Molecular dynamics study of helium bubble pressure in tungsten

Nuclear Instruments and Methods in Physics Research B 352 (2015) 104–106

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Molecular dynamics study of helium bubble pressure in tungsten Jiechao Cui, Min Li, Jun Wang, Qing Hou ⇑ Key Lab for Radiation Physics and Technology, Institute of Nuclear Science and Technology, Sichuan University, Chengdu 610061, China

a r t i c l e

i n f o

Article history: Received 26 June 2014 Received in revised form 4 December 2014 Accepted 9 December 2014 Available online 30 December 2014 Keywords: Molecular dynamics simulations Helium bubble Tungsten Stress field

a b s t r a c t Molecular dynamics simulations were performed to calculate the stress field in a tungsten matrix containing a nano-scale helium bubble. A helium bubble in tungsten is found to consist of a core and an interface of finite thickness of approximately 0.6 nm. The core contains only helium atoms that are uniformly distributed. The interface is composed of both helium and tungsten atoms. In the periphery region of the helium bubble, the stress filed is found to follow the stress formula based on the elasticity theory of solid. The pressure difference between both sides of the interface can be well described by the Young– Laplace equation for the core size of a helium bubble as small as 0.48 nm. A comparison was performed between the pressure in the helium bubble core and the pressure in pure helium. For a core size larger than 0.3 nm, the pressure in the core of a helium bubble is in good agreement with the pressure in pure helium of the same helium density. These results provide guidance to larger scale simulation methods, such as in kinetic Monte Carlo methods and rate theory. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction In the field of nuclear materials, the formation, growth, coalescence and release of helium bubbles in metals have received great attention. The stress field is an important factor driving the evolution of the micro-structure in the helium-bubble and metal system. The Young–Laplace (Y–L) equation, a relationship between the internal pressure and the surface tension for spherical bubbles [1], in combination with the state equation of helium had been adopted for simulating the evolution of helium bubbles in materials [2,3]. However, the applicability of the Y–L equation for bubbles at the nano-scale is unclear, considering that the Y–L equation was originally deduced based on the stress concept in continuum mechanics, while at the nano-scale, the interaction distance between atoms tends to be comparable with the bubble size; in addition, the definition of bubble size is ambiguous. Matsumoto and Tanaka [4] proved that the Y–L equation is valid for an argon gas bubble with a radius in the range of 1.7–5 nm in argon liquid. A similar conclusion was drawn by Park et al. [5] for an argon bubble in argon liquid. In argon liquid, the pair-wise potential between atoms was applicable. However, in metals, the elasticity plays a role, and the interaction between metal atoms should be described by many-body potentials [6,7]. As a result, revisiting the validation of Y–L equation for a nano-scale helium bubble in metals is warranted. On the aspect of the state equation, it is difficult to directly ⇑ Corresponding author. Tel.: +86 028 8541 2104; fax: +86 028 8541 0252. E-mail address: [email protected] (Q. Hou). http://dx.doi.org/10.1016/j.nimb.2014.12.025 0168-583X/Ó 2014 Elsevier B.V. All rights reserved.

measure the state equation of helium in a nano-bubble. Using Electron Energy-Loss Spectroscopy (EELS), Fréchard et al. [8] measured the average helium density in helium bubbles and then converted the density to the bubble pressure by using a state equation for pure helium that can be directly measured experimentally [9,10]. A question to be addressed is whether the helium density in a nano-bubble is uniform and whether the pressure of the bubble is consistent with the pressure of pure helium of the same density. In the present paper, we will present the results of the molecular dynamics (MD) simulations of stress in a tungsten matrix containing a nano-scale helium bubble. The structure around the bubble will be analyzed. The characteristics of the pressures inside and outside of a bubble will be discussed. By comparing the pressure in a helium bubble and the pressure calculated for pure helium of same density at various temperatures, the validation of the method of using the pressure of pure helium in place of a helium bubble will be explored.

2. Model and methods In our MD simulations, a Finnis–Sinclair-type potential obtained by Ackland and Thetford was used to describe the interactions between tungsten atoms [11]. The exp-6 potential [12] was used for He–He interactions, and a pair-wise potential given by Wang et al. was used for the He–W interactions [13]. A bcc tungsten box of size 50 a0  50 a0  50 a0 was first constructed, where a0 = 0.31652 nm is the lattice constant of tungsten.

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Our previous simulations of helium bubble growth in Ti [14] and in W [15] indicated that metal atoms eject from their lattice positions during the growth of a helium bubble and the cluster formed by the ejected atoms moved away for the bubble in a short time when the number of the helium atoms reaches approximately twice the number of the ejected atoms. Based on this observation, we created a helium bubble by removing a given number of tungsten atoms that were closest to the center of the box and then filled in the cavity with helium atoms twice the number of the removed tungsten atoms; in other words, the initial density of helium is 4 per lattice unit cell. Periodic boundary conditions were applied in the x-, yand z-directions in all simulations. The simulations were performed in the NVT scheme at a given temperature. After the system reaches equilibrium, the system remained relaxing for a time and the virial stress in a spherical region denoted by Xr was obtained by the ensemble average of instant virial stress in the region [16]:

PðrÞ ¼

1 V Xr

* NXr kB T þ

X ri  f i

+ ð1Þ

i2Xr

where r and V denote the radius and volume of region Xr, respectively, NXr is the number of atoms in Xr, kB is the Boltzmann

constant, and T is the temperature of the system. ri and fi represent the positions of and forces on the atoms in region Xr, respectively. 3. Results and discussion Using a GPU-based MD package [17], simulations were performed for helium bubbles containing various number of helium atoms NHe from 100 to 8000. The local structures were analyzed by the adaptive-common-neighbor-analysis (a-CNA) method of Stukowski [18]. Fig. 1(a) shows a cross-section of a box in the case of NHe = 8000. Visually, an interface of finite thickness is clearly observed. The interface thickness is found to be approximately 0.6 nm, almost independent of NHe. We denote the region enclosed by the inner surface of the interface as the core of the bubble. In the core are helium atoms that are uniformly and randomly distributed. The region beyond the outer surface of the interface, which we call the periphery region of the bubble, contains only tungsten atoms remaining in bcc structure. The Wigner–Seitz volume of atoms was calculated and the average number density in spherical shells was obtained; the results corresponding to Fig. 1(a) are shown in Fig. 1(b). In the periphery region (r > 2.8 nm), the density tends to be the density of bcc tungsten. In the core (r < 2.2 nm), the density of helium is higher than the initial density when the

(a) 180 Virial stress(kbar)

160 Virial stress(kbar)

140 120 100

60 N He=800 Eq. (2) 30

NHe=800

80

0

NHe=2000

60

2

4 r(nm)

NHe=5000

6

NHe=8000

40

W

20 0

1

2

3

4

5

6

7

r(nm) Fig. 2. P(r) vs. r for the helium bubble having different sizes. W denotes the stress in a prefect tungsten crystal and blue arrow points to the outer surface of the interface, beyond which is the periphery region of the bubble. The inset displays the stress fields in the periphery region obtained by simulations and by Eq. (2) for NHe = 800. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

4.5 4.0 3.5 3.0

150

2.5 2.0 1.5 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 r(nm)

Fig. 1. (a) a cross-section of a simulation box with NHe = 8000. Red color denotes tungsten atoms with bcc local structure, dark yellow color denotes tungsten atoms with disordered local structure, black denotes helium atoms with disordered local structure; (b) average number of atoms per lattice unit cell in spherical shells, for NHe = 8000. All the centers of the spherical shells are located in the middle of the box. Black color denotes helium atoms and red color denotes tungsten atoms. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Pressure difference(kbar)

-3

Average number of atoms(lattice )

(b)

bubble pressure fitted curve

100

50

0.8

1.6

2.4

The core radius R(nm) Fig. 3. Pressure difference PHe(R)-Po vs. R. Dots: the pressure calculated by the MD simulation; solid line: the Y–L equation fitting to the MD data.

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Table 1 Comparison between the pressure in a helium bubble and the pressure calculated for pure helium at T = 500 K. Ppure denotes the pressure of pure helium. NHe Density /lattice PHe/kbar Ppure/kbar

3

100

200

500

800

2000

5000

8000

5.72 279.96 ± 7.62 288.28

5.25 226.92 ± 2.00 229.95

4.84 179.06 ± 1.32 177.70

4.68 161.80 ± 1.01 159.72

4.48 138.24 ± 0.72 139.66

4.31 123.46 ± 0.54 123.08

4.24 117.89 ± 0.38 117.57

helium bubble was created, indicating a compression of helium in the core. For a smaller value of NHe, the density of helium in the core is higher, resulting in a higher pressure in the core, as will be demonstrated later. Here, we note that the core does not contain all helium atoms. Some helium atoms exist in the interface, which is actually composed of both helium and tungsten atoms. The smaller the bubble is, the higher the fraction of helium atoms in the interface is. For example, in the case of NHe = 100, the core contains 20 helium atoms only, whereas 80 helium atoms are in the interface that has an inner radius 0.3 nm and an outer radius 0.62 nm. Fig. 2 shows the stress field P(r) at T = 500 K obtained for difference NHe. The standard deviation of the stress over the ensemble average is also drawn. The smaller r is, the more uncertain the number of atoms in Xr becomes; thus, the standard deviation is larger. Even so, P(r) has a plateau in the core of the bubble until r reaches the inner side of the interface. In the interface, P(r) exhibits a sharp peak initially and then decreases quickly. On the outer surface of the interface, a knee point of P(r) is observed, after which P(r) decreases smoothly to the pressure of the tungsten crystal. We found that P(r) in the periphery region of the bubble can be represented by the following equation:

PðrÞ  Pw ¼

R3o r3

ðPo  Pw Þ

ð2Þ

where Ro is outer radius of the interface, Po is the pressure on the outer surface of the interface, and Pw is the pressure of pure tungsten. Eq. (2) is the stress field predicted in elasticity theory. The inset in Fig. 2 compares the simulation data and P(r) calculated by Eq. (2) for NHe = 800. For other NHe values from 100 to 8000, the simulation data and P(r) calculated by Eq. (2) are also in good agreement. The agreement between the simulation results and theoretical results indicates that the periphery region of the bubble is elastic. Fig. 3 shows the pressure PHe on the inner surface vs. the core radius R. The pressure difference between both sides of the interface is found to be well described by the Y–L equation [1]:

PHe ðRÞ  Po ¼

2c R

ð3Þ

with c = 3.89 J/m2, which is in good agreement with the surface energy of pure tungsten [19]. Eq. (3) is valid for R as small as 0.48 nm, i.e., a helium bubble containing 200 helium atoms. After determining the helium densities in bubble cores, the pressures of pure helium with the same densities were calculated. A comparison between the pressure PHe in a helium bubble and the pressure of pure helium is listed in Table. 1. The pressure in a helium bubble in a tungsten matrix is in good agreement with

the pressure of pure helium of the same density, even for the number of helium atoms in a bubble down to 100, or R = 0.3 nm. 4. Conclusions To summarize, the structure and stress filed of nano-scale helium bubbles in a tungsten matrix were studied using MD simulations. A helium bubble consists of a core and an interface of finite thickness of approximately 0.6 nm. The Y–L equation can well describe the pressure difference of the bubble core and the outside pressure of the bubble for bubble cores as small as 0.48 nm. For a bubble containing 100 or more helium atoms, the pressure in a bubble core is in good agreement with the pressure of pure helium of the same density. However, the deviation between them will increase with decreasing bubble size. The results of the present paper are instructive for the establishment of models in Monte Carlo methods and the rate theory of the evolution of the microstructures of helium bubble-matrix systems. Acknowledgements This work was supported partly by the National Magnetic Confinement Fusion Program of China (2013GB109002 and 2011GB110005). References [1] S. Thompson, K. Gubbins, J. Walton, R. Chantry, J. Rowlinson, J. Chem. Phys. 81 (1984) 530. [2] Q. Hou, Y.L. Zhou, J. Wang, A.H. Deng, J. Appl. Phys. 107 (2010) 084901. [3] J.H. Evans, R.E. Galindo, A.v. Veen, Nucl. Instr. Meth. Res. B 217 (2004) 276. [4] M. Matsumoto, K. Tanaka, Fluid Dyn. Res. 40 (2008) 546. [5] S. Park, J. Weng, C. Tien, Int. J. Heat Mass Trans. 44 (2001) 1849. [6] M.W. Finnis, J.E. Sinclair, Philos. Mag. A 50 (1984) 45. [7] M.S. Daw, M.I. Baskes, Phys. Rev. B 29 (1984) 6443. [8] S. Fréchard, M. Walls, M. Kociak, J. Chevalier, J. Henry, D. Gorse, J. Nucl. Mater. 393 (2009) 102. [9] W. Nellis, N. Holmes, A. Mitchell, R. Trainor, G. Governo, M. Ross, D. Young, Phys. Rev. Lett. 53 (1984) 1248. [10] P. Loubeyre, R. LeToullec, J. Pinceaux, H. Mao, J. Hu, R. Hemley, Phys. Rev. Lett. 71 (1993) 2272. [11] G.J. Ackland, R. Thetford, Philos. Mag. A 56 (1987) 15. [12] D.A. Young, A.K. McMahan, M. Ross, Phys. Rev. B 24 (1981) 5119. [13] J. Wang, Y.L. Zhou, M. Li, Q. Hou, J. Nucl. Mater. 427 (2012) 290. [14] J. Wang, Q. Hou, T. Sun, X. Long, X. Wu, S. Luo, J. Appl. Phys. 102 (2007) 093510. [15] J. Wang, private communication. [16] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, 1987. [17] Q. Hou, M. Li, Y. Zhou, J. Cui, Z. Cui, J. Wang, Comput. Phys. Commun. 184 (2013) 2091. [18] A. Stukowski, Model. Simul. Mater. Sci. Eng. 20 (2012) 045021. [19] M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B 54 (1996) 4519.