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Grain boundary migration in nanocrystalline Ni under constant shear strains and its mechanism
Computational Materials Science 176 (2020) 109530
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Grain boundary migration in nanocrystalline Ni under constant shear strains and its mechanism
T
⁎
Xinhua Yanga,b, Jie Lia, , Peng Wangc a
School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, 1037 Luoyu Road, 430074 Wuhan, China c College of Engineering, Huazhong Agricultural University, Wuhan 430070, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Nanocrystalline Ni Grain boundary migration Stress redistribution Molecular dynamics
The grain boundary (GB) migration behavior of nanocrystalline Ni during shear dwelling was simulated using the molecular dynamics method. It was found that further GB migration could take place and be completed within a very short period of time when the external shear load is kept unchanged. The atomic shear stress redistribution was demonstrated to be the driving force. Both the dwelling strain and temperature have considerable complicated influences on the further GB migration. The local atomic shear stress difference around the GB was proposed to characterize the unevenness of atomic shear stress. A threshold condition was given in terms of the local atomic shear stress difference, namely only when it is larger than its threshold value would the further GB migration occur. The threshold value was quantitatively related to the temperature. The effects of dwelling strain and temperature on the further GB migration can be well explained.
1. Introduction Compared to coarse-grained materials, nanocrystalline materials reveal better physical and mechanical properties in many respects, such as higher strength and hardness [1–3]. This is attributed to the fact that grain boundaries (GBs) account for a larger proportion in them. Moreover, GBs were proved to play an important role in plastic deformation [2,4–7]. As the relevant research progressed, more GB complexions were discovered, enriching the understanding of material properties [8–10]. Grain growth is ubiquitous in nanocrystalline materials. As one of its important mechanisms, the GB migration was observed in the theoretical analyses [11] and a large number of experiments [12–16]. It is usually driven by the shear stress [13,14,16] and also closely related to the thermally assisted diffusion mechanism [17]. Normal GB migration perpendicular to a GB plane is usually accompanied by GB sliding along the tangential direction of the GB plane [18,19]. Many investigations were carried out on the shear-coupled GB motion in bicrystals [20–28]. By comparison, it was found that the molecular dynamics (MD) simulations are very consistent with the theoretical analyses [29] and experiments [30,31] for the bicrystals. However, the GB motion occurring in nanocrystals is significantly different from that occurring in bicrystals due to the pinning effect of triple junctions [11,32,33]. In nanocrystalline materials, the normal migration displacement is generally non-uniform along a GB and easy ⁎
to saturate [34]. Dislocation nucleation and propagation could change the stress distribution in grain interiors in the GB motion process. In addition, the interaction between dislocations and GBs could disorder the elementary structural units of GBs to prevent GB migration [32,34]. It is worth noting that materials generally are viscoelastic. When a certain constant load is applied for a period of time, which is called dwelling, creep deformation would possibly occur. Plastic deformation is highly time-dependent for nanocrystalline materials, so creep and stress relaxation can be observed even at room temperature [35–37]. GB diffusion, GB sliding and grain rotation are considered to be predominant mechanisms of creep deformation in nanocrystals [38,39]. The plastic deformation is dominated by dislocations and GB diffusion at different stages of stress relaxation [40], and the interactions between mobile dislocations and GBs are involved in the stress relaxation process [41]. Naturally, the macroscopic viscosity can be related to GBs mechanisms. Stress-driven grain growth during dwelling was observed in some indentation experiments of nanocrystalline Cu [42–44] even in cryogenic temperature [43]. This indicates that GB migration could occur in a constant external load condition. The MD indentation simulations were performed on nanocrystals as well [45,46]. Huang et al. [46] argued that GB motion dominates the deformation during dwelling for a small grain size. Tucker and Foiles [45] found that grain growth mechanisms are very complex in the case of dwelling, including GB
Corresponding author. E-mail address: [email protected] (J. Li).
https://doi.org/10.1016/j.commatsci.2020.109530 Received 15 September 2019; Received in revised form 6 January 2020; Accepted 7 January 2020 0927-0256/ © 2020 Elsevier B.V. All rights reserved.
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X. Yang, et al.
Fig. 2. Loading history curves for different dwelling strains.
600 ps to realize shear dwelling. The loading history curves are plotted in Fig. 2 for different dwelling strains. Strong inertial effects are possibly observed if a shear stress is imposed suddenly in MD simulations [52,53]. To reduce these effects, an equivalent quasi-static loading should be imposed and the system should be relaxed with a sufficient number of MD time-steps at each loading step [54]. Fig. 3(a) shows the nominal-shear-stress vs. nominalshear-strain curves at a strain rate of 2 × 108 s−1 for different numbers of relaxation steps. The stress–strain curve almost does not change when the relaxation step number is larger than 1000. It means a little inertial effect. Then we fixed the relaxation step number at 2000 and change the strain rate from 5 × 107 s−1 to 1 × 108 s−1, 2 × 108 s−1, 4 × 108 s−1, and 1 × 109 s−1. Fig. 3(b) shows their nominal-shearstress vs. nominal-shear-strain curves. The loading rate has a slight effect on the stress–strain response when it is less than 4 × 108 s−1, which was also confirmed in Ref. [50]. The three-dimensional visualization software OVITO [55] was used to visualize the calculation results. The common neighbor analysis method [56] was used to identify the atomic configurations. All the atoms were colored according to their configurations. Face-centered cubic (FCC) atoms are marked blue, hexagonal close packed (HCP)
Fig. 1. Initial configuration of nanocrystalline Ni model with a horizontal Σ9 < 1 1 0 > {2 2 1} symmetric tilt GB.
relaxation, atomic rearrangement, etc. To this day, however, GB migration during dwelling is not fully understood yet. How does GB migration occur in nanocrystals during dwelling? What factor is the predominant driving force? How do load and temperature affect GB migration? Their answers are important for in-depth understanding of GB migration in a dwelling condition. In this paper, the MD method is used to simulate the GB migration behavior of nanocrystalline Ni at various dwelling strains and temperatures, in an attempt to answer the above questions. 2. Model and computational method A quasi-three-dimensional atomic model of nanocrystalline Ni with − idealized [110] texture was constructed, as shown in Fig. 1. The outline − − dimensions of the model in the X [1 1 4], Y [221] and Z [110] directions are 49.3 nm, 49.3 nm and 2.0 nm, respectively. As the main observation object, a Σ9 < 1 1 0 > {2 2 1} symmetric tilt GB with a length of 16 nm flatly lies in the middle of the model. The large-scale atomic/molecular massively parallel simulator (LAMMPS) [47] was used for all MD simulations in conjunction with the embedded atom method potential for Ni [48]. This potential was demonstrated to give a good evaluation of many material properties, including the elastic constants, the vacancy formation and migration energies, the stacking fault energies, and the surface energies [48]. Thus, it was successfully used for many numerical investigations of GB migration behaviors [23,34,49–51]. The periodic boundary conditions were imposed in the X and Z directions, while free boundary conditions were applied in the Y direction. Unless otherwise stated, the canonical NVT ensemble with constant number of atoms, volume and temperature was employed in the computational system. The simulation steps are given as follows. (1) The model was relaxed at room temperature (300 K) for 100 ps, and then relaxed again for 100 ps to reach the initial equilibrium state after the temperature was slowly reduced to 10 K. (2) Several layers of atoms in the top of the upper grain and in the bottom of the lower grain are set as two slabs. The displacement was applied at the top slab in the X direction to induce the shear deformation, while the bottom slab was fixed. A constant strain rate of 2 × 108 s−1 was used in all the simulations. (3) When the nominal shear strain reached a given value, the strain was kept for
Fig. 3(a). Nominal-shear-stress vs. nominal-shear-strain curves at a strain rate of 2 × 108 s−1 for different numbers of relaxation steps. 2
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Fig. 3(b). Nominal-shear-stress vs. nominal-shear-strain curves for a relaxation step number of 2000 at different strain rates. Fig. 5. Atomic configurations of the observed GB at different dwelling time points of (a) 0, (b) 10 ps, (c) 12 ps, (d) 20 ps, (e) 24 ps and (f) 600 ps.
atoms are marked light blue, and the remaining atoms are marked magenta.
To check whether the further GB migration would occur in a bicrystal during dwelling, a Ni bicrystal model with the same geometrical, loading and computational conditions was simulated. Its GB migration displacement vs. time curve is also plotted in Fig. 4 by a red solid line. It can be found that the curve always remains horizontal after the dwelling starts. This means that no further GB migration occurs in a bicrystal during dwelling. Accordingly, it is possible that further GB migration only occurs in a nanocrystal during dwelling. It is known that GB migration is generally achieved by transformation of elementary structural units [23–27,29,49,50], so that GB configuration constantly evolves in a process of GB migration. To analyze the GB migration mechanism of nanocrystalline Ni during dwelling, Fig. 5 shows the GB atomic configurations at six dwelling time points shown in Fig. 4. The six time points (a), (b), (c), (d), (e), and (f) correspond to the dwelling time of 0, 10 ps, 12 ps, 20 ps, 24 ps and 600 ps, respectively. For the sake of observation, all the FCC atoms are removed and the remaining atoms are colored according to their Y coordinates in Fig. 5. It is found that the ordered |EE| structural units [23,34,57] stay in the middle of the GB in Fig. 5(a), but the GB configurations are unordered at the two ends near the triple junctions. Two symmetric steps are formed at the joints between the ordered and unordered units, as marked by the black arrows in Fig. 5(a). They are called GB disconnections [58,59]. At the beginning of dwelling, the two disconnections move in opposite directions towards the triple junctions. As they are far away from each other, a new pair of disconnections is generated at the middle of the GB, as marked by the red arrows in Fig. 5(b). At the same time, the first migration begins to occur, which corresponds to the ‘slip’ behavior at the time point (b) shown in Fig. 4. Subsequently, the two new disconnections constantly move towards both the ends of the GB, as shown in Fig. 5(c) and (d), which corresponds to the ‘stick’ behavior from the time point (c) to (d) in Fig. 4. After this migration, new disconnections nucleate and move quickly, as marked by the green arrows in Fig. 5(e). As a result, the GB completes its second migration between the time point (d) and (e), and there is a transformation of elementary structural units of |EE|, as shown in Fig. 5(d) and (e). The corresponding ‘slip’ behavior can be also seen in Fig. 4. After the time point (e), the GB configuration, as shown in Fig. 5(f), remains substantially unchanged. It means that the further GB migration is stopped. Obviously, the external load is kept unchanged during dwelling, so
3. Results and discussion 3.1. Characteristics and mechanism of further GB migration during dwelling Some atoms in the intermediate region of the target GB were tracked in the simulations and their average Y-axis coordinate variations were considered as the GB migration displacement. In order to investigate characteristics of GB migration behavior during dwelling, the 6.5% dwelling strain case was taken as an example here. The GB migration displacement vs. time curve is plotted in Fig. 4. For comparison, the loading history curve represented by the nominal shear strain is also plotted in Fig. 4 by a blue dashed line. It can be seen that the target GB does not stop migrating at once after the dwelling starts. The twice migrations take place in a very short interval from the time point (a) to (e) shown in Fig. 4. After the further GB migration is completed, the curve keeps straight. It means that the GB migration is stopped.
Fig. 4. GB migration displacement vs. time curves for a nanocrystal and a bicrystal, where dwelling time points of (a) 0, (b) 10 ps, (c) 12 ps, (d) 20 ps, (e) 24 ps and (f) 600 ps are marked. 3
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Fig. 6. Atomic shear stress contours at the dwelling time points of (a) 0, (b) 10 ps, (c) 12 ps, (d) 20 ps, (e) 24 ps and (f) 600 ps.
what is the force to drive the GB to migrate further? In order to answer this question, Fig. 6 shows the atomic shear stress contours at the six different dwelling time points in the same scaling ratio. It is known that the atomic stress can almost be completely relaxed by GB motion in bicrystals [34]. Different from the bicrystal case, it is found from Fig. 6(a) that the atomic shear stress distribution is uneven at the beginning of dwelling. There is higher stress concentrated within the grains but lower stress near the observed GB, as marked by black dotted ellipses in Fig. 6. It can be found that the area of the lower stress region increases from the time point (a) to (b), but then decreases from (b) to (c) and (d). From the time point (d) to (e), however, the lower stress region area increases again, but the stress around the triple junctions is significantly reduced. Consequently, the stress distribution is more uniform in Fig. 6(f) than in Fig. 6(e). It indicates that the GB disconnection nucleation significantly relaxes the stress near the GB, as shown in Fig. 6(b) and (e). But the disconnection movement has little effect on the stress distribution, as shown in Fig. 6(c) and (d). It is consistent with the fact that it only requires lower stress to move a preexisting disconnection than to nucleate and form a new one [60]. Accordingly, the disconnection nucleation usually corresponds to a significant change in the shear stress distribution. In other words, the uneven shear stress distribution can drive the GB disconnection to nucleate. And then the atomic shear stress distribution becomes uniform after the GB migration. When there is not enough shear stress difference to drive a new disconnection to nucleate, the GB migration will come to an end. Does the atomic shear stress redistribution during dwelling change the nominal shear stress of the computational system? Fig. 7 plots the curves of the nominal shear stress and the further GB migration displacement vs. the dwelling time. The time points corresponding to (b), (c), (d) and (e) in Fig. 4 are marked with black dashed lines in Fig. 7. It can be seen that the stress-time curve decreases monotonically before point (e) but slightly fluctuate after it. It also proves that GB motion is a plastic relaxation mechanism [12]. This descend stage can be roughly divided into three sections. The stress descend is very sharp in Sections I and III but gentle in Section II. Section I corresponds to the interval between the time points (a) and (b) shown in Fig. 4, Section II corresponds to the interval from the time point (b) to (d), and Section III corresponds to the interval between the time points (d) and (e). There is a new pair of disconnections nucleating in Sections I and III but no in
Fig. 7. Curves of nominal shear stress and further GB migration displacement vs. dwelling time.
Section II. Thus, the nucleation of GB disconnections can obviously reduce the nominal shear stress during dwelling, but the movement of disconnections has little effect on the stress. 3.2. Threshold condition of further GB migration To investigate the conditions under which the further GB migration occurs, the dwelling strain was changed from 3% to 7% with a constant interval of 0.5%. Fig. 8 plots the curves of the further GB migration displacement vs. the dwelling time for different dwelling strains. It can be seen that no GB migration occurs for the dwelling strains less than 4%. When the dwelling strain lies between 4.5% and 7%, however, the GB migration is rapidly completed in a short period of time and then almost becomes unchanged. There are different migration displacements for different dwelling strains. As the dwelling strain reaches 7%, no GB migration occurs during dwelling again. In order to evaluate the effect of dwelling strain on the GB migration quantitatively, more dwelling strain cases with an increment of 0.1% 4
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Fig. 8. Further GB migration displacement vs. dwelling time curves for different dwelling strains.
Fig. 10. Variation of local atomic shear stress difference with dwelling time at 6.5% dwelling strain.
away from the observed GB. Area B is near the observed GB. The average atomic shear stresses in areas A and B, τA and τB , were recorded [63–66]. The stress difference Δτ = τA − τB was used for characterizing the heterogeneity degree of atomic shear stress quantitatively and called the local atomic shear stresses difference. Fig. 10 shows the variation of Δτ with dwelling time at 6.5% dwelling strain. As the dwelling time goes by, Δτ has two sharp drops, which begin at points (a) and (b) shown in Fig. 10, respectively. The two drops correspond to the first and second GB migrations. After each drop, Δτ enters into a repeated fluctuation process. This suggests that Δτ can be treated as the driving force of further GB migration. Fig. 11 shows the variations of τA , τB , and Δτ corresponding to point (a) shown in Fig. 10 with the dwelling strain. It can be seen that τA has a considerable increase trend with increasing dwelling strain as a whole, while τB rises at a moderate pace. As a result, their difference Δτ has also an increase trend similar to τA . However, Δτ has a sudden drop after the dwelling strain exceeds 6.8%. How should it be explained why the GB sometimes migrates but sometimes does not migrate for different dwelling strains? We assume that the GB must overcome a certain resistance to migrate. The GB migration resistance is characterized
Fig. 9. Scatter plot of further GB migration displacement vs. dwelling strain as well as nominal shear stress vs. dwelling strain curve.
from 3% to 7% were considered. Fig. 9 shows their further migration displacements during dwelling. In this figure, a dotted line is used to show the variation trend of the migration displacement with the dwelling strain. The migration displacement goes up and down with increasing dwelling strain but has a rough increase trend as a whole. At the dwelling strain of about 5.5% and 6.8%, the GB migration displacement reaches its peak values. And the further migration occurs only at 4.5% dwelling strain when the dwelling strain is less than 5%. For comparison, Fig. 9 also plots the nominal shear stress vs. dwelling strain curve by a blue solid line. It can be found that there is a rough correspondence between the two curves. Therefore, the GB migration possibly takes place only when the average shear stress reaches a certain value. It may be because more intragranular stress can be transferred to the GB region. Moreover, it is noted that the peaks in the nominal shear stress curve is related to dislocation nucleation [32,34,61,62]. In general, GB motion is more likely to occur than intragranular dislocation slip [12,61], so atomic stress redistribution is inclined to drive the GB to migrate further during dwelling. In the following, the relation between atomic stress redistribution and further GB migration would be quantitatively analyzed. Two areas with the same size were selected for stress observation, as shown in Fig. 1. Area A is near the triple junction in the upper grain and 40 Å
Fig. 11. Variations of local averaged atomic shear stresses and their difference with dwelling strain. 5
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Fig. 12. GB atomic configurations at dwelling strain of 7% for different dwelling time of (a) 0, (b) 2 ps, (c) 4 ps, (d) 6 ps and (e) 600 ps.
Fig. 14. Curves of nominal shear stress versus dwelling time for NPT and NVT ensembles at 6.5% dwelling strain with temperature varying from 10 K to 200 K.
with the threshold value of local atomic shear stress difference as Δτth , which is sketched in Fig. 11 by a red dotted line by whether the further migration occurs or not. The following threshold condition can be proposed for further GB migration during dwelling. Only when
Δτ > Δτth
atoms instead of GB migration. 3.3. Dependence of the threshold on temperature
(1)
the GB will migrate further during dwelling. When the dwelling strain exceeds 6.8%, Δτ rapidly drops and is lower than Δτth so that the GB can no longer migrate further. Fig. 12 exhibits the GB atomic configurations at 7% dwelling strain for different dwelling time. The GB structure is disordered at the beginning of dwelling and almost the same in the whole dwelling process, so the GB cannot migrate from beginning to end [23,34,57]. As is well known, the stress redistribution could also be realized through initiation and development of dislocations [32,61]. The deformation mechanism would even shift from GB migration to dislocation nucleation under the proper conditions [67,68]. The percentage of HCP atoms is related to the number of dislocations [69]. Fig. 13 plots the percentage of HCP atoms vs. dwelling time curves for different dwelling strains. For 7% dwelling strain, the number of HCP atoms sharply increases in a very short time as soon as the dwelling process starts. It proves that at a dwelling strain larger than 6.8%, the rapid relaxation of atomic shear stress is caused by a sharp increment in HCP
It is well known that temperature has a significant effect on plastic deformation in nanocrystals during dwelling [40,41,70]. To evaluate the effect of temperature on further GB migration driven by the atomic stress redistribution, the dwelling strain was fixed at 6.5%, but the temperature was changed from 1 K to 10 K, 50 K, 100 K, 150 K and 200 K. For an NVT ensemble, temperature change would cause internal stresses in the computational system. In order to evaluate the influence of internal stresses on the computational results, Fig. 14 compares the nominal shear stress vs. dwelling time curves for NPT and NVT ensembles when the temperature varies from 10 K to 200 K. It reveals that the internal stresses have a slight effect. For the computational models with the free boundary conditions in the Y-axis direction, the NVT ensemble is more convenient. Fig. 15 plots the variation curves of further GB migration displacement with dwelling time for different temperatures. It was found that no further GB migration occurs during dwelling at 1 K, 150 K and 200 K, while further GB migration is completed within 25 ps at 10 K but
Fig. 13. Curves of percentage of HCP atoms vs. dwelling time for different dwelling strains.
Fig. 15. Curves of further GB migration displacement vs. dwelling time at different temperature for 6.5% dwelling strain. 6
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squares, red circles and blue triangles, respectively. The stress differences at points (a) and (b) can drive the GB to migrate for the first and last times respectively, while that at point (c) cannot drive the GB to migrate once again. The stress difference at points (a) is used to fit the curve of Δτ , as shown by a black line in Fig. 17, for the first migration. As the GB migration resistance, the threshold value of local atomic shear stress difference Δτth should lie in between the stress differences at points (b) and (c). Therefore, some stress differences very close to each other at points (b) and (c) for different temperatures are selected to fit the curve of Δτth pairwisely, as shown in Fig. 17 by a solid magenta line. For the dwelling strain of 6.5%, when the temperature is between 5 K and 70 K, 2
Δτth = −0.8953(T − 5) 3 + 0.1978T + 3.359
(2)
where T represents temperature. However, when it lies between 70 K and 150 K,
Δτth = 1.243 × 10−7T 4 − 4.628 × 10−5T 3 + 6.322 × 10−3T 2 − 0.3593T Fig. 16. Scatter plot of further GB migration displacement vs. temperature for 6.5% dwelling strain.
+ 9.791
(3)
within 10 ps at 50 K and 100 K. Apparently, temperature has a considerable effect on further GB migration during dwelling. For in-depth investigation on the temperature effect, some numerical simulations were added when temperature varied from 0 to 200 K with an increment of 10 K. Fig. 16 draws the variation trend of the further GB migration displacement with temperature by a purple dotted line. When the temperature is elevated from 10 K to 130 K, the GB migration displacement firstly increases and then decreases. The further GB migration displacement reaches the maximum value near 4 Å when the temperature lies in between 20 K and 60 K, but no further GB migration occurs when the temperature is higher than 130 K. Although temperature variation significantly influences GB migration displacement, it does not change the GB motion mechanism [71]. Thus, even at high temperature, further GB migration during dwelling should also be driven by the atomic stress redistribution. To investigate the connection between further GB migration and atomic stress redistribution, the average atomic shear stresses of areas A and B and their difference are captured at different temperatures. Points (a), (b) and (c) in Fig. 10 correspond to the maximum differences before the first further GB migration, before and after the last further GB migration. These differences at different temperatures are plotted in Fig. 17 by gray
It can be found that the temperature has a slight effect on the stress difference Δτ before migration, but a considerable effect on the threshold value Δτth . Δτth goes down from 0 to 30 K and then up after 30 K. As a result, Inequality (1) is satisfied between about 5 K and 135 K so further GB migration could occur. However, Inequality (1) is not satisfied in the other intervals so the GB cannot migrate further. There is a bigger gap between Δτ and Δτth when the temperature lies between about 20 K and 60 K, so that a larger GB migration displacement could take place more than two times. This also proves that, as a mechanism of grain growth, GB motion is a thermally-activated process [72]. Why does Δτth increase with the elevated temperature after the temperature is higher than 30 K? It is worth noting that the temperature also has a considerable effect on dislocation [73,74]. Fig. 18 shows the variation curves of the HCP atoms percentage with the dwelling time for different temperatures. It is found that the number of HCP atoms always changes little at 1 K and 10 K, but rapidly increases within a short period of time when the temperature is elevated, especially after the temperature is higher than 50 K. The increment in the number of HCP atoms could relax stress around the GB, so that a higher stress difference is needed to drive the GB to migrate further. As a result, the resistance to GB migration Δτth is increased. Moreover, it can be found that Eq. (2) consists of three items. The first term represents the driving force required for GB migration at a
Fig. 17. Local shear stress difference and its threshold value at different temperatures for 6.5% dwelling strain.
Fig. 18. Curves of percentage of HCP atoms vs. dwelling time at different temperatures for 6.5% dwelling strain. 7
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given temperature, which is similar in form to the driving force of GB migration in bicrystals [75]. In other words, the stress relaxed by the GB motion decreases as the temperature increases. The second term represents the stress relaxed by the occurrence of dislocations in nanocrystals as temperature increases, which increases with the temperature. The remaining constant value is related to material properties, GB configuration and other factors. As the atomic thermal activation becomes more intense, the effect of temperature on the threshold value is more complicated. So the form of Eq. (3) is not straightforward. In general, Eq. (2) can be well matched to the above analyses when the temperature is lower than 70 K.
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4. Conclusions In this paper, the molecular dynamics method was used to simulate the further GB migration behavior of nanocrystalline Ni during shear dwelling. The effects of dwelling strain and temperature on further GB migration were evaluated and some internal mechanisms were investigated. The following conclusions are obtained. 1) Further GB migration driven by non-uniform atomic stress distribution could occur in nanocrystalline Ni during shear dwelling. The dwelling strain and temperature have significant complicated effects on further GB migration. 2) The GB must overcome the resistance characterized by the threshold value of local atomic shear stress difference to migrate further. A threshold condition is proposed for further GB migration. Only when the local atomic shear stress difference is larger than its threshold value could the GB migrate further during shear dwelling. The effect of temperature on the threshold value of local atomic shear stress difference is quantitatively described. The effects of dwelling strain and temperature on further GB migration can be well explained with this threshold condition. CRediT authorship contribution statement Xinhua Yang: Conceptualization, Investigation, Funding acquisition, Project administration, Writing - review & editing. Jie Li: Methodology, Investigation, Formal analysis, Data curation, Validation, Writing - original draft. Peng Wang: Investigation, Validation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant numbers: 11772137 and 11572135) and the Fundamental Research Funds for Central Universities (No. 2662017QD018). References [1] K.S. Kumar, H. Van Swygenhoven, S. Suresh, Mechanical behavior of nanocrystalline metals and alloys, Acta Materialia 51 (19) (2003) 5743–5774. [2] M.A. Meyers, A. Mishra, D.J. Benson, Mechanical properties of nanocrystalline materials, Prog. Mater. Sci. 51 (4) (2006) 427–556. [3] M. Dao, L. Lu, R. Asaro, J. Dehosson, E. Ma, Toward a quantitative understanding of mechanical behavior of nanocrystalline metals, Acta Mater. 55 (12) (2007) 4041–4065. [4] Z. Shan, E.A. Stach, J.M.K. Wiezorek, J.A. Knapp, D.M. Follstaedt, S.X. Mao, Grain boundary-mediated plasticity in nanocrystalline nickel, Science 305 (5684) (2004) 654–657. [5] D. Farkas, Atomistic simulations of metallic microstructures, Curr. Opin. Solid State Mater. Sci. 17 (6) (2013) 284–297.
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relaxation tests and simulations, Acta Mater. 108 (2016) 252–263. [41] Y. Wang, A. Hamza, E. Ma, Temperature-dependent strain rate sensitivity and activation volume of nanocrystalline Ni, Acta Mater. 54 (10) (2006) 2715–2726. [42] K. Zhang, J.R. Weertman, J.A. Eastman, The influence of time, temperature, and grain size on indentation creep in high-purity nanocrystalline and ultrafine grain copper, Appl. Phys. Lett. 85 (22) (2004) 5197–5199. [43] K. Zhang, J.R. Weertman, J.A. Eastman, Rapid stress-driven grain coarsening in nanocrystalline Cu at ambient and cryogenic temperatures, Appl. Phys. Lett. 87 (6) (2005). [44] P.L. Gai, K. Zhang, J. Weertman, Electron microscopy study of nanocrystalline copper deformed by a microhardness indenter, Scr. Mater. 56 (1) (2007) 25–28. [45] G.J. Tucker, S.M. Foiles, Molecular dynamics simulations of rate-dependent grain growth during the surface indentation of nanocrystalline nickel, Mater. Sci. Eng. A 571 (2013) 207–214. [46] C.-C. Huang, T.-C. Chiang, T.-H. Fang, Grain size effect on indentation of nanocrystalline copper, Appl. Surf. Sci. 353 (2015) 494–498. [47] P. Steve, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117 (1995) 1–19. [48] Y. Mishin, D. Farkas, M.J. Mehl, D.A. Papaconstantopoulos, Interatomic potentials for monoatomic metals from experimental data and ab initio calculations, Phys. Rev. B 59 (1999) 3393–3407. [49] P. Wang, X. Yang, D. Peng, Molecular dynamics investigation of the grain boundary migration hysteresis of nanocrystalline Ni under cyclic shear loading, Modell. Simul. Mater. Sci. Eng. 25 (2) (2017). [50] P. Wang, X. Yang, D. Peng, Effects of cyclic loading performance on grain boundary motion of nanocrystalline Ni, Metallur. Mater. Trans. A 48 (10) (2017) 4977–4989. [51] D. Farkas, A. Frøseth, H.V. Swygenhoven, Grain boundary migration during room temperature deformation of nanocrystalline Ni, Scr. Mater. 55 (8) (2006) 695–698. [52] D. Rodney, Molecular dynamics simulation of screw dislocations interacting with interstitial frank loops in a model FCC crystal, Acta Mater. 52 (3) (2004) 607–614. [53] H. Fan, J.A. El-Awady, Q. Wang, Towards further understanding of stacking fault tetrahedron absorption and defect-free channels – A molecular dynamics study, J. Nucl. Mater. 458 (2015) 176–186. [54] F. Shuang, P. Xiao, F. Ke, Y. Bai, Efficiency and fidelity of molecular simulations relevant to dislocation evolutions, Comput. Mater. Sci. 139 (2017) 266–272. [55] A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool, Modell. Simul. Mater. Sci. Eng. 18 (1) (2010) 015012. [56] D. Faken, H. Jonsson, Systematic analysis of local atomic structure combined with 3D computer graphics, Comput. Mater. Sci. 2 (1994) 279–286. [57] F. Sansoz, J.F. Molinari, Mechanical behavior of Σ tilt grain boundaries in nanoscale Cu and Al: A quasicontinuum study, Acta Mater. 53 (7) (2005) 1931–1944. [58] J.P. Hirth, R.C. Pond, Steps, dislocations and disconnections as interface defects relating to structure and phase transformations, Acta Mater. 44 (12) (1996)
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Grain boundary migration in nanocrystalline Ni under constant shear strains and its mechanism
T
⁎
Xinhua Yanga,b, Jie Lia, , Peng Wangc a
School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, 1037 Luoyu Road, 430074 Wuhan, China c College of Engineering, Huazhong Agricultural University, Wuhan 430070, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Nanocrystalline Ni Grain boundary migration Stress redistribution Molecular dynamics
The grain boundary (GB) migration behavior of nanocrystalline Ni during shear dwelling was simulated using the molecular dynamics method. It was found that further GB migration could take place and be completed within a very short period of time when the external shear load is kept unchanged. The atomic shear stress redistribution was demonstrated to be the driving force. Both the dwelling strain and temperature have considerable complicated influences on the further GB migration. The local atomic shear stress difference around the GB was proposed to characterize the unevenness of atomic shear stress. A threshold condition was given in terms of the local atomic shear stress difference, namely only when it is larger than its threshold value would the further GB migration occur. The threshold value was quantitatively related to the temperature. The effects of dwelling strain and temperature on the further GB migration can be well explained.
1. Introduction Compared to coarse-grained materials, nanocrystalline materials reveal better physical and mechanical properties in many respects, such as higher strength and hardness [1–3]. This is attributed to the fact that grain boundaries (GBs) account for a larger proportion in them. Moreover, GBs were proved to play an important role in plastic deformation [2,4–7]. As the relevant research progressed, more GB complexions were discovered, enriching the understanding of material properties [8–10]. Grain growth is ubiquitous in nanocrystalline materials. As one of its important mechanisms, the GB migration was observed in the theoretical analyses [11] and a large number of experiments [12–16]. It is usually driven by the shear stress [13,14,16] and also closely related to the thermally assisted diffusion mechanism [17]. Normal GB migration perpendicular to a GB plane is usually accompanied by GB sliding along the tangential direction of the GB plane [18,19]. Many investigations were carried out on the shear-coupled GB motion in bicrystals [20–28]. By comparison, it was found that the molecular dynamics (MD) simulations are very consistent with the theoretical analyses [29] and experiments [30,31] for the bicrystals. However, the GB motion occurring in nanocrystals is significantly different from that occurring in bicrystals due to the pinning effect of triple junctions [11,32,33]. In nanocrystalline materials, the normal migration displacement is generally non-uniform along a GB and easy ⁎
to saturate [34]. Dislocation nucleation and propagation could change the stress distribution in grain interiors in the GB motion process. In addition, the interaction between dislocations and GBs could disorder the elementary structural units of GBs to prevent GB migration [32,34]. It is worth noting that materials generally are viscoelastic. When a certain constant load is applied for a period of time, which is called dwelling, creep deformation would possibly occur. Plastic deformation is highly time-dependent for nanocrystalline materials, so creep and stress relaxation can be observed even at room temperature [35–37]. GB diffusion, GB sliding and grain rotation are considered to be predominant mechanisms of creep deformation in nanocrystals [38,39]. The plastic deformation is dominated by dislocations and GB diffusion at different stages of stress relaxation [40], and the interactions between mobile dislocations and GBs are involved in the stress relaxation process [41]. Naturally, the macroscopic viscosity can be related to GBs mechanisms. Stress-driven grain growth during dwelling was observed in some indentation experiments of nanocrystalline Cu [42–44] even in cryogenic temperature [43]. This indicates that GB migration could occur in a constant external load condition. The MD indentation simulations were performed on nanocrystals as well [45,46]. Huang et al. [46] argued that GB motion dominates the deformation during dwelling for a small grain size. Tucker and Foiles [45] found that grain growth mechanisms are very complex in the case of dwelling, including GB
Corresponding author. E-mail address: [email protected] (J. Li).
https://doi.org/10.1016/j.commatsci.2020.109530 Received 15 September 2019; Received in revised form 6 January 2020; Accepted 7 January 2020 0927-0256/ © 2020 Elsevier B.V. All rights reserved.
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Fig. 2. Loading history curves for different dwelling strains.
600 ps to realize shear dwelling. The loading history curves are plotted in Fig. 2 for different dwelling strains. Strong inertial effects are possibly observed if a shear stress is imposed suddenly in MD simulations [52,53]. To reduce these effects, an equivalent quasi-static loading should be imposed and the system should be relaxed with a sufficient number of MD time-steps at each loading step [54]. Fig. 3(a) shows the nominal-shear-stress vs. nominalshear-strain curves at a strain rate of 2 × 108 s−1 for different numbers of relaxation steps. The stress–strain curve almost does not change when the relaxation step number is larger than 1000. It means a little inertial effect. Then we fixed the relaxation step number at 2000 and change the strain rate from 5 × 107 s−1 to 1 × 108 s−1, 2 × 108 s−1, 4 × 108 s−1, and 1 × 109 s−1. Fig. 3(b) shows their nominal-shearstress vs. nominal-shear-strain curves. The loading rate has a slight effect on the stress–strain response when it is less than 4 × 108 s−1, which was also confirmed in Ref. [50]. The three-dimensional visualization software OVITO [55] was used to visualize the calculation results. The common neighbor analysis method [56] was used to identify the atomic configurations. All the atoms were colored according to their configurations. Face-centered cubic (FCC) atoms are marked blue, hexagonal close packed (HCP)
Fig. 1. Initial configuration of nanocrystalline Ni model with a horizontal Σ9 < 1 1 0 > {2 2 1} symmetric tilt GB.
relaxation, atomic rearrangement, etc. To this day, however, GB migration during dwelling is not fully understood yet. How does GB migration occur in nanocrystals during dwelling? What factor is the predominant driving force? How do load and temperature affect GB migration? Their answers are important for in-depth understanding of GB migration in a dwelling condition. In this paper, the MD method is used to simulate the GB migration behavior of nanocrystalline Ni at various dwelling strains and temperatures, in an attempt to answer the above questions. 2. Model and computational method A quasi-three-dimensional atomic model of nanocrystalline Ni with − idealized [110] texture was constructed, as shown in Fig. 1. The outline − − dimensions of the model in the X [1 1 4], Y [221] and Z [110] directions are 49.3 nm, 49.3 nm and 2.0 nm, respectively. As the main observation object, a Σ9 < 1 1 0 > {2 2 1} symmetric tilt GB with a length of 16 nm flatly lies in the middle of the model. The large-scale atomic/molecular massively parallel simulator (LAMMPS) [47] was used for all MD simulations in conjunction with the embedded atom method potential for Ni [48]. This potential was demonstrated to give a good evaluation of many material properties, including the elastic constants, the vacancy formation and migration energies, the stacking fault energies, and the surface energies [48]. Thus, it was successfully used for many numerical investigations of GB migration behaviors [23,34,49–51]. The periodic boundary conditions were imposed in the X and Z directions, while free boundary conditions were applied in the Y direction. Unless otherwise stated, the canonical NVT ensemble with constant number of atoms, volume and temperature was employed in the computational system. The simulation steps are given as follows. (1) The model was relaxed at room temperature (300 K) for 100 ps, and then relaxed again for 100 ps to reach the initial equilibrium state after the temperature was slowly reduced to 10 K. (2) Several layers of atoms in the top of the upper grain and in the bottom of the lower grain are set as two slabs. The displacement was applied at the top slab in the X direction to induce the shear deformation, while the bottom slab was fixed. A constant strain rate of 2 × 108 s−1 was used in all the simulations. (3) When the nominal shear strain reached a given value, the strain was kept for
Fig. 3(a). Nominal-shear-stress vs. nominal-shear-strain curves at a strain rate of 2 × 108 s−1 for different numbers of relaxation steps. 2
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Fig. 3(b). Nominal-shear-stress vs. nominal-shear-strain curves for a relaxation step number of 2000 at different strain rates. Fig. 5. Atomic configurations of the observed GB at different dwelling time points of (a) 0, (b) 10 ps, (c) 12 ps, (d) 20 ps, (e) 24 ps and (f) 600 ps.
atoms are marked light blue, and the remaining atoms are marked magenta.
To check whether the further GB migration would occur in a bicrystal during dwelling, a Ni bicrystal model with the same geometrical, loading and computational conditions was simulated. Its GB migration displacement vs. time curve is also plotted in Fig. 4 by a red solid line. It can be found that the curve always remains horizontal after the dwelling starts. This means that no further GB migration occurs in a bicrystal during dwelling. Accordingly, it is possible that further GB migration only occurs in a nanocrystal during dwelling. It is known that GB migration is generally achieved by transformation of elementary structural units [23–27,29,49,50], so that GB configuration constantly evolves in a process of GB migration. To analyze the GB migration mechanism of nanocrystalline Ni during dwelling, Fig. 5 shows the GB atomic configurations at six dwelling time points shown in Fig. 4. The six time points (a), (b), (c), (d), (e), and (f) correspond to the dwelling time of 0, 10 ps, 12 ps, 20 ps, 24 ps and 600 ps, respectively. For the sake of observation, all the FCC atoms are removed and the remaining atoms are colored according to their Y coordinates in Fig. 5. It is found that the ordered |EE| structural units [23,34,57] stay in the middle of the GB in Fig. 5(a), but the GB configurations are unordered at the two ends near the triple junctions. Two symmetric steps are formed at the joints between the ordered and unordered units, as marked by the black arrows in Fig. 5(a). They are called GB disconnections [58,59]. At the beginning of dwelling, the two disconnections move in opposite directions towards the triple junctions. As they are far away from each other, a new pair of disconnections is generated at the middle of the GB, as marked by the red arrows in Fig. 5(b). At the same time, the first migration begins to occur, which corresponds to the ‘slip’ behavior at the time point (b) shown in Fig. 4. Subsequently, the two new disconnections constantly move towards both the ends of the GB, as shown in Fig. 5(c) and (d), which corresponds to the ‘stick’ behavior from the time point (c) to (d) in Fig. 4. After this migration, new disconnections nucleate and move quickly, as marked by the green arrows in Fig. 5(e). As a result, the GB completes its second migration between the time point (d) and (e), and there is a transformation of elementary structural units of |EE|, as shown in Fig. 5(d) and (e). The corresponding ‘slip’ behavior can be also seen in Fig. 4. After the time point (e), the GB configuration, as shown in Fig. 5(f), remains substantially unchanged. It means that the further GB migration is stopped. Obviously, the external load is kept unchanged during dwelling, so
3. Results and discussion 3.1. Characteristics and mechanism of further GB migration during dwelling Some atoms in the intermediate region of the target GB were tracked in the simulations and their average Y-axis coordinate variations were considered as the GB migration displacement. In order to investigate characteristics of GB migration behavior during dwelling, the 6.5% dwelling strain case was taken as an example here. The GB migration displacement vs. time curve is plotted in Fig. 4. For comparison, the loading history curve represented by the nominal shear strain is also plotted in Fig. 4 by a blue dashed line. It can be seen that the target GB does not stop migrating at once after the dwelling starts. The twice migrations take place in a very short interval from the time point (a) to (e) shown in Fig. 4. After the further GB migration is completed, the curve keeps straight. It means that the GB migration is stopped.
Fig. 4. GB migration displacement vs. time curves for a nanocrystal and a bicrystal, where dwelling time points of (a) 0, (b) 10 ps, (c) 12 ps, (d) 20 ps, (e) 24 ps and (f) 600 ps are marked. 3
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Fig. 6. Atomic shear stress contours at the dwelling time points of (a) 0, (b) 10 ps, (c) 12 ps, (d) 20 ps, (e) 24 ps and (f) 600 ps.
what is the force to drive the GB to migrate further? In order to answer this question, Fig. 6 shows the atomic shear stress contours at the six different dwelling time points in the same scaling ratio. It is known that the atomic stress can almost be completely relaxed by GB motion in bicrystals [34]. Different from the bicrystal case, it is found from Fig. 6(a) that the atomic shear stress distribution is uneven at the beginning of dwelling. There is higher stress concentrated within the grains but lower stress near the observed GB, as marked by black dotted ellipses in Fig. 6. It can be found that the area of the lower stress region increases from the time point (a) to (b), but then decreases from (b) to (c) and (d). From the time point (d) to (e), however, the lower stress region area increases again, but the stress around the triple junctions is significantly reduced. Consequently, the stress distribution is more uniform in Fig. 6(f) than in Fig. 6(e). It indicates that the GB disconnection nucleation significantly relaxes the stress near the GB, as shown in Fig. 6(b) and (e). But the disconnection movement has little effect on the stress distribution, as shown in Fig. 6(c) and (d). It is consistent with the fact that it only requires lower stress to move a preexisting disconnection than to nucleate and form a new one [60]. Accordingly, the disconnection nucleation usually corresponds to a significant change in the shear stress distribution. In other words, the uneven shear stress distribution can drive the GB disconnection to nucleate. And then the atomic shear stress distribution becomes uniform after the GB migration. When there is not enough shear stress difference to drive a new disconnection to nucleate, the GB migration will come to an end. Does the atomic shear stress redistribution during dwelling change the nominal shear stress of the computational system? Fig. 7 plots the curves of the nominal shear stress and the further GB migration displacement vs. the dwelling time. The time points corresponding to (b), (c), (d) and (e) in Fig. 4 are marked with black dashed lines in Fig. 7. It can be seen that the stress-time curve decreases monotonically before point (e) but slightly fluctuate after it. It also proves that GB motion is a plastic relaxation mechanism [12]. This descend stage can be roughly divided into three sections. The stress descend is very sharp in Sections I and III but gentle in Section II. Section I corresponds to the interval between the time points (a) and (b) shown in Fig. 4, Section II corresponds to the interval from the time point (b) to (d), and Section III corresponds to the interval between the time points (d) and (e). There is a new pair of disconnections nucleating in Sections I and III but no in
Fig. 7. Curves of nominal shear stress and further GB migration displacement vs. dwelling time.
Section II. Thus, the nucleation of GB disconnections can obviously reduce the nominal shear stress during dwelling, but the movement of disconnections has little effect on the stress. 3.2. Threshold condition of further GB migration To investigate the conditions under which the further GB migration occurs, the dwelling strain was changed from 3% to 7% with a constant interval of 0.5%. Fig. 8 plots the curves of the further GB migration displacement vs. the dwelling time for different dwelling strains. It can be seen that no GB migration occurs for the dwelling strains less than 4%. When the dwelling strain lies between 4.5% and 7%, however, the GB migration is rapidly completed in a short period of time and then almost becomes unchanged. There are different migration displacements for different dwelling strains. As the dwelling strain reaches 7%, no GB migration occurs during dwelling again. In order to evaluate the effect of dwelling strain on the GB migration quantitatively, more dwelling strain cases with an increment of 0.1% 4
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Fig. 8. Further GB migration displacement vs. dwelling time curves for different dwelling strains.
Fig. 10. Variation of local atomic shear stress difference with dwelling time at 6.5% dwelling strain.
away from the observed GB. Area B is near the observed GB. The average atomic shear stresses in areas A and B, τA and τB , were recorded [63–66]. The stress difference Δτ = τA − τB was used for characterizing the heterogeneity degree of atomic shear stress quantitatively and called the local atomic shear stresses difference. Fig. 10 shows the variation of Δτ with dwelling time at 6.5% dwelling strain. As the dwelling time goes by, Δτ has two sharp drops, which begin at points (a) and (b) shown in Fig. 10, respectively. The two drops correspond to the first and second GB migrations. After each drop, Δτ enters into a repeated fluctuation process. This suggests that Δτ can be treated as the driving force of further GB migration. Fig. 11 shows the variations of τA , τB , and Δτ corresponding to point (a) shown in Fig. 10 with the dwelling strain. It can be seen that τA has a considerable increase trend with increasing dwelling strain as a whole, while τB rises at a moderate pace. As a result, their difference Δτ has also an increase trend similar to τA . However, Δτ has a sudden drop after the dwelling strain exceeds 6.8%. How should it be explained why the GB sometimes migrates but sometimes does not migrate for different dwelling strains? We assume that the GB must overcome a certain resistance to migrate. The GB migration resistance is characterized
Fig. 9. Scatter plot of further GB migration displacement vs. dwelling strain as well as nominal shear stress vs. dwelling strain curve.
from 3% to 7% were considered. Fig. 9 shows their further migration displacements during dwelling. In this figure, a dotted line is used to show the variation trend of the migration displacement with the dwelling strain. The migration displacement goes up and down with increasing dwelling strain but has a rough increase trend as a whole. At the dwelling strain of about 5.5% and 6.8%, the GB migration displacement reaches its peak values. And the further migration occurs only at 4.5% dwelling strain when the dwelling strain is less than 5%. For comparison, Fig. 9 also plots the nominal shear stress vs. dwelling strain curve by a blue solid line. It can be found that there is a rough correspondence between the two curves. Therefore, the GB migration possibly takes place only when the average shear stress reaches a certain value. It may be because more intragranular stress can be transferred to the GB region. Moreover, it is noted that the peaks in the nominal shear stress curve is related to dislocation nucleation [32,34,61,62]. In general, GB motion is more likely to occur than intragranular dislocation slip [12,61], so atomic stress redistribution is inclined to drive the GB to migrate further during dwelling. In the following, the relation between atomic stress redistribution and further GB migration would be quantitatively analyzed. Two areas with the same size were selected for stress observation, as shown in Fig. 1. Area A is near the triple junction in the upper grain and 40 Å
Fig. 11. Variations of local averaged atomic shear stresses and their difference with dwelling strain. 5
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Fig. 12. GB atomic configurations at dwelling strain of 7% for different dwelling time of (a) 0, (b) 2 ps, (c) 4 ps, (d) 6 ps and (e) 600 ps.
Fig. 14. Curves of nominal shear stress versus dwelling time for NPT and NVT ensembles at 6.5% dwelling strain with temperature varying from 10 K to 200 K.
with the threshold value of local atomic shear stress difference as Δτth , which is sketched in Fig. 11 by a red dotted line by whether the further migration occurs or not. The following threshold condition can be proposed for further GB migration during dwelling. Only when
Δτ > Δτth
atoms instead of GB migration. 3.3. Dependence of the threshold on temperature
(1)
the GB will migrate further during dwelling. When the dwelling strain exceeds 6.8%, Δτ rapidly drops and is lower than Δτth so that the GB can no longer migrate further. Fig. 12 exhibits the GB atomic configurations at 7% dwelling strain for different dwelling time. The GB structure is disordered at the beginning of dwelling and almost the same in the whole dwelling process, so the GB cannot migrate from beginning to end [23,34,57]. As is well known, the stress redistribution could also be realized through initiation and development of dislocations [32,61]. The deformation mechanism would even shift from GB migration to dislocation nucleation under the proper conditions [67,68]. The percentage of HCP atoms is related to the number of dislocations [69]. Fig. 13 plots the percentage of HCP atoms vs. dwelling time curves for different dwelling strains. For 7% dwelling strain, the number of HCP atoms sharply increases in a very short time as soon as the dwelling process starts. It proves that at a dwelling strain larger than 6.8%, the rapid relaxation of atomic shear stress is caused by a sharp increment in HCP
It is well known that temperature has a significant effect on plastic deformation in nanocrystals during dwelling [40,41,70]. To evaluate the effect of temperature on further GB migration driven by the atomic stress redistribution, the dwelling strain was fixed at 6.5%, but the temperature was changed from 1 K to 10 K, 50 K, 100 K, 150 K and 200 K. For an NVT ensemble, temperature change would cause internal stresses in the computational system. In order to evaluate the influence of internal stresses on the computational results, Fig. 14 compares the nominal shear stress vs. dwelling time curves for NPT and NVT ensembles when the temperature varies from 10 K to 200 K. It reveals that the internal stresses have a slight effect. For the computational models with the free boundary conditions in the Y-axis direction, the NVT ensemble is more convenient. Fig. 15 plots the variation curves of further GB migration displacement with dwelling time for different temperatures. It was found that no further GB migration occurs during dwelling at 1 K, 150 K and 200 K, while further GB migration is completed within 25 ps at 10 K but
Fig. 13. Curves of percentage of HCP atoms vs. dwelling time for different dwelling strains.
Fig. 15. Curves of further GB migration displacement vs. dwelling time at different temperature for 6.5% dwelling strain. 6
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squares, red circles and blue triangles, respectively. The stress differences at points (a) and (b) can drive the GB to migrate for the first and last times respectively, while that at point (c) cannot drive the GB to migrate once again. The stress difference at points (a) is used to fit the curve of Δτ , as shown by a black line in Fig. 17, for the first migration. As the GB migration resistance, the threshold value of local atomic shear stress difference Δτth should lie in between the stress differences at points (b) and (c). Therefore, some stress differences very close to each other at points (b) and (c) for different temperatures are selected to fit the curve of Δτth pairwisely, as shown in Fig. 17 by a solid magenta line. For the dwelling strain of 6.5%, when the temperature is between 5 K and 70 K, 2
Δτth = −0.8953(T − 5) 3 + 0.1978T + 3.359
(2)
where T represents temperature. However, when it lies between 70 K and 150 K,
Δτth = 1.243 × 10−7T 4 − 4.628 × 10−5T 3 + 6.322 × 10−3T 2 − 0.3593T Fig. 16. Scatter plot of further GB migration displacement vs. temperature for 6.5% dwelling strain.
+ 9.791
(3)
within 10 ps at 50 K and 100 K. Apparently, temperature has a considerable effect on further GB migration during dwelling. For in-depth investigation on the temperature effect, some numerical simulations were added when temperature varied from 0 to 200 K with an increment of 10 K. Fig. 16 draws the variation trend of the further GB migration displacement with temperature by a purple dotted line. When the temperature is elevated from 10 K to 130 K, the GB migration displacement firstly increases and then decreases. The further GB migration displacement reaches the maximum value near 4 Å when the temperature lies in between 20 K and 60 K, but no further GB migration occurs when the temperature is higher than 130 K. Although temperature variation significantly influences GB migration displacement, it does not change the GB motion mechanism [71]. Thus, even at high temperature, further GB migration during dwelling should also be driven by the atomic stress redistribution. To investigate the connection between further GB migration and atomic stress redistribution, the average atomic shear stresses of areas A and B and their difference are captured at different temperatures. Points (a), (b) and (c) in Fig. 10 correspond to the maximum differences before the first further GB migration, before and after the last further GB migration. These differences at different temperatures are plotted in Fig. 17 by gray
It can be found that the temperature has a slight effect on the stress difference Δτ before migration, but a considerable effect on the threshold value Δτth . Δτth goes down from 0 to 30 K and then up after 30 K. As a result, Inequality (1) is satisfied between about 5 K and 135 K so further GB migration could occur. However, Inequality (1) is not satisfied in the other intervals so the GB cannot migrate further. There is a bigger gap between Δτ and Δτth when the temperature lies between about 20 K and 60 K, so that a larger GB migration displacement could take place more than two times. This also proves that, as a mechanism of grain growth, GB motion is a thermally-activated process [72]. Why does Δτth increase with the elevated temperature after the temperature is higher than 30 K? It is worth noting that the temperature also has a considerable effect on dislocation [73,74]. Fig. 18 shows the variation curves of the HCP atoms percentage with the dwelling time for different temperatures. It is found that the number of HCP atoms always changes little at 1 K and 10 K, but rapidly increases within a short period of time when the temperature is elevated, especially after the temperature is higher than 50 K. The increment in the number of HCP atoms could relax stress around the GB, so that a higher stress difference is needed to drive the GB to migrate further. As a result, the resistance to GB migration Δτth is increased. Moreover, it can be found that Eq. (2) consists of three items. The first term represents the driving force required for GB migration at a
Fig. 17. Local shear stress difference and its threshold value at different temperatures for 6.5% dwelling strain.
Fig. 18. Curves of percentage of HCP atoms vs. dwelling time at different temperatures for 6.5% dwelling strain. 7
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given temperature, which is similar in form to the driving force of GB migration in bicrystals [75]. In other words, the stress relaxed by the GB motion decreases as the temperature increases. The second term represents the stress relaxed by the occurrence of dislocations in nanocrystals as temperature increases, which increases with the temperature. The remaining constant value is related to material properties, GB configuration and other factors. As the atomic thermal activation becomes more intense, the effect of temperature on the threshold value is more complicated. So the form of Eq. (3) is not straightforward. In general, Eq. (2) can be well matched to the above analyses when the temperature is lower than 70 K.
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4. Conclusions In this paper, the molecular dynamics method was used to simulate the further GB migration behavior of nanocrystalline Ni during shear dwelling. The effects of dwelling strain and temperature on further GB migration were evaluated and some internal mechanisms were investigated. The following conclusions are obtained. 1) Further GB migration driven by non-uniform atomic stress distribution could occur in nanocrystalline Ni during shear dwelling. The dwelling strain and temperature have significant complicated effects on further GB migration. 2) The GB must overcome the resistance characterized by the threshold value of local atomic shear stress difference to migrate further. A threshold condition is proposed for further GB migration. Only when the local atomic shear stress difference is larger than its threshold value could the GB migrate further during shear dwelling. The effect of temperature on the threshold value of local atomic shear stress difference is quantitatively described. The effects of dwelling strain and temperature on further GB migration can be well explained with this threshold condition. CRediT authorship contribution statement Xinhua Yang: Conceptualization, Investigation, Funding acquisition, Project administration, Writing - review & editing. Jie Li: Methodology, Investigation, Formal analysis, Data curation, Validation, Writing - original draft. Peng Wang: Investigation, Validation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant numbers: 11772137 and 11572135) and the Fundamental Research Funds for Central Universities (No. 2662017QD018). References [1] K.S. Kumar, H. Van Swygenhoven, S. Suresh, Mechanical behavior of nanocrystalline metals and alloys, Acta Materialia 51 (19) (2003) 5743–5774. [2] M.A. Meyers, A. Mishra, D.J. Benson, Mechanical properties of nanocrystalline materials, Prog. Mater. Sci. 51 (4) (2006) 427–556. [3] M. Dao, L. Lu, R. Asaro, J. Dehosson, E. Ma, Toward a quantitative understanding of mechanical behavior of nanocrystalline metals, Acta Mater. 55 (12) (2007) 4041–4065. [4] Z. Shan, E.A. Stach, J.M.K. Wiezorek, J.A. Knapp, D.M. Follstaedt, S.X. Mao, Grain boundary-mediated plasticity in nanocrystalline nickel, Science 305 (5684) (2004) 654–657. [5] D. Farkas, Atomistic simulations of metallic microstructures, Curr. Opin. Solid State Mater. Sci. 17 (6) (2013) 284–297.
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