. B 75 (2007) 100102(R). [14] E.E. Gruber, J. Appl. Phys. 38 (1967) 243. [15] R. Rajaraman, B. Viswanathan, M.C. Valsakuma, K.P. Gopinathan, Phys. Rev. B 50 (1994) 597. [16] G. Boisvert, L.J. Lewis, Phys. Rev. B 54 (1996) 2880. [17] U. Kürpick, A. Kara, T.S. Rahman, Phys. Rev. Lett. 78 (1997) 1086. [18] A. Steltenpohl, N. Memmel, Phys. Rev. Lett. 84 (2000) 1728. [19] H.L. Heinisch, H. Trinkaus, B.N. Singh, J. Nucl. Mater. 367–370 (2007) 332. [20] F. Cleri, V. Rosato, Phys. Rev. B 48 (1993) 22. [21] R.M. Nieminen, in: S.E. Donnelly, J.H. Evans (Eds.), Fundamental Aspects of Inert Gases in Solids, Plenum, New York, 1991. [22] J. Wang, Q. Hou, T.Y. Sun, Z.C. Wu, X.G. Long, X.C. Wu, S.Z. Luo, Chin. Phys. Lett. 23 (2006) 1666. [23] J. Wang, Q. Hou, T. Sun, X. Long, X. Wu, S. Luo, J. Appl. Phys. 102 (2007) 093510. [24] Q. Hou, M. Hou, L. Bardotti, B. Prével, P. Mélinon, A. Perez, Phys. Rev. B 62 (2000) 2825.