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Irradiation-initiated plastic deformation in prestrained single-crystal copper
Nuclear Instruments and Methods in Physics Research B 368 (2016) 60–65
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Irradiation-initiated plastic deformation in prestrained single-crystal copper Bo Li a,b, Liang Wang b,c, Wu-Rong Jian b,d, Jun-Cheng E b,c, Hong-Hao Ma a,⇑, Sheng-Nian Luo b,c,* a
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, PR China The Peac Institute of Multiscale Sciences, Chengdu, Sichuan 610031, PR China c Key Laboratory of Advanced Technologies of Materials, Ministry of Education, Southwest Jiaotong University, Chengdu, Sichuan 610031, PR China d Department of Engineering Mechanics, South China University of Technology, Guangzhou, Guangdong 510640, PR China b
a r t i c l e
i n f o
Article history: Received 8 August 2015 Received in revised form 20 October 2015 Accepted 8 December 2015
Keywords: Irradiation Prestrain Dislocations Twinning Voids
a b s t r a c t With large-scale molecular dynamics simulations, we investigate the response of elastically prestrained single-crystal Cu to irradiation as regards the effects of prestrain magnitude and direction, as well as PKA (primary knock-on atom) energy. Under uniaxial tension, irradiation induces such defects as Frenkel pairs, stacking faults, twins, dislocations, and voids. Given the high dislocation concentration, twins and quad-stacking faults form through overlapping of different stacking faults. Voids nucleate via liquid cavitation, and dislocations around void play a lesser role in the void nucleation and growth. Dislocation density increases with increasing prestrain and PKA energy. At a given prestrain, there exists a critical PKA energy for dislocation activation, which decreases with increasing prestrain and depends on crystallographic direction of the applied prestrain. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Materials used in nuclear engineering have to endure extreme amounts of irradiation, and point defects (interstitials and vacancies) are created as a result [1–4]. Those defects can aggregate to form clusters, stacking fault tetrahedra and voids, causing degradation of materials integrity and performances [5–8]. To ensure long-time and safe service of nuclear reactors, it is necessary to understand microstructure evolution of materials under irradiation in detail. Collision cascade occurs over lengths of nanometers (nm) and times of picoseconds (ps). Computer simulations have the advantage in inquiring irradiation damage evolution in real time and in situ with high spatial resolutions. Molecular dynamics simulations is one of the most efficient implements, and has been widely used for studying irradiation damage. In nuclear engineering, materials often suffer applied strain due to irradiation induced void swelling and transmutation, as well as thermal and mechanical loads. Previous studies found that applied strains have distinct effects on, e.g., Frenkel pair production efficiency, defect cluster size, void density, and size range, under ⇑ Corresponding authors at: Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, PR China (H.-H. Ma), The Peac Institute of Multiscale Science, Chengdu, Sichuan 610031, PR China (S.-N. Luo). E-mail addresses: [email protected] (H.-H. Ma), [email protected] (S.-N. Luo). http://dx.doi.org/10.1016/j.nimb.2015.12.011 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.
irradiation [9–15]. However, most of them concentrated on low strain (small stress) conditions without extensive structure rearrangement. Local high strains or stresses in materials may also be induced by hydride precipitates, accumulation of He atoms, and so on. Recently, Di et al. [16] and Zolnikov et al. [17] observed that voids and twins formed after irradiation when high prestrains were applied to Zr and Fe. However, it is desirable to examine in detail, and in other types of materials, microstructure evolutions as well as the mechanisms of deformation, and void nucleation and growth under prestrain conditions. In this work, we investigate irradiation induced microstructure evolution in prestrained Cu, a model system for face-centered-cubic metals widely used in fundamental studies of irradiation damage [18–23]. We explore the effects of prestrain magnitude (e), PKA (primary knock-on atom) energy (EPKA ) and prestrain direction on radiation damage. Dislocation emissions and void nucleation are observed, and stacking fault interaction can lead to the formation of twins and quadstacking faults. Voids nucleate and grow via cavitation of melts induced by thermal spike during cascade. The relation between critical PKA energy (EcPKA ) and prestrain magnitude is established, and depends on crystallographic direction of the applied prestrain. This article is organized as follows. Section 2 addresses the methodology of MD simulations and data analysis, followed by results and detailed discussions on microstructure evolution in Section 3. Conclusions are presented in Section 4.
B. Li et al. / Nuclear Instruments and Methods in Physics Research B 368 (2016) 60–65
2. Methods We perform MD simulations using the Large-scale Atomic/ Molecular Massively Parallel Simulator (LAMMPS) [24] with an accurate embedded-atom-method (EAM) potential of Cu [25]. This widely used EAM potential has well established accuracy in describing, e.g., defect formation energy, stacking fault energy and melting point [25,26]. When interatomic distance is smaller than 0.5 Å, this EAM potential is smoothly splined to the Ziegler– Biersack–Littmark (ZBL) potential [27]. The coordinate system for irradiation simulations of single crystal Cu is defined in Fig. 1. Since temperature has remarkable effects on irradiation damages [17,22,28,29], sufficiently large system sizes are necessary. In our simulations, the initial configuration dimensions are about 540 Å along x-, y- and z-axes corresponding to 13 million atoms. The initial configuration is first equilibrated at 300 K and zero pressure with the constant-pressure–tempera ture ensemble for 50 ps to achieve thermal equilibrium. A tensile strain is then applied along one axis, while the stresses along the other two axes are fixed at zero. The new configuration is then subjected to extra equilibration at 300 K with constant-volumetemperature ensemble for 50 ps. To investigate the effects of prestrain magnitude and direction, prestrain varies from 0% to 5%, and three loading or prestrain directions are explored:
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0 and ½1 1 2. Since the yield strains along those three ½1 1 1; ½1 1 directions are about 7.8%, 7.4% [30] and 7.5% at 300 K on MD time and length scales, respectively, the final configurations are in elastically prestrained states. PKA is introduced into the system by assigning an atom specific kinematic energy corresponding to the desired PKA energy. For each prestrain/PKA energy condition, twenty simulations with different recoil directions and random PKA positions are carried out. A total of 1000 runs are conducted. Microcanonical ensemble (NVE) is used during irradiation simulations. The time step is fixed at 0.005 fs within the first 2 ps period of simulation, and then it is increased from 0.005 fs to 1 fs, as long as no atoms are displaced by more than 0.05 Å within a single time step. The total run durations are up to 50 ps. All simulations are conducted under three-dimensional (3D) periodic boundary conditions.
Fig. 3. Radiation-induced damage for different prestrains at t ¼ 50 ps (a), 7 ps (b) and 7 ps (c). Color-coding of atoms is based on their local structures. FCC: facecentered cubic; BCC: body-centered cubic; HCP: hexagonal close-packed. Prestrain direction: ½1 1 1; PKA energy: 5 keV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 1. Coordinate system for simulations of prestrained single crystal Cu. The cluster in the center denotes cascade region.
Fig. 2. Cascade volume vs. prestrain for different prestrain directions. PKA energy: 5 keV.
Fig. 4. Radiation-induced defects for different prestrains after collision cascading (t = 7 ps). Prestrain direction: ½1 1 1; PKA energy: 5 keV.
Fig. 5. Time evolutions of dislocation densities for different prestrains. Prestrain direction: ½1 1 1; PKA energy: 5 keV.
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We perform 3D binning analysis [31,32] to resolve spatially such physical properties as stress tensor (rij ). The binning width is 5 Å. To calculate rij within each bin, we remove its center-of i (i ¼ x; y and z), or apply corrections: mass velocity, v Drij ¼ ðNm=VÞv i v j , where m is the atomic mass, V is the bin volume, and N is the number of atoms in the volume under consideration. To describe the diffusion characteristics of cascade region, the mean squared displacements are calculated as MSD ¼ hjrðtÞ rð0Þj2 i, where h i denotes averaging over ensemble only, and r is the atomic position. The diffusion coefficient (D) is related to MSD via the Einstein relation, MSD ¼ 6Dt. Visualization and partial post-processing work are performed using a software package OVITO [33]. The dislocation extraction algorithm (DXA), which generates continuous lines corresponding to identified dislocations contained in an arbitrary crystal, is used to visualize and analyze dislocation evolution [34]. Twins, intrinsic and quad-stacking faults are identified via neighbor distance analysis (NDA) [35]. Common neighbor analysis (CNA) is also adopted to characterize local structures [36]. 3. Results and discussion 3.1. Prestrain magnitude dependence of irradiation damage The case of prestrain along the x-axis and 5 keV PKA energy is chosen to elucidate prestrain magnitude dependence of irradiation damage. We first examine the cascade region, which is represented by the disordered region during cascade (e.g., the biggest cluster in Fig. 1). The biggest volume of the disordered region achieved within 1 ps is cascade volume (V c ). As shown in Fig. 2, cascade volume increases with increasing prestrain magnitude. Fig. 3 shows snapshots after collision cascade for three typical prestrains. There is no dislocation nucleation at e ¼ 1% (Fig. 3a). However, the number of stable Frenkel pairs, N F ¼ 12, is more than
its counterpart for e ¼ 0 (unstrained, N F ¼ 10). This results is consistent with previous studies [11,15]. When prestrain increases to 3% or larger, dislocations nucleate and grow (Fig. 3b and c), indicating that prestrain facilitates dislocation formation. The corresponding defect structures are presented in Fig. 4. Sessile stair rod dislocations form as indicated by the dashed arrow (Fig. 4a), resulting from dislocation interactions. Full dislocations also appear with trailing partial dislocation motion as denoted by the solid arrow (Fig. 4b). At higher strains, more dislocations nucleate and grow simultaneously, and their propagation velocities are higher. Therefore, dislocation density (qd ) grows faster at higher strains (Fig. 5). Nevertheless, no slip system on ð1 1 1Þ (perpendicular to the prestrain direction) is activated, because there is no resolved prestress on this plane. Irradiation-induced twinning in prestrained Fe and V was reported by Zolnikov et al. [17], but the relevant details were not presented. Twinning also occurs in prestrained Cu (e ¼ 5%) irradiated at 5 keV (Fig. 6). Since qd is very high, twins form via overlapping of different stacking faults [37,38], different from wellknown pole mechanism [39,40]. In this pole mechanism, a twin forms through one partial dislocation climbing a screw dislocation pole to adjacent slip planes. Twinning evolution is illustrated schematically with color horizontal bars in Fig. 6. Two extended partial dislocations with one atom-layer offset glide toward each other, and the intrinsic stacking faults of the two dislocations in the overlapping region combine to form a twin. When another extended dislocation joins the overlapped pair, the twin broadens (Fig. 6d). Quad-stacking faults also form in a similar way, except that the stacking faults involved are offset initially by two atom layers. Besides dislocations, voids nucleate and grow at e ¼ 5%. The black curve in Fig. 7a presents the corresponding void volume evolution at 5 keV. There are three distinct stages identified, namely, incubation stage (0–3 ps), fast growth stage (3–13 ps) and stable stage (13 ps later) with small oscillations. Previous studies suggested that voids grow via dislocation glide and related
Fig. 6. Snapshots of defect evolution after collision cascading for e ¼ 5%. The horizontal bars illustrate schematically the microstructure features within the dashed rectangles related to twins; yellow, blue and red bars denote intrinsic stacking faults, twins and quad-stacking faults, respectively. Prestrain direction: ½1 1 1; PKA energy: 5 keV.
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B. Li et al. / Nuclear Instruments and Methods in Physics Research B 368 (2016) 60–65
Fig. 9. Critical PKA energy required for initiating dislocation for different prestrains along different prestrain directions.
Table 1 Fitting parameters for different prestrain directions (Eq. (1)). Loading direction
E0 (keV)
a (keV)
b
½1 1 1 0 ½1 1 ½1 1 2
0.001 0.005 0.020
64.5 64.5 64.5
0.71 0.80 0.69
cascade region can only be identified after collision cascading ends. Before 13 ps, D is 6:1 109 m2 =s, comparable to that for a liquid
Fig. 7. Time evolutions of void volume (a), MSD of the cascade region (b) and dislocation density (c) for e ¼ 5%. Insets in (a) denote voids. Prestrained direction: ½1 1 1.
[43]. Then, D decreases to 7:1 1010 m2 =s, typical for solids, as a result of cooling of the liquid and partial crystallization through heat conduction. Since the tensile stress is smaller than the tensile strength of the solid, void growth within the liquid terminates after crystallization. The transition point of the MSD slopes (13 ps; crystallization) corresponds to the slowdown of void growth (Fig. 7a and b). Therefore, void nucleation and growth are dominated by liquid cavitation, and dislocation around the void plays a lesser role. 3.2. Effects of PKA energy on irradiation damage
Fig. 8. Atomic configuration snapshots for different PKA energies after irradiation at t ¼ 50 ps (a), t ¼ 7 ps (b) and t ¼ 7 ps (c). Prestrained direction: ½1 1 1; prestrain magnitude: 5%.
deformation which provide means to transport atoms away from the cascade region [16], while cavitation appears to play a more important role in the void formation as seen from our simulations. Cavitation occurs when the external tensile stress exceeds the tensile strength of a liquid, followed by tensile stress relaxation [41,42]. During cavitation, the atomic distances decrease around the void (equivalent to compression when the stress changes from tension to close zero), accommodating void formation. Since thermal spike formed during cascade leads to local melting, it is natural to ask which factor dominates void nucleation and growth, atoms transportation or liquid cavitation. To this end, we calculate MSDs of the cascade region. Note that time origin is at 2 ps, because a
PKA energy is another important factor in irradiation damage. We explore in this section the effects of PKA energy on irradiation damage with a prestrain, e = 5%, applied along the x-axis. The cascade volumes for EPKA ¼ 1, 5 and 10 keV are 22.19, 97.63 and 193.98 nm3, respectively. Therefore, cascade volume increases with increasing PKA energy. Given that cavitation probability is proportional to liquid volume, more voids can nucleate at higher PKA energy. For example, two voids nucleate and then grow when EPKA increases from 5 keV to 10 keV (Figs. 7a and 8c). Therefore, void volume increases faster than its counterpart for EPKA ¼ 5 keV. On the contrary, no void nucleation occurs for EPKA ¼ 1 keV (Fig. 8b), because of the small cascade volume. Evolutions of dislocation density for EPKA ¼ 1, 5 and 10 keV plotted in Fig. 7(c) reveal their growth rates increase with PKA energy at the same time. Comparing to Fig. 5, it is found that dislocation density is more sensitive to prestrain magnitude than PKA energy. When Ep decreases to 0.025 keV, dislocation nucleation is absent, indicating that there exists a minimum PKA energy (critical PKA energy, EcPKA ) beyond which dislocations can nucleate and grow for a given prestrain. We obtain an EcPKA e curve in Fig. 9, which can be described as
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B. Li et al. / Nuclear Instruments and Methods in Physics Research B 368 (2016) 60–65
(b) and [1 1 0] (c). The tensile stresses are lower in Fig. 10. Stress maps on x–z planes at t ¼ 4 ps. The stress components are along the loading directions are [1 1 1] (a), [1 1 2] 2 the center due to cascading (red). The stresses are in GPa. Prestrain: 5%; PKA energy: 5 keV. Cross-section dimensions: 300 300 Å . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
e EcPKA ¼ E0 þ a exp : b
ð1Þ
The fitting parameters E0 , a and b are listed in Table 1. It clearly shows that the higher the prestrain magnitude, the lower the critical PKA energy is. For e ¼ 3% along the x-axis, EcPKA is 0.95 keV, and for e ¼ 5%, it deceases approximately by a factor of 20 times to 0.055 keV. When e approaches the elastic limit, EcPKA becomes zero as expected. 3.3. Prestrain direction effects on irradiation damage Since the mechanical properties of surface-free single crystal Cu are anisotropic, it is worthwhile to simulate cascade under different prestrained directions and show how this anisotropy affects irradiation damage. To this end, prestrain is applied on another 0 and ½1 1 2. two directions, ½1 1 For different prestrain directions, the atomic structures are different, so cascading characteristics are different including cascade volume. In addition, the elastic moduli are direction-dependent. 0 and ½1 1 2 Under identical prestrains, the prestresses along ½1 1 are smaller than that along ½1 1 1. Consequently, the tensile stres 0 and ½1 1 2 are too small to inises in the cascade region along ½1 1 tiate liquid cavitation, as opposed to the [1 1 1] case (Fig. 10). The EcPKA e curves for different prestrain directions are shown in Fig. 9. Since dislocation emissions are the results of prestrain coupled with individual cascade, EcPKA displays anisotropy regarding different prestrain directions. EcPKA is the largest for prestrain 0 at identical prestrains and its decreasing rate with along ½1 1 increasing prestain is the smallest, while the EcPKA e curves for prestrain along the other two directions are close to each other. Generally, EcPKA decreases with increasing prestrain and becomes zero when e approaches the elastic limit. Extrapolation to zero strain yields the critical PKA energy of 64.5 keV, regardless of prestrain direction. We also calculate EcPKA at the same prestress (4 GPa), being 2.5, 0, and ½1 1 2, respec1.2 and 1.65 keV for loading along ½1 1 1; ½1 1 tively. EcPKA is related to loading direction via Schmid factor: the higher the maximum absolute Schmid factor, the lower is the critical PKA energy. 4. Conclusions Molecular dynamics simulations have been performed on single-crystal Cu to investigate the effects of applied strains on irradiation damage. Besides Frenkel pair formation, dislocations and voids nucleation and growth are observed at high prestrain magnitudes under irradiation. As two different parallel stacking faults move toward each other and overlap, twins (and quad-stacking
faults) can form, and this mechanism is different from the conventional pole mechanism. At high prestrains (e.g., 5%), liquid cavitation dominates void nucleation and growth, while dislocation around void plays a lesser role. At a certain prestrain, there exists a critical PKA energy, below which dislocation glide cannot be activated. EcPKA decreases with increasing prestrain magnitude, and displays anisotropy regarding different prestrain directions. We establish the relation between EcPKA and prestrain for given loading directions; extrapolation to zero strain yields the critical PKA energy of 64 keV, regardless of the prestrain direction.
Acknowledgements This work was partially supported by NSFC and NSAF (Nos. 51174183, 51374189 and 11472253) of China.
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Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Irradiation-initiated plastic deformation in prestrained single-crystal copper Bo Li a,b, Liang Wang b,c, Wu-Rong Jian b,d, Jun-Cheng E b,c, Hong-Hao Ma a,⇑, Sheng-Nian Luo b,c,* a
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, PR China The Peac Institute of Multiscale Sciences, Chengdu, Sichuan 610031, PR China c Key Laboratory of Advanced Technologies of Materials, Ministry of Education, Southwest Jiaotong University, Chengdu, Sichuan 610031, PR China d Department of Engineering Mechanics, South China University of Technology, Guangzhou, Guangdong 510640, PR China b
a r t i c l e
i n f o
Article history: Received 8 August 2015 Received in revised form 20 October 2015 Accepted 8 December 2015
Keywords: Irradiation Prestrain Dislocations Twinning Voids
a b s t r a c t With large-scale molecular dynamics simulations, we investigate the response of elastically prestrained single-crystal Cu to irradiation as regards the effects of prestrain magnitude and direction, as well as PKA (primary knock-on atom) energy. Under uniaxial tension, irradiation induces such defects as Frenkel pairs, stacking faults, twins, dislocations, and voids. Given the high dislocation concentration, twins and quad-stacking faults form through overlapping of different stacking faults. Voids nucleate via liquid cavitation, and dislocations around void play a lesser role in the void nucleation and growth. Dislocation density increases with increasing prestrain and PKA energy. At a given prestrain, there exists a critical PKA energy for dislocation activation, which decreases with increasing prestrain and depends on crystallographic direction of the applied prestrain. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Materials used in nuclear engineering have to endure extreme amounts of irradiation, and point defects (interstitials and vacancies) are created as a result [1–4]. Those defects can aggregate to form clusters, stacking fault tetrahedra and voids, causing degradation of materials integrity and performances [5–8]. To ensure long-time and safe service of nuclear reactors, it is necessary to understand microstructure evolution of materials under irradiation in detail. Collision cascade occurs over lengths of nanometers (nm) and times of picoseconds (ps). Computer simulations have the advantage in inquiring irradiation damage evolution in real time and in situ with high spatial resolutions. Molecular dynamics simulations is one of the most efficient implements, and has been widely used for studying irradiation damage. In nuclear engineering, materials often suffer applied strain due to irradiation induced void swelling and transmutation, as well as thermal and mechanical loads. Previous studies found that applied strains have distinct effects on, e.g., Frenkel pair production efficiency, defect cluster size, void density, and size range, under ⇑ Corresponding authors at: Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, PR China (H.-H. Ma), The Peac Institute of Multiscale Science, Chengdu, Sichuan 610031, PR China (S.-N. Luo). E-mail addresses: [email protected] (H.-H. Ma), [email protected] (S.-N. Luo). http://dx.doi.org/10.1016/j.nimb.2015.12.011 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.
irradiation [9–15]. However, most of them concentrated on low strain (small stress) conditions without extensive structure rearrangement. Local high strains or stresses in materials may also be induced by hydride precipitates, accumulation of He atoms, and so on. Recently, Di et al. [16] and Zolnikov et al. [17] observed that voids and twins formed after irradiation when high prestrains were applied to Zr and Fe. However, it is desirable to examine in detail, and in other types of materials, microstructure evolutions as well as the mechanisms of deformation, and void nucleation and growth under prestrain conditions. In this work, we investigate irradiation induced microstructure evolution in prestrained Cu, a model system for face-centered-cubic metals widely used in fundamental studies of irradiation damage [18–23]. We explore the effects of prestrain magnitude (e), PKA (primary knock-on atom) energy (EPKA ) and prestrain direction on radiation damage. Dislocation emissions and void nucleation are observed, and stacking fault interaction can lead to the formation of twins and quadstacking faults. Voids nucleate and grow via cavitation of melts induced by thermal spike during cascade. The relation between critical PKA energy (EcPKA ) and prestrain magnitude is established, and depends on crystallographic direction of the applied prestrain. This article is organized as follows. Section 2 addresses the methodology of MD simulations and data analysis, followed by results and detailed discussions on microstructure evolution in Section 3. Conclusions are presented in Section 4.
B. Li et al. / Nuclear Instruments and Methods in Physics Research B 368 (2016) 60–65
2. Methods We perform MD simulations using the Large-scale Atomic/ Molecular Massively Parallel Simulator (LAMMPS) [24] with an accurate embedded-atom-method (EAM) potential of Cu [25]. This widely used EAM potential has well established accuracy in describing, e.g., defect formation energy, stacking fault energy and melting point [25,26]. When interatomic distance is smaller than 0.5 Å, this EAM potential is smoothly splined to the Ziegler– Biersack–Littmark (ZBL) potential [27]. The coordinate system for irradiation simulations of single crystal Cu is defined in Fig. 1. Since temperature has remarkable effects on irradiation damages [17,22,28,29], sufficiently large system sizes are necessary. In our simulations, the initial configuration dimensions are about 540 Å along x-, y- and z-axes corresponding to 13 million atoms. The initial configuration is first equilibrated at 300 K and zero pressure with the constant-pressure–tempera ture ensemble for 50 ps to achieve thermal equilibrium. A tensile strain is then applied along one axis, while the stresses along the other two axes are fixed at zero. The new configuration is then subjected to extra equilibration at 300 K with constant-volumetemperature ensemble for 50 ps. To investigate the effects of prestrain magnitude and direction, prestrain varies from 0% to 5%, and three loading or prestrain directions are explored:
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0 and ½1 1 2. Since the yield strains along those three ½1 1 1; ½1 1 directions are about 7.8%, 7.4% [30] and 7.5% at 300 K on MD time and length scales, respectively, the final configurations are in elastically prestrained states. PKA is introduced into the system by assigning an atom specific kinematic energy corresponding to the desired PKA energy. For each prestrain/PKA energy condition, twenty simulations with different recoil directions and random PKA positions are carried out. A total of 1000 runs are conducted. Microcanonical ensemble (NVE) is used during irradiation simulations. The time step is fixed at 0.005 fs within the first 2 ps period of simulation, and then it is increased from 0.005 fs to 1 fs, as long as no atoms are displaced by more than 0.05 Å within a single time step. The total run durations are up to 50 ps. All simulations are conducted under three-dimensional (3D) periodic boundary conditions.
Fig. 3. Radiation-induced damage for different prestrains at t ¼ 50 ps (a), 7 ps (b) and 7 ps (c). Color-coding of atoms is based on their local structures. FCC: facecentered cubic; BCC: body-centered cubic; HCP: hexagonal close-packed. Prestrain direction: ½1 1 1; PKA energy: 5 keV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 1. Coordinate system for simulations of prestrained single crystal Cu. The cluster in the center denotes cascade region.
Fig. 2. Cascade volume vs. prestrain for different prestrain directions. PKA energy: 5 keV.
Fig. 4. Radiation-induced defects for different prestrains after collision cascading (t = 7 ps). Prestrain direction: ½1 1 1; PKA energy: 5 keV.
Fig. 5. Time evolutions of dislocation densities for different prestrains. Prestrain direction: ½1 1 1; PKA energy: 5 keV.
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We perform 3D binning analysis [31,32] to resolve spatially such physical properties as stress tensor (rij ). The binning width is 5 Å. To calculate rij within each bin, we remove its center-of i (i ¼ x; y and z), or apply corrections: mass velocity, v Drij ¼ ðNm=VÞv i v j , where m is the atomic mass, V is the bin volume, and N is the number of atoms in the volume under consideration. To describe the diffusion characteristics of cascade region, the mean squared displacements are calculated as MSD ¼ hjrðtÞ rð0Þj2 i, where h i denotes averaging over ensemble only, and r is the atomic position. The diffusion coefficient (D) is related to MSD via the Einstein relation, MSD ¼ 6Dt. Visualization and partial post-processing work are performed using a software package OVITO [33]. The dislocation extraction algorithm (DXA), which generates continuous lines corresponding to identified dislocations contained in an arbitrary crystal, is used to visualize and analyze dislocation evolution [34]. Twins, intrinsic and quad-stacking faults are identified via neighbor distance analysis (NDA) [35]. Common neighbor analysis (CNA) is also adopted to characterize local structures [36]. 3. Results and discussion 3.1. Prestrain magnitude dependence of irradiation damage The case of prestrain along the x-axis and 5 keV PKA energy is chosen to elucidate prestrain magnitude dependence of irradiation damage. We first examine the cascade region, which is represented by the disordered region during cascade (e.g., the biggest cluster in Fig. 1). The biggest volume of the disordered region achieved within 1 ps is cascade volume (V c ). As shown in Fig. 2, cascade volume increases with increasing prestrain magnitude. Fig. 3 shows snapshots after collision cascade for three typical prestrains. There is no dislocation nucleation at e ¼ 1% (Fig. 3a). However, the number of stable Frenkel pairs, N F ¼ 12, is more than
its counterpart for e ¼ 0 (unstrained, N F ¼ 10). This results is consistent with previous studies [11,15]. When prestrain increases to 3% or larger, dislocations nucleate and grow (Fig. 3b and c), indicating that prestrain facilitates dislocation formation. The corresponding defect structures are presented in Fig. 4. Sessile stair rod dislocations form as indicated by the dashed arrow (Fig. 4a), resulting from dislocation interactions. Full dislocations also appear with trailing partial dislocation motion as denoted by the solid arrow (Fig. 4b). At higher strains, more dislocations nucleate and grow simultaneously, and their propagation velocities are higher. Therefore, dislocation density (qd ) grows faster at higher strains (Fig. 5). Nevertheless, no slip system on ð1 1 1Þ (perpendicular to the prestrain direction) is activated, because there is no resolved prestress on this plane. Irradiation-induced twinning in prestrained Fe and V was reported by Zolnikov et al. [17], but the relevant details were not presented. Twinning also occurs in prestrained Cu (e ¼ 5%) irradiated at 5 keV (Fig. 6). Since qd is very high, twins form via overlapping of different stacking faults [37,38], different from wellknown pole mechanism [39,40]. In this pole mechanism, a twin forms through one partial dislocation climbing a screw dislocation pole to adjacent slip planes. Twinning evolution is illustrated schematically with color horizontal bars in Fig. 6. Two extended partial dislocations with one atom-layer offset glide toward each other, and the intrinsic stacking faults of the two dislocations in the overlapping region combine to form a twin. When another extended dislocation joins the overlapped pair, the twin broadens (Fig. 6d). Quad-stacking faults also form in a similar way, except that the stacking faults involved are offset initially by two atom layers. Besides dislocations, voids nucleate and grow at e ¼ 5%. The black curve in Fig. 7a presents the corresponding void volume evolution at 5 keV. There are three distinct stages identified, namely, incubation stage (0–3 ps), fast growth stage (3–13 ps) and stable stage (13 ps later) with small oscillations. Previous studies suggested that voids grow via dislocation glide and related
Fig. 6. Snapshots of defect evolution after collision cascading for e ¼ 5%. The horizontal bars illustrate schematically the microstructure features within the dashed rectangles related to twins; yellow, blue and red bars denote intrinsic stacking faults, twins and quad-stacking faults, respectively. Prestrain direction: ½1 1 1; PKA energy: 5 keV.
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Fig. 9. Critical PKA energy required for initiating dislocation for different prestrains along different prestrain directions.
Table 1 Fitting parameters for different prestrain directions (Eq. (1)). Loading direction
E0 (keV)
a (keV)
b
½1 1 1 0 ½1 1 ½1 1 2
0.001 0.005 0.020
64.5 64.5 64.5
0.71 0.80 0.69
cascade region can only be identified after collision cascading ends. Before 13 ps, D is 6:1 109 m2 =s, comparable to that for a liquid
Fig. 7. Time evolutions of void volume (a), MSD of the cascade region (b) and dislocation density (c) for e ¼ 5%. Insets in (a) denote voids. Prestrained direction: ½1 1 1.
[43]. Then, D decreases to 7:1 1010 m2 =s, typical for solids, as a result of cooling of the liquid and partial crystallization through heat conduction. Since the tensile stress is smaller than the tensile strength of the solid, void growth within the liquid terminates after crystallization. The transition point of the MSD slopes (13 ps; crystallization) corresponds to the slowdown of void growth (Fig. 7a and b). Therefore, void nucleation and growth are dominated by liquid cavitation, and dislocation around the void plays a lesser role. 3.2. Effects of PKA energy on irradiation damage
Fig. 8. Atomic configuration snapshots for different PKA energies after irradiation at t ¼ 50 ps (a), t ¼ 7 ps (b) and t ¼ 7 ps (c). Prestrained direction: ½1 1 1; prestrain magnitude: 5%.
deformation which provide means to transport atoms away from the cascade region [16], while cavitation appears to play a more important role in the void formation as seen from our simulations. Cavitation occurs when the external tensile stress exceeds the tensile strength of a liquid, followed by tensile stress relaxation [41,42]. During cavitation, the atomic distances decrease around the void (equivalent to compression when the stress changes from tension to close zero), accommodating void formation. Since thermal spike formed during cascade leads to local melting, it is natural to ask which factor dominates void nucleation and growth, atoms transportation or liquid cavitation. To this end, we calculate MSDs of the cascade region. Note that time origin is at 2 ps, because a
PKA energy is another important factor in irradiation damage. We explore in this section the effects of PKA energy on irradiation damage with a prestrain, e = 5%, applied along the x-axis. The cascade volumes for EPKA ¼ 1, 5 and 10 keV are 22.19, 97.63 and 193.98 nm3, respectively. Therefore, cascade volume increases with increasing PKA energy. Given that cavitation probability is proportional to liquid volume, more voids can nucleate at higher PKA energy. For example, two voids nucleate and then grow when EPKA increases from 5 keV to 10 keV (Figs. 7a and 8c). Therefore, void volume increases faster than its counterpart for EPKA ¼ 5 keV. On the contrary, no void nucleation occurs for EPKA ¼ 1 keV (Fig. 8b), because of the small cascade volume. Evolutions of dislocation density for EPKA ¼ 1, 5 and 10 keV plotted in Fig. 7(c) reveal their growth rates increase with PKA energy at the same time. Comparing to Fig. 5, it is found that dislocation density is more sensitive to prestrain magnitude than PKA energy. When Ep decreases to 0.025 keV, dislocation nucleation is absent, indicating that there exists a minimum PKA energy (critical PKA energy, EcPKA ) beyond which dislocations can nucleate and grow for a given prestrain. We obtain an EcPKA e curve in Fig. 9, which can be described as
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(b) and [1 1 0] (c). The tensile stresses are lower in Fig. 10. Stress maps on x–z planes at t ¼ 4 ps. The stress components are along the loading directions are [1 1 1] (a), [1 1 2] 2 the center due to cascading (red). The stresses are in GPa. Prestrain: 5%; PKA energy: 5 keV. Cross-section dimensions: 300 300 Å . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
e EcPKA ¼ E0 þ a exp : b
ð1Þ
The fitting parameters E0 , a and b are listed in Table 1. It clearly shows that the higher the prestrain magnitude, the lower the critical PKA energy is. For e ¼ 3% along the x-axis, EcPKA is 0.95 keV, and for e ¼ 5%, it deceases approximately by a factor of 20 times to 0.055 keV. When e approaches the elastic limit, EcPKA becomes zero as expected. 3.3. Prestrain direction effects on irradiation damage Since the mechanical properties of surface-free single crystal Cu are anisotropic, it is worthwhile to simulate cascade under different prestrained directions and show how this anisotropy affects irradiation damage. To this end, prestrain is applied on another 0 and ½1 1 2. two directions, ½1 1 For different prestrain directions, the atomic structures are different, so cascading characteristics are different including cascade volume. In addition, the elastic moduli are direction-dependent. 0 and ½1 1 2 Under identical prestrains, the prestresses along ½1 1 are smaller than that along ½1 1 1. Consequently, the tensile stres 0 and ½1 1 2 are too small to inises in the cascade region along ½1 1 tiate liquid cavitation, as opposed to the [1 1 1] case (Fig. 10). The EcPKA e curves for different prestrain directions are shown in Fig. 9. Since dislocation emissions are the results of prestrain coupled with individual cascade, EcPKA displays anisotropy regarding different prestrain directions. EcPKA is the largest for prestrain 0 at identical prestrains and its decreasing rate with along ½1 1 increasing prestain is the smallest, while the EcPKA e curves for prestrain along the other two directions are close to each other. Generally, EcPKA decreases with increasing prestrain and becomes zero when e approaches the elastic limit. Extrapolation to zero strain yields the critical PKA energy of 64.5 keV, regardless of prestrain direction. We also calculate EcPKA at the same prestress (4 GPa), being 2.5, 0, and ½1 1 2, respec1.2 and 1.65 keV for loading along ½1 1 1; ½1 1 tively. EcPKA is related to loading direction via Schmid factor: the higher the maximum absolute Schmid factor, the lower is the critical PKA energy. 4. Conclusions Molecular dynamics simulations have been performed on single-crystal Cu to investigate the effects of applied strains on irradiation damage. Besides Frenkel pair formation, dislocations and voids nucleation and growth are observed at high prestrain magnitudes under irradiation. As two different parallel stacking faults move toward each other and overlap, twins (and quad-stacking
faults) can form, and this mechanism is different from the conventional pole mechanism. At high prestrains (e.g., 5%), liquid cavitation dominates void nucleation and growth, while dislocation around void plays a lesser role. At a certain prestrain, there exists a critical PKA energy, below which dislocation glide cannot be activated. EcPKA decreases with increasing prestrain magnitude, and displays anisotropy regarding different prestrain directions. We establish the relation between EcPKA and prestrain for given loading directions; extrapolation to zero strain yields the critical PKA energy of 64 keV, regardless of the prestrain direction.
Acknowledgements This work was partially supported by NSFC and NSAF (Nos. 51174183, 51374189 and 11472253) of China.
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