Development of a tight-binding model for Cu and its application to a ...

J Mater Sci (2015) 50:5684–5693 DOI 10.1007/s10853-015-9097-7

Development of a tight-binding model for Cu and its application to a Cu-heat-sink under irradiation Wenyi Ding1 • Haiyan He1 • Bicai Pan1

Received: 3 January 2015 / Accepted: 12 May 2015 / Published online: 30 May 2015 Ó Springer Science+Business Media New York 2015

Abstract An environment-dependent tight-binding potential model for copper within the framework of quantum theory is developed. Our benchmark calculations indicate that this model has good performance in describing the elastic property, the stability and the vibrational property of bulk copper, as well as in handling the clusters, the surfaces and the defective Cu systems. By combining this model with molecular dynamics, we study how the evolution of structural defects arising from the irradiation of the energetic particles influences the mechanical and the thermal properties of the copper-heat-sinks in fusion reactors. Based on our simulations, the heat blockade in the irradiated Cuheat-sinks is predicted. This finding is valuable for the development of wall materials in fusion reactors.

Introduction Copper (Cu), with excellent thermal conductivity, is widely used in ordinary life, as well as is an important material for the devices working under extreme environment. Typically, Cu has been regarded as a good heat-sink material that links with the plasma facing materials (PFMs) in the Tokamak fusion reactor walls [1–3]. As we know, both the PFMs and the Cu-heat-sink material in a fusion reactor are irradiated by some energetic particles from the plasma core, and consequently their mechanical & Bicai Pan [email protected] 1

Key Laboratory of Strongly-Coupled Quantum Matter Physics, Department of Physics and Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China

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and thermal properties are degraded to some extent. So, it is important to assess the performance of the Cu-heat-sink by studying how the physical properties of the Cu-heatsink correlate with the evolution of the structural defects induced by the irradiation of the energetic particles. Clearly, this is a dynamical event occurring on the atomic scale, which is quite difficult to be investigated in experiments so far. Alternatively, theoretical simulations are helpful to treat this concern. This is because simulations on the atomistic scale can provide valuable results such as the evolution of structural defects in the time scale of picoseconds. Previously, simulations about the irradiation damage in Cu-related materials have been predominantly performed at the level of empirical potentials [4–11], such as the embedded atom model (EAM) and the tight-binding potential based on the second-moment approximation (TBSMA). For example, Rubia et al. [4, 5] studied the defect production in high-energy displacement cascades for Cu. They found that the local melting at the centre of the cascade arising from the irradiation persisted for several picoseconds. Foreman et al. [6, 7] also simulated the irradiation damage cascades in Cu, and proposed a new mechanism for the vacancy interstitial separation. In addition, the annealing mechanism of irradiation damage near grain boundaries in Cu was investigated by Bai and the co-workers [8], and they found that the grain boundaries could serve as effective sinks for some of the interstitial atoms and vacancies. It is noted that all of these efforts did not concern the heat transport in the Cu-heatsinks, although the heat transport of Cu-heat-sinks is also a key property for its potential application to fusion reactors. This is due to the fact that the heat transport in a metal is predominantly carried out by electrons, and these empirical potentials cannot treat the transport of electrons at all.

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As we know, ab initio simulations can provide all of the concerns above within the framework of quantum theory. However, the ab initio treatment about the irradiated events in a material requires huge computational demand, and thus it is impossible to have a long-time simulation for a large system at the ab initio level. It is noted that a tightbinding (TB) model that counts for the spirit of quantum theory may describe the bonding integrals using simple formula as well as may employ a minimum (or smaller) basis set [12] to largely reduce the computational demand. Hence, a TB potential model can be used to fast simulate the properties relevant to electronic structures of the irradiated materials. Basically, a TB model is suitable to study the materials in which the bonding character between atoms is localised, such as the semiconductor materials with covalent bonds [13–17] and some of transition metals possessing the localised d orbitals [18–22]. Generally, the bonding character between the atoms in a material is altered by the bonding environment around the bonds. So far the bonding environment has been taken into account in some of previous TB models. For example, the TB models for Ge [16] and for some metals [19] contained the bonding environment described using the atom density; the TB model for Mo included the bonding environment using a screening function [18]. Tang and co-authors not only used the screening function but also introduced effective bond lengths to reflect the bonding environment in their TB model for C [13], largely improving the quality of the TB potential model for carbon related to the early TB version. Later on, such an environment-dependent correction was employed in the TB models for Si [15] and SiMo systems [23]. It is noted that the effectiveness of the environmentdependent correction in these models is dependent on the expression of the bonding environment. Developing a more effective expression for the bonding environment is still an issue. More recently, we have proposed a simpler formula to describe the environment correction for Ge [17], and the transferability of this new TB model is better than the previous ones. In the present work, we employ the simple formula we proposed [17] to Cu system, so as to reliably handle some properties of Cu-heat-sink under irradiation of highly energetic neutrons within the framework of quantum theory. Our benchmark calculations indicate that our proposed TB model has good performance in dealing with the structures, the electronic structures and the stabilities of either the defect free crystals or the defective crystals. Moreover, the calculated phonon dispersion and the elastic properties of the crystal match those from experiments [24, 25] and from the density functional theory (DFT) calculations very well. In addition, the evaluated melting

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temperature of bulk Cu from our tight-binding molecular dynamics (TBMD) simulations is well consistent with the experimental value [26]. More importantly, we apply this potential model to simulate the Cu-heat-sinks irradiated by energetic neutrons. The processes of the defect formation and the defect annihilation in the irradiated Cu system are revealed, during which the evolution of both the stress and the thermal conductivity of the irradiated Cu-heat-sinks are studied.

Method In a TB potential model, the total energy (E) of a system is written as E ¼ Eband þ Erep :

ð1Þ

Here, Eband is the energy of the electronic band structure, defined as the sum of eigenvalues of the electron-occupied P energy levels k , namely, Eband ¼ 2 occ k k . The eigenvalues k of the system are obtained by solving the secular equation   ð2Þ HTB  k STB ck ¼ 0; where the HTB is the Hamiltonian matrix, STB is the overlap matrix and ck is the corresponding coefficient matrix. Similar to the previous treatment [15, 17], the orthogonal basis set is employed here, and thus the overlap matrix STB in formula (2) is approximated to be a unit matrix. As for the Hamiltonian matrix HTB , each of its elements is expressed as a linear combination of bonding integrals Vll0 m [12]. According to the scheme proposed by Slater and Koster [12], a bonding integral could be expressed as an empirical formula in which the value of bonding integral is a function of distance between the related two atoms. This is the so-called twocentre approximation. In this approximation, however, the neighbouring atoms around each bond in a solid are not involved. This leads to the poor transferability of the potential usually. Basically, the strength of a chemical bond in a realistic material is relevant to its neighbouring atoms to some extent. To account for such bonding environment, several schemes have been proposed [13, 17–19]. A typical scheme is that an effective bond length Rij [13, 15, 17] was introduced into the formula of the bonding integral for atoms i and j, namely     ð3Þ Vll0 m ðrij Þ ¼ a1 Raij2 exp a3 Raij4 fc Rij : The effective bond length Rij is obtained by scaling the real bond length rij

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  Rij ¼ rij 1 þ d1 D þ d2 D2 þ d3 D3 :

ð4Þ

Here, D counts for the concerned bonding environment. Unlike the previous expressions [13, 15], D is expressed in this model as 2 D ¼ arctan nij ; p

ð5Þ

with nij ¼

X k

  ! rij rik þ rjk b2 : exp b1 rik þ rjk rij

0

ð6Þ

Both b1 and b2 are parameters. Clearly, nij depends not only on the separation between atoms i and j, but also on the relative positions of their common neighbours. According to the expression (6), the maximum effect arising from an environmental atom k on the bond length occurs when the atom k locates at the centre site between atoms i and j. In order to fast decay the strength of a bond that has a long bond length, a smooth cutoff function fc ðRij Þ is introduced into formula (3), which is !1    fc ðRij Þ ¼ 1 þ exp Rij  rcut =w : ð7Þ ˚ is the cutoff distance for Cu In above formula, rcut ¼ 6:0 A ˚ system and w ¼ 0:1 A is the range in which the cutoff function works effectively. Overall, the value of Rij is dependent on the environmental atoms {k}, and hence the environment-dependent feature is contained in each bonding integral. For our concerned systems, we consider the atomic orbitals of s, px , py , pz , dxy , dyz , dzx , dx2 y2 and d3z2 r2 on each atom. So, ll0 m=ssr, spr, ppr, ppp, sdr, pdr, pdp, ddr, ddp or ddd. As for the onsite terms hiajHjiai in the Hamiltonian matrix, they are approximated to be the eigenvalues of the related atomic orbitals of atom i. In our model, the onsite terms s , p and d are chosen to be 2.408, 4.00 and 5.00 eV respectively. In addition, by calculating the pairwise potential /ðrij Þ, which has the same formula as that of the bonding integral Vll0 m , we have the repulsive energy 0 1 X X Erep ¼ f@ /ðrij ÞA: ð8Þ i

jð6¼iÞ

Here, f is a polynomial function f ðxÞ ¼ c0 þ c1 x þ c2 x2 þ c3 x3 þ c4 x4 :

ð9Þ

When the system is irradiated, some atoms in the system have large kinetic energy, causing collision between some of the atoms around them. During the collision, the distances between the related atoms become very small, and

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the repulsive interaction between them is very strong. In this case, the interaction between these atoms cannot be described correctly using the pairwise potential above. Instead, such short-range repulsive interaction is usually expressed by the Ziegler–Biersack–Littmark (ZBL) interatomic potential [27]. In terms of this consideration, the effective pairwise potential in our potential is revised as        0 /mod ðrij Þ ¼ / ðrij Þ 1  F rij þ / rij F rij ; ð10Þ where / ðrij Þ is the ZBL interatomic potential, and Fðrij Þ is the fermi function, which is expressed as !1    ; ð11Þ Fðrij Þ ¼ 1 þ exp bf Rij  rf 1

˚ and rf ¼ 1:3 A. ˚ Here, the fermi function with bf ¼ 20 A 0 ensures that /ðrij Þ joins / ðrij Þ smoothly. The parameters in our model are achieved by fitting the electronic band structures and the cohesive energy curves for different phases, such as the face-centred cubic (FCC) structure and several artificial structures including the body-centred cubic (BCC), diamond, hexagonal close packing (HCP), b-W (A15), a-Mn (A12), b-Mn (A13) and the simple cubic (SC) structures. We stress that these selected phases in our fitting have different bonding environments, which will improve the transferability of our TB potential. In our treatment, the band structures and the cohesive energy curves of these considered systems are obtained by performing DFT calculations [28, 29], where the projector-augmented wave (PAW) method [30] in conjunction with the Perdew–Wang (PW91) [31] functional is used. The resulting parameters in our model are listed in Tables 1 and 2. Figure 1 shows each Vll0 m and the pairwise potential as a function of distance rij between atoms i and j. Clearly, each bonding integral and the pairwise potential smoothly reach zero before rij ¼ rcut .

Testing results Properties of bulk copper The band structures of different phases from present TB calculations are displayed in Fig. 2. From this figure, we observe that our calculated valence bands and some conduction bands near the Fermi level match those from DFT calculations very well. However, the conduction bands which are far from the Fermi level are in disagreement with the DFT bands. This is mainly attributed to the small basis set we used in our TB potential. It is noted that this potential is generated for studying the ground state

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Table 1 The parameters in our model a2

a1

a3

a4

b1

b2

d1

d2

d3

Vssr

3.981210

1.641614

0.010300

3.2350

0.08422

6.2891

0.14680

0.067412

0.200000

Vspr

4.279389

1.512800

0.003256

3.8325

0.08422

6.2891

0.14680

0.080000

0.149834

Vppr

8.064037

1.423000

0.002526

4.0000

0.08422

6.2891

0.14680

0.080000

0.020000

Vppp

0.625000

1.605000

0.016500

4.0500

0.08422

6.2891

0.14680

0.364997

0.200000

Vsdr

3.247739

1.833692

0.010350

3.2000

0.08422

6.2891

0.14680

0.235000

0.200000

Vpdr

4.107218

1.781660

0.010000

3.2000

0.08422

6.2891

0.14680

0.220163

0.020000

Vpdp

1.936082

2.104600

0.010626

3.2000

0.08422

6.2891

0.14680

0.195313

0.200000

Vddr

7.680902

2.848120

0.020350

2.8900

0.08422

6.2891

0.14680

0.080000

0.020000

Vddp

6.239943

3.095827

0.023211

3.0118

0.08422

6.2891

0.14680

0.123552

0.200000

Vddd

0.746681

2.827904

0.020363

2.8900

0.08422

6.2891

0.14680

0.079401

0.020000

/

57.12080

4.49792

0.010224

3.8350

0.10733

6.4089

0.24173

0.098327

0.122753

The parameters of a1 are in unit of eV, and the other parameters are dimensionless

Table 2 The coefficients in the polynomial function f ðxÞ

c0 (eV)

c4 (eV3 )

0.0003676

0.000004

51.8897

0.5

0.016302

54.6390

0.513165

0.0

1.0

Energy (eV)

c3 (eV2 )

x [ 22:0

ssσ spσ ppσ ppπ sdσ pdσ

pdπ ddσ ddπ ddδ φ(r)

0.0

-1.0

2.0

c2 (eV1 )

x  22:0

2.0

-2.0

c1

4.0

6.0

r (Å)

Fig. 1 The bonding integrals Vll0 m and pairwise potential /ðrÞ as a function of separation between two atoms

properties of Cu-based materials, which are relevant to the valence bands only. So, the difference of the excited states between our TB calculations and the DFT calculations does not alter the effectiveness of our TB potential describing the ground state properties of Cu. The cohesive energy as a function of the nearestneighbour distance for each considered phase is shown in Fig. 3. Clearly, our TB energy curves are in good agreement with the related ones from DFT calculations except for some high-energy phases. Such agreement implies that our potential can correctly predict the relative stabilities of the different phases. Furthermore, we compute the elastic constants C11 , C12 and C44 of bulk Cu. As

0.0

0.0

listed in Table 3, our calculated values are quite close to the related ones from experiments and to that from other computations. As we know, some properties of a material are essentially correlated with the lattice dynamics. In our benchmark tests, we compute the phonon dispersion of bulk Cu. As shown in Fig. 4, our obtained phonon dispersion agrees with that obtained from both the experiment [24] and the DFT calculations. In order to examine the performance of our TB potential at finite temperatures, we perform TBMD simulations on a solid–liquid coexistence structure [38]. In details, the half part of the supercell that consists of 864 Cu atoms is fixed at the initial solid structure, while the other part of the supercell is heated to 2500K so that this part is molten. Thus, we have a solid–liquid coexistence structure. Based on this solid–liquid coexistence structure, we perform MD simulations at different temperatures. At each temperature we carry out 22000 MD steps, with the time interval of 1 fs. Among these MD steps, the first 12000 MD steps run for the constant pressure ensemble (NPT) simulations so as to reach the thermal equilibrium, and the followed 10000 steps for the microcanonical ensemble (NVE) simulations. During the NVE MD simulations, the coordinates of atoms are extracted to calculate the meansquared displacement (MSD) [39] of all atoms. Here, the MSD is defined as MSDðtÞ ¼

N 1X jri ðtÞ  ri ð0Þj2 ; N i¼1

ð12Þ

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Fig. 2 The electronic energy bands of considered crystalline structures for Cu. Dotted curves are from the present TB calculations and the solid curves from the DFT calculations. The Fermi levels are at 0.0 eV

FCC structure

BCC stucture

0

Energy (eV)

Energy (eV)

0

-5

-10

Γ

X

W

-10

Γ

L

-5

K

Γ

H

HCP structure

Energy (eV)

Energy (eV)

P

0

-5

Γ

-2.0

M

Γ

K

A

H

-5

-10

A

Γ

X

X

K

M

R

Γ

M

X

R

8

FCC BCC HCP A12 A13 A15 SC Diamond

-2.5

L

TB Exp. DFT

6

ν(THz)

Energy(eV)

H

A15 structure

0

-10

Γ

N

-3.0

4

2

-3.5 2.1

2.4

2.7

0

Nearest-neighbor distance(Å)

Fig. 3 The binding energies as a function of nearest-neighbour distance for copper for different crystalline structures. The symbols are from our TB calculations, and the solid curves from the DFT calculations

Table 3 Elastic constants (in GPa) for the bulk Cu and formation energies (in eV) of point defects calculated using the present TB model

Γ

Γ

L

Fig. 4 Phonon dispersion curves calculated from the present TB potential (solid lines). The results indicated with triangles and pluses are respectively from the DFT calculations and experiment [24], which are used for comparison

Parameter

TB

Exp.

EAM or TB-SMA

NRL-TB [19]

C11 (Mbar)

1.63

1.700 [25]

1.76 [37]

1.61

C12 (Mbar)

1.36

1.225 [25]

1.25 [37]

1.08

C44 (Mbar)

0.72

0.758 [25]

0.82 [37]

0.55

EVf EIf EIf EIf

(eV)

1.75

1.27 [32],1.28 [33]

1.25 [37]

1.18

(eV) (100-dumbbell)

2.39

2.8–4.2 [34]

3.30 [37], 2.652 [35], 3.05 [36]

(eV) (Crowdion)

2.40

2.8–4.2 [34]

2.856 [35]

(eV) (Octahedral)

2.46

2.8–4.2 [34]

2.902 [35]

The results from the experiments [25, 32–34], from EAM [35, 36], TB-SMA [37] and NRL-TB calculations [19] are also listed for comparison

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30

20

2

MSD (Å )

1400 K 1600 K

10

0

2

4

8

6

10

Time (ps)

Fig. 5 The MSD varying with time for bulk copper at 1400 and 1600 K

where ri ðtÞ and ri ð0Þ are the position of the ith atom at time t and at the initial state, respectively. Fig. 5 displays the computed MSD. Clearly, the evolution of MSD vs time at 1600 K has a liquid behaviour, while that at 1400 K is not yet. So the melting point of Cu is about 1500  100 K, being quite close to the experimental [26] value of 1357 K. Stacking faults, surfaces and clusters In a realistic material, there exist point defects such as single vacancies and interstitial defects inevitably. These defects influence the mechanical and the electronic as well as the thermal properties of the material to some extent. Because of this, it is necessary to examine the ability of our potential model in handling these defects in bulk Cu. In our treatment, either a single vacancy or an interstitial Cu atom inside a cubic supercell consisting of 864 atoms is considered. As listed in Table 3, the formation energy of a vacancy, EVf , and the formation energy of the most stable self-interstitial defect (100-dumbbell), EIf , obtained from present potential model are 1.75 and 2.39 eV, respectively. These formation energies agree with those from the experiments [32–34] and from the EAM or TB-SMA calculations. Additionally, the stacking faults (SFs) in Cu are concerned. For the bulk Cu in the FCC lattice, the normal stacking sequence along the h111i direction is ABCABC. . .. In our testing calculations, three kinds of the stacking faults, such as the intrinsic SF, the extrinsic SF and the twin type SF [41]), are studied. Table 4 lists the formation Table 4 The formation energies for three kinds of stacking faults TB

EAM [41]

Exp. [40]

DFT

45

energies of these stacking faults from our TB calculations. Our calculated energies are in excellent agreement with those from the experiments [40] and from the DFT and the EAM [41] calculations. Surfaces of a material exhibit some unusual properties with respect to the bulk. This is because the bonding environment around an atom at surface is different from that in bulk. In present work, three kinds of surfaces, (100), (110) and (111), are fully relaxed by using the present TB potential. Table 5 shows our calculated surface energies. Apparently, our calculated values match those from both the experiment [42] and our DFT calculations. So, our TB model is also suitable to treat the surfaces of Cu. This agreement together with the results mentioned above strongly supports that our TB model is reliable to handle the defective Cu systems. To go further, we perform our testing calculations on the small-sized Cun (n ¼ 2  10) clusters. Table 6 summarises the resulting energies and the symmetries of our calculated clusters. The results available from experiments [43] and from the DFT calculations [44] are also listed in Table 6 for comparison. Clearly, both the relative stabilities and the symmetries of Cun (n = 3 - 8) clusters predicted by the present TB potential are all the same as those either from the experiments or from the DFT calculations, except for our binding energies of smaller-sized isomers which are overestimated relative to the related ones reported in experiments.

TBMD simulation for the irradiated Cu-heat-sink In a Tokamak fusion reactor, an energetic neutron from the plasma core can penetrate the PFMs into the Cu-heat-sinks, then collide with a Cu atom there. This causes the scattered Cu atom deviate from its initial position. This deviated atom is the so-called primary knock-on atom (PKA). The PKA collides with some of its neighbouring atoms, and then these neighbouring atoms collide with their neighbours sequentially. With this sequence, many Cu atoms deviate from their lattice sites, and thus the energetic neutron induces very complicate structural damage in the Cu-heat-sinks. In an irradiated event, a PKA which obtains

Table 5 The surface energies of the copper (100), (110) and (111) surfaces TB

DFT

NRL-TB[19]

Exp. [42] 1790

Intrinsic

64

78.1

37.9

ð111Þ

1922

1284

1730

Extrinsic

68

78.5

40.3

ð001Þ

2005

1426

1930

Twin type

38

39.3

26.8

ð110Þ

2187

1504

2040

The energy is in unit of mJ=m2

The energy is in unit of mJ=m2

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Table 6 The binding energies (BE) (eV/atom) and the point group (PG) of small-sized clusters obtained from the present TB calculations Cun

TB

DFT [44]

Exp. [43]

BE

PG

BE

PG

BE

2

1.36

D1h

1.04

D1h

1:02  0:08

3

1.37

C2v

0.98

C2v

1:06  0:12

4

1.59

D2

1.40

D2

1:48  0:14

5

1.64

C2

1.49

C2

1:55  0:15

6

1.81

D3h

1.67

D3h

1:71  0:18

7

1.82

D5h

1.72

D5h

1:85  0:22

8

1.91

C2v

1.81

C2v

2:00  0:23

9

1.92

C2v

1.80

Cs

10

1.96

D4d

1.84

D4d

The results from the experiment [43] and from DFT calculations [44] are listed for comparison

large kinetic energy from an energetic neutron induces many structural defects, and these defects evolute in a large space. In the present work, a large supercell containing 1372 Cu atoms is chosen, and a constrained MD scheme [45] is employed for the case of PKA with large kinetic energy. Figure 6 concisely describes the main feature of the constrained MD scheme, in which a proper free MD region around the PKA is selected and a thermostat region is outside the free MD region. In our simulations, the temperature of the thermostat region is set to be 300 K using the Berendsen temperature control technique [46].

On the atomistic scale, how to assess the damage caused by energetic particle irradiation in metals is a complicated issue [9, 10, 47–49]. To describe the structural damage of the material, several parameters are analysed. The typical parameters are (1) the number of ‘hot’ atoms (Nhot ) whose kinetic energies are larger than 1 eV, (2) the number of atoms (Ndispl ) [10, 49] that displace by more than half the nearest-neighbour distance away from their initial positions and (3) the number of vacancies (Nva ) [9, 10]. Since the ‘hot’ atoms are energetically active, they probably deflect from their initial positions and collide with other atoms, producing many new ‘hot’ atoms nearby. So, Nhot increases at the initial stage of the cascade. As the collision extends to a wide region, many ‘hot’ atoms transfer part of their energies to the other atoms, and accordingly some of the ‘hot’ atoms cool down, corresponding to the decrement of Nhot . This behaviour in the evolution of Nhot is indeed observed in our simulations. As displayed in Fig. 7a, Nhot increases abruptly during the initial 0.1 ps, and then drops to a constant. During the period of Nhot decreasing, the local region around the PKA is found to get reinstated gradually, with some of the displaced atoms occupying the lattice sites. This is reflected in the evolution of Ndispl shown in Fig. 7. As for Nva , it increases quickly at the initial stage of the cascade. With cooling down or recrystallization of the local region around the PKA, Nva decreases in about 2.2 ps. Finally, a few number of vacancies remain in the system only. These findings are quite similar to those reported from previous studies [4, 5, 10]. We note that the previous studies [4, 5, 10] mainly focused on how the irradiated defects evolute, but paid less attention to the evolution of

Number

(a)

60 Nva Nhot Ndispl

40 20

κ/κ0

(b)1.0 κxx/κxx0 κyy/κyy0

0.5

κzz/κzz0

Stress(GPa)

(c) 10

Pxx Pyy

5

Pzz

0

Fig. 6 The sketch of the constrained MD scheme used in atomistic simulations of irradiation effects in solid. The atoms in the areas which are hatched are allowed to move without any constraints (free MD), while the velocities of atoms at the thermostat region are scaled to allow for energy dissipation

123

1

2

3

4

Time (ps)

Fig. 7 The evolution of a Ndispl , Nhot , Nva , b thermal conductivity and c stress tensor as a function of time, for the the cases of PKA moving along the [001] directions at initial stage

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the mechanical and the thermal properties of the system with the changes of the structural defects. Since the irradiation causes the evolution of structural defects in the system in a very short time interval, the system is not at a stationary state but at a nonequilibrium state, at which the mechanical and the thermal properties of the Cu-heat-sink should be impacted harmfully. Therefore, this state of the material working in a nuclear reactor can be regarded as an ‘‘emergence state’’. We now study the thermal and the mechanical properties of the irradiated Cu material with evolution of defects using our TB potential model. Firstly, we pay our attention to the thermal conductivity of the irradiated Cu. As we know, the freely moving electrons transfer not only electric current but also thermal energy in Cu, and the thermal conductivity of a metal approximately tracks electrical conductivity according to the Wiedemann– Franz law. In general, the Boltzmann equation can describe the change of electron distribution induced by external fields or by various defects. Since the mechanism of electron scattering in a metal is very complex, it is very cumbersome to obtain an exact solution of the Boltzmann transport equation except for very simple systems. Usually, the relaxation time approximation is used to simplify the calculations about the Boltzmann equation. This approach has been demonstrated to reasonably evaluate the electrical transport properties of many bulk thermoelectric compounds [50–52]. Based on above approximation, the thermal conductivity of a material can be written as   1 pkB 2 j¼ rT; ð13Þ 3 e with the components of the electrical conductivity tensor   Z of0 dk rab ðTÞ ¼ 2e2 sva ðkÞvb ðkÞ : ð14Þ oðkÞ ð2pÞ3 Here 1 vðkÞ ¼ rk ðkÞ: 3

ð15Þ

In above equations, j, kB , r, s and e stand for the thermal conductivity tensor, the Boltzmann constant, the electrical conductivity tensor, the electronic relaxation time and the electron charge, respectively. k is the wave vector, and va ðkÞða ¼ x; y; zÞ is the ath component of the group velocity vðkÞ. In our calculations, with selecting s ¼ 4:3  1013 s, the thermal conductivity of the Cu system at temperature 300K is computed to be 399 W=ðm  KÞ, quite close to the experimental value 401 W=ðm  KÞ [26]. Furthermore, we compute the thermal conductivity of the irradiated system with a 500 eV PKA that initially moves along [001] direction. As is displayed in Fig. 7b, the thermal conductivity (jx ; jy ; jz ) of the system strikingly

decreases with accumulation of structural defects. When the number of defects reaches the maximum value, the thermal conductivity of the defective system is reduced to the minimum value. Apparently, the minimum value of the thermal conductivity is about a quarter of that before the cascade. This observation strongly indicates that the thermal conductivity of the Cu-heat-sink degrades largely at the ‘‘emergency case’’ arising from the accumulated structural defects in the system. In addition, our calculations show that although the PKA is moving along [001] direction initially, the evolution of thermal conductivity in [100] and [010] directions also has the same feature as that in the [001] direction. So, the local thermal property of the Cu-heat-sink is influenced by the irradiation of energetic particles in different directions. More importantly, the big degradation of the thermal conductivity heavily blocks the heat transport in the Cuheat-sink, although such a ‘‘heat blockade’’ occurs in a very short period. One may imagine that, in a reactor, if the initial kinetic energy of the PKA becomes large, and if energetic neutrons continuously irradiate the Cu-heat-sink, the defective region may extensively expand. Under this situation, the property of the heat transport in the Cu-heatsink should be degraded dramatically. Beside the thermal conductivity, the stress tensors, which can be used to analyse the instantaneous performance in the mechanics of the irradiated system during the cascade, are evaluated in the form of virial formula as given below ! 1 X 1XX Pab ¼ mðti Þa ðti Þb þ ðrij Þa ðf ij Þb ð16Þ X 2 i i6¼j i where X is the volume of the system, m is mass of the related atom, f ij is the force exerted on the atom i by the atom j, rij ¼ ri  rj , ðti Þa ða ¼ x; y; zÞ is the ath component of the velocity of atom i. As shown in Fig. 7c, Pzz is as high as 12 GPa, while Pxx and Pyy are nearly zero at the initial stage. Such a feature mainly correlates to the fact that the PKA starts to move along [001] direction. After the initial stage, the cascade starts. During the cascade, the PKA transfers parts of its kinetic energy and momentum to its neighbouring atoms via collision, and thus the local structure around the PKA in the system gets distorted heavily. Such a distorted region quickly extends in all directions, including along x and y directions. Undoubtedly, during the structural evolution, the increment of the strains in both x and y directions accompanies with partly releasing strains in z direction. So, the values of both Pxx and Pyy increase while the value of Pzz decreases at the initial stage of the cascade. This scenario is kept until the number of structural defects in the system reduce to a certain amount.

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What we are concerned here is that the irradiation of highly energetic particles can strikingly decrease the mechanical property of the local region around the PKA at a very short time interval. This, together with the influence on the thermal property as discussed above, should be considered in realistic application of the Cu-heat-sink in reactor.

Conclusions We develop an environment-dependent TB potential model for Cu. This potential model correctly predicts the relative stabilities, mechanical properties, phonon frequencies and the melting behaviour of bulk Cu. Tests for point defects, stacking faults, surfaces and clusters show that our potential model has an ability in handling the defective systems. Furthermore, by performing TBMD simulations, the structural damage from the collision of highly energetic neutron is reasonably described, where the correlation of the structural damage with the initial direction and the energy of the PKA is studied. Particularly, our simulations indicate that the high density of defects arising from the irradiation considerably degrades the performance in the heat transport and in the mechanical property of the Cuheat-sink. Acknowledgements This work is supported by the National Magnetic Confinement Fusion Science Program with No, of 2013GB107004, the National Science Foundation of China (Grant No. NSFC11275191 and NSFC11105140). The supercomputer centre of USTC is acknowledged for computational support.

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