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Atomic simulation of bcc niobium Σ 5〈001〉310 grain boundary under shear deformation
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Atomic simulation of bcc niobium R5h001if310g grain boundary under shear deformation Bo-Wen Huang a, Jia-Xiang Shang a,⇑, Zeng-Hui Liu a, Yue Chen b a
School of Materials Science and Engineering, Beihang University, Beijing 100191, People’s Republic of China Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA
b
Received 8 March 2014; received in revised form 19 May 2014; accepted 19 May 2014 Available online 1 July 2014
Abstract The shear behaviors of grain boundaries are investigated using molecular dynamics simulations. The R5h001if310g symmetric tilt grain boundary (GB) of body-centered cubic (bcc) Nb is investigated and the simulations are conducted under a series of shear directions at a wide range of temperatures. The results show that the GB shearing along ½1 30, which is perpendicular to the tilt axis, has a coupled motion behavior. The coupling factor is predicted using Cahn’s model. The critical stress of the coupling motion is found to decrease 1 direction, which is parallel to the tilt axis, exponentially with increasing temperature. The GB under shear deformation along the ½00 has a pure sliding behavior at most of the temperatures investigated. The critical stress of sliding is found to be much larger than that of the coupled motion at the same temperature. At very low temperatures, pure sliding is not observed, and dislocation nucleating and extending is found on GBs. We observed mixed behaviors when the shear direction is between ½1 30 and ½00 1. The transition region between GB coupled motion and pure sliding is determined. If the shear angles between the shear direction and the tilt axis are larger than a certain value, the GB has a coupled motion behavior similar to the ½1 30 direction. A GB with a shear angle smaller than the critical angle exhibits mixed mechanisms at low temperatures, such as dislocation, atomic shuffle and GB distortion, whereas for the ½00 1-like GB pure sliding is the dominating mechanism at high temperatures. The stresses to activate the coupling and gliding motions are analyzed for shear deformations along different directions at various temperatures. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Grain boundary; Shear deformation; Molecular dynamics; Bcc
1. Introduction Grain boundaries (GBs) play a significant role in determining the mechanical properties of polycrystalline materials[1,2], e.g. nanocrack healing by migrating GBs [3]. Numerous investigations have been conducted in order to understand GB structures, energies and thermodynamics. Molecular dynamics (MD) simulation is a powerful method, providing a comprehensive understanding of the microscopic mechanisms of GB movements [4–8], plastic ⇑ Corresponding author. Tel./fax: +86 10 8231 6500.
deformations [9,10] and other atomic behaviors [11–13]. One of a GB’s most important features is its mobility, which depends on the GB crystallography and external conditions. It has been found that the normal GB motion is often coupled to the tangential translation of grains (referred to as coupled GB motion). Stress-induced GB motions have been studied in great detail, both in simulations and experimentally. GB motions can be classified into different groups based on analysis of the atomic structure evolution [14–16], which includes GB coupled motion, GB sliding, grain rotation and dislocation emission. GB coupled motion was first observed in experiments by Li et al. [17] in small-angle Zn GBs. Molteni and
E-mail address: [email protected] (J.-X. Shang). http://dx.doi.org/10.1016/j.actamat.2014.05.047 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268
Shiga et al. investigated the tilt and twist GB motions in Ge, Al and Ni using density functional theory and found that the sliding and migration motions exhibit a stick–slip behavior [18,19]. The Frank–Bilby equation defines the dislocation of an interface between two crystals. The solutions of this equation are meaningful only for low-angle tilt GBs [20], though recently the equation has been extended to high-angle tilt GBs [21,22]. Cahn et al. [23] proposed a geometric model to predict the GB coupled motion. The model considers the upper grain as a parallelogram. In particular, one face of the parallelogram is along the GB plane, while another one is normal to the tilt axis, and the third one is parallel to the slip plane. To calculate the coupling factor b, which is defined as the ratio of the normal to the lateral motion, the parallelogram is first sheared along the direction parallel to the slip plane by a magnitude of B (the sum of the Burgers vectors). Then the parallelogram is rotated clockwise by h (tilt angle) to align the upper grain with the lower grain. The model gives two opposite signs for the coupling factor, which refer to the different slip mechanisms. For the slip plane of GB dislocation along the f100g plane: b ¼ 2tanðh=2Þ
ð1Þ
For the slip plane of GB dislocation along the f110g plane: b ¼ 2tanðp=4 h=2Þ
ð2Þ
It can be seen that the coupling factor only depends on the misorientations. Caillard et al. [7] have further proposed a model of shear-migration coupling for general GBs. A series of MD simulations of the h100i tilt GBs in Cu have been performed by Cahn et al. [5,6,21], confirming the above models. The coupling factors predicted by the geometric theory show excellent agreements with MD simulations and experimental data [24–27]. In addition, three typical tilt GBs (h100i; h110i and h111i) in Ni have been investigated [28,29] using a synthetic driving force method. It was found that the GBs exhibit excellent agreements with the theory for most of the h100i boundaries, whereas the h110i and h111i boundaries did not obey the previous models. Wan and Wang [30] studied the R9h001if221g GB in Cu, Al and Ni with different shear directions at room temperature. They observed various GB motions including coupling and sliding, while the motion types that are activated depended on the shear directions and materials. In another report, occasional sliding has been observed during coupled motions as the strain rate increases [6]. In other words, coupling and sliding take place once the stress has accumulated to a critical level. If the critical stress for coupling is larger than that needed to activate sliding, sliding motion will be dominant; otherwise, coupled motion will take place. In certain conditions, e.g. low temperatures or along some particular shear directions, if the critical stresses for coupling or sliding exceed the yield stress, dislocations accompanied by atomic shuffles will take place.
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Most of the investigations of GB motions focus on materials with face-centered cubic (fcc) structure such as Ni, Al and Cu. The GB motions of body-centered cubic (bcc) metals have rarely been studied except for the bccFe R5h001if310g symmetric tilt GB, which has been investigated using MD under shear deformation from 1 to 600 K. The nucleation and gliding of partial GB dislocations were found in GB migration [31]. Do the GBs in bcc materials exhibit a behavior similar to that seen in fcc materials? Does the geometric theory hold valid in metals with bcc structure? What is the relation between GB motions, temperatures and shear directions? To answer these questions, we choose the R5h001if310g symmetric tilt GB of niobium (Nb) with bcc structure as a model in this paper. Nb is a refractory metal, and one of the most important elements in superalloys with promising applications. The GB is constructed using the coincidence site lattice (CSL) model. By applying shear loads parallel to the GB plane at a wide range of temperatures and different directions, we observe that there are various kinds of GB motion which depend on the shear conditions. We have shown that the shear deformation along the ½1 30 axis is consistent with the prediction of geometric theory. Shear deformation along the ½001 axis displays a GB pure sliding behavior. Other simulations whose directions are between the ½130 and ½001 axes show a mixed behavior. A transition region between the coupling and sliding motions, which depends on the temperatures and directions, is described. The paper is organized as follows. In Section 2, the general atomistic simulation method is described. In Section 3, the results and discussions are presented. Finally, a summary of the present work and conclusions are given in Section 4. 2. Simulation methodology The shear deformation of the Nb R5h001if310g GB is investigated in this paper. Simulation models are constructed using the CSL method. We create the GB model by concatenating two separate grains with specific crystallographic orientations. The orientation of the lower grain is shown in Fig. 1. The upper grain is built by rotating the lower grain around the [310] axis for 180°. The size of the simulation model is approximately 10 nm 10 nm 20 nm, which contains about 120,000 atoms. The periodic boundary condition (PBC) is applied along three dimensions to mimic the bulk material conditions. In order to avoid the interference of the second GB which is caused by the PBC, we fix the atoms on the top and bottom layers of the model. The fixed atoms are located in their perfect lattice positions. We define the thickness of each fixed region as twice of the cut-off distance. Two grains undergo a rigid-body translation within the GB plane to find the position with the lowest energy. A conjugate gradient algorithm is applied for energy minimization in this work. Once the optimized structure is obtained, we then run MD simulations for sufficiently long times to ensure
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Fig. 1. Schematic illustration of the GB simulation model that used in this work. The crystallographic directions are defined with reference to the lower grain. The ½00 1 axis is the tilt axis.
the stability of the model. The initial equilibrium GB structure is shown in Fig. 2, which is viewed along the tilt axis. According to their CSL notation and the GB normal direction, this GB is termed R5h001if310g; its structure was observed experimentally in Ref. [32]. In Fig. 2, the structural unit, which contains six atoms located on the GB plane, is outlined and the orientation vectors for both the GB period and normal directions are given. The black and white atoms define two adjacent atomic planes along the tilt axis. It should be noted that each kite-like structure unit is composed of atoms belonging to two neighboring atomic planes. MD simulations are then performed to deform the GB model at a constant shear strain rate of 1108 s1 . The
shear direction is parallel to the GB plane. Table 1 lists the shear directions and the included angles between the ½001 tilt axis and the shear directions. The shear process deforms the simulation box as a whole. Stresses on all directions except the shear direction are allowed to relax during the simulations. A canonical ensemble (NPT) with the Nose–Hoover thermostat is applied [33]. Stress is calculated using the standard viral expression. We adopt the Nb embedded atom method potential [34] which has been used in other studies [35,36]. The accuracy of this potential is confirmed by testing the elastic constants, melting point, lattice constant and thermal expansion. Common neighbor analysis (CNA) is used to display the atomic structure. MD simulations are realized using the LAMMPS code [37].
Fig. 2. The R5h001if310g symmetric tilt GB structure in Nb at 0 K. Filled and empty circles represent the Nb atoms in two adjacent atomic planes. The structural units are outlined in solid blue lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Table 1 The shear directions, critical stresses and GB motion types at 100 K. Direction ½001 ½1330 ½1320 ½1315
½1310
½136
½395
½131
½130
Angle h Critical stress (GPa) Motion type
17.55° 2.80 Mix
27.79° 2.20 Couple
62.21° 1.18 Couple
72.45° 1.10 Couple
90° 1.22 Couple
0° 3.50 Slide
6.02° 5.00 Mix
8.99° 4.80 Mix
11.90° 4.10 Mix
3. Result and discussion 3.1. Shear along the ½1 30 direction Shear deformation is applied along the ½1 30 direction with a constant strain rate. The ½1 30 direction is located on the GB plane and is perpendicular to the ½00 1 tilt axis. We perform MD simulations at different temperatures to analyze the mechanisms of GB motion. Fig. 3 shows the stress–time relations at temperatures between 1 and 2400 K (the bulk melting point 2750 K). It is obvious that the curves display a strong temperature dependence. The yield stress decreases exponentially with increasing temperature as shown in the inset of Fig. 3. At temperatures between 1 and 2100 K, the stress–time curves display a sawtooth behavior. The stress drops when the critical value is reached. The magnitude of the dropping and the period of each curve are temperature independent between 300 and 2100 K. Ivanov [6] reported that the grain size in the normal direction affects the stress behavior during shear deformation. At temperatures below 100 K, the stress–time curves display larger stress drops and longer periods. The atoms at temperatures below 100 K are difficult to move; thus the GBs can accumulate and release more elastic energy in one period. When the temperature is above 2400 K, the stress is close to zero. After checking the GB structure at 2400 K using the radial distribution function (RDF), we find that the GB has already premelted, and this explains the abnormal behavior of the stress–time relation at 2400 K. In order to understand the relation between GB motion and stress, we plotted the time dependency of the shear
stress and GB displacement at 300 K (see Fig. 4). The GB position is tracked by the CNA computation, which gives values of 3 and 5 for atoms in the bcc lattice and the GB region, respectively [38]. It is obvious that the GB migration curve displays a regular serrated profile, which is the so-called “stick–slip” behavior. While the “stick” stages correspond to elastic straining, the “slip” stages relate to certain structural transformations. During the elastic deformation stage, the shear stress increases almost linearly. After the critical stress is reached, the GB rapidly moves to a new position. As the deformation continues, the stress drops after GB migration; this is then followed by a new increase in the stress until the next peak. Each peak of the stress correlates exactly with an increment of the GB motion. One possible explanation is that the GB becomes mechanically unstable as the deformation continues and requires some mechanism to release the excess energy. GB migration is one of the mechanisms that is likely to occur in this condition. The accompanying grain translation produces a permanent shear deformation. This type of GB motion is trapped in one energy minimum until it loses stability and jumps to a new minimum. Fig. 5 shows the GB migrations at different temperatures. It can be seen that the GB migration is similar for most of the temperatures considered here. An obvious stick–slip behavior is seen in our simulations at low temperatures. The displacement steps firstly decrease and then maintain at a certain value as the temperature increases. Thermal fluctuation tends to smooth out the discontinuous stick–slip behavior. The smallest displacement step is confirmed to be the distance between two neighboring atomic planes in the GB region.
Fig. 3. Stress–time curves for shear along the ½130 direction at temperatures between 1 and 2400 K. The simulations are performed with a constant strain rate of 1 108 s1 . The inset of the plot shows the exponential fitting of the critical stress.
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Fig. 4. The shear stress and the GB migration as functions of time for shearing along ½1 30 at 300 K. The process of migration exhibits a stick– slip behavior. The blue line is the stress–time curve and the red line represents the GB displacement which normal to the GB plane. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Migrations of the GB as functions of time at different temperatures for shearing along ½1 30. Migrations exhibit stick–slip behavior at most temperatures. The GB exhibits a triple jumps and double jumps at 1 and 100 K, respectively. At 2400 K, the GB shows random motions.
At 100 K, the GB jumps by twice the smallest height and the drop in stress is also doubled. The atoms cannot move easily at this low temperature; therefore, the accumulated elastic strain energy is so large that one single jump cannot release it completely. It is expected that at temperatures below 100 K, higher multiple jumps may be observed. Simulation at 1 K shows a triple-distance migration which confirms this expectation. At 2400 K, the migration is interrupted because of the GB pre-melting. These phenomena are consistent with a recent publication that reports the transition from stick–slip to driven Brownian dynamics for fcc materials [5]. We measure the GB displacement on the GB plane to study whether sliding is also occurring.The magnitude of sliding is calculated using relative positions between two
predefined atoms located on different sides of the GB. The relative distance between this two marked atoms is equal to zero at time ðtÞ ¼ 0. In all of our simulations, GB sliding along the ½1 30 direction is found to accompany GB migration. The temperature dependence of sliding (see Fig. 6) has a behavior similar to that of migration, i.e. stick–slip behavior. The GB migration and sliding velocities can be obtained from Figs. 5 and 6 (displacement–time curves), respectively. Our results indicate that the GB velocity is independent of temperature, and is determined by the shear rate and GB structure. The shear rate is also referred to as the sliding velocity by other authors [5,6,21]. Considering that the velocities of migration and sliding remain constant, we assume that a coupled relation exists in these two motions. Fig. 7(a) shows the GB motions in both of the ½130 and ½310 directions at 100 K. It can be seen that the migration and the sliding motions take place simultaneously, which means they couple perfectly. The coupled motion is similar to other stick–slip motions, e.g. the tip movements in atomic force microscopy. The coupling factor b for GB introduced by Cahn et al. [21] is defined as the ratio between the GB sliding velocity (v//) and the GB migration velocity (vn). From the geometric model of coupling, the perfect coupling factor of our GB is b ¼ 2tanðp=4 h=2Þ ¼ 0:99. Our simulations give a consistent result that b ¼ v== =vn 1. In the range of 1–2100 K, b is found to be practically independent of temperature and in good agreement with its geometrical value (see Fig. 7(b)). This indicates that the coupling is nearly perfect for this temperature range. When the temperature is above 2400 K, the coupling behavior is destroyed by GB pre-melting. It was mentioned in a previous publication for fcc Cu [21] that the GB coupling behavior is determined solely by the GB geometric structure. In other words, the coupling factor is fixed once the GB structure is determined.
Fig. 6. Sliding of the GB as functions of time at different temperatures for shearing along ½130. Sliding exhibits stick–slip behavior at most temperatures. The stick–slip behavior is weakened as temperature increases.
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deformation of the units. This process can be viewed as the glide of the parallel arrays of the GB dislocations along ½001. The coupling factor can be obtained, which is the ratio of the unit displacement length on plane to that in the vertical direction. 3.2. Shear along the ½001 direction
Fig. 7. The GB coupled motion for shearing along ½130. (a) The GB migration and sliding as functions of time at 100 K. The blue line describes migration and the red line represents the sliding process. (b) The coupling factor b as a function of temperature. The dashed line is the b predicted from the geometric model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
As our simulations reveal similar behaviors in Nb, the same rule may also apply to other bcc materials. However, more GBs in other bcc materials need to be further tested in order to confirm this. To study the structure evolution of the GB, snapshots of the atomic positions are recorded at fixed time-step intervals. Fig. 8 shows a plot of the vector field of the R5h001if310g GB, which is viewed along the ½001 direction. It records the position variation before and after one migration. The green unit (abcdef) belongs to the lower grain and is interlocked with the kite-like blue unit (ABCDEF). The green unit is a slightly distorted version of the perfect lattice. Importantly, the green and blue units are topologically identical and can transform to each other by relatively small atomic displacements. At each step of boundary motion, the green unit changes its shape and transforms to a kite-like blue unit, whereas the blue unit simultaneously transforms to an orange unit. Note that the latter is a mirror reflection of the green unit. As a result, the GB position shifts one step down while the upper grain translates to the right to accommodate the
The ½001 direction is parallel to the tilt axis. We apply shear deformation along this direction as was done for the ½130 direction. Several phenomena that are different from those we have discussed in the ½130 direction are observed. It is determined that the GB motion is pure sliding along the ½001 direction. Stress–time curves at different temperatures are shown in Fig. 9. The critical stress is much larger than that of the ½130 direction, which means pure sliding along the ½001 direction is harder to activate than coupled motion. The critical stress does not strongly depend on temperature in the range 100–300 K. A periodic sliding behavior similar to the GB coupled motions is observed at most temperatures except 1 K. Based on the analysis on the variance of the GB position, it is found that there is no displacement perpendicular to the GB plane. The GB can only slide along the ½001 direction within the GB plane. Similar to the ½130 direction, the GB sliding velocities display temperature independence and remain constant. The shear stress and the GB in-plane displacement at 900 K are shown as functions of time in Fig. 10. The sliding along ½001 can also be regarded as a stick–slip process. The GB initially stays in a stick stage, and it accumulates strain throughout the crystal. The slip process takes place when strain reaches the critical level. Simulation at T = 1 K shows an abnormal behavior. In order to understand the mechanisms in detail, we monitor the crystal structure evolution at this temperature. Fig. 11 shows the deformation along the ½001 direction at 1 K. It can be seen that a partial dislocation nucleates and is emitted from the GB, due to its lower activation energy compared to pure GB sliding at this temperature. Once the partial dislocation moves toward the fixed boundary, the region occupied by the stacking fault will extend. Meanwhile, detailed analysis shows that limited GB sliding and local atomic shuffle also take place simultaneously. The nucleation and gliding of partial GB dislocations has previously been observed in bcc Fe at low temperature [31]. 3.3. Shear between the ½130 and ½001 directions In this section, we conduct a series of simulations with shear directions between the ½130 and ½001. The specific information about the shear directions are summarized in Table 1. We have divided the simulations into two groups based on the results at T = 100 K. The first group represents shearing along the ½136; ½395 and ½131 directions. These simulations behave similarly to the ½1 30 direction (coupled motion). The second group consists of simulations
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Fig. 8. Vector field of atomic displacements for shearing along ½130 at 300 K. The black dots represent the original positions of the atoms, while the red arrows show the atomic trajectories. The blue and green lines represent the GB structural units before and after migration takes place, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. The stress–time relations for shearing along the ½001 direction at temperatures between 1 and 2100 K. The simulations are performed with a constant strain rate of 1 108 s1 . The inset shows the exponential fitting of the critical stress.
direction as Fig. 10. The shear stress and the GB sliding in the ½001 functions of time for shearing along ½001 direction at 900 K.
½1 ½1 and ½1 directions; these along the ½1 330; 320; 315 310 show complex mechanisms at low temperatures and ½001like pure sliding at high temperatures.
In the first group, we have studied the ½136 and ½ 5 39 directions in detail. The ½136 direction is about 27:79 clockwise to the ½001 direction and the ½395 direction is 62:21 anticlockwise to the ½001 direction. Two kinds of simulations behave similarity. The GB migrates and slides with stick–slip behavior which is similar to the deformation along the ½130 direction. Furthermore, the GB in-plane movements do not only take place in the shear direction, but also in the vertical direction. The GB can move along the ½136 and ½395 directions simultaneously. As we have discussed previously, the GB exhibits a coupled motion and a pure sliding when it is sheared along the ½1 30 and ½001 directions, respectively. Thus, we have decomposed the velocities of sliding into the ½001 and ½130 directions. Fig. 12 shows how we have decomposed the velocity and stress. For shear along ½136, the resolved velocity in ½00 1 is close to zero and the resolved velocity in ½130 is 3.81 m s1. The GB migration velocity is 3.85 m s1. Therefore, the coupling factor is 0.99, which is consistent with that of the ½130 direction. A similar decomposition is applied to
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Fig. 11. Selected snapshots of dislocation nucleation, motion and atomic shuffle on the GB at 1 K. Atoms are colored according to the centrosymmetry parameters.
Fig. 12. Decomposition of the ½136 displacement into the ½001 and ½130 directions.
the critical stress. The resolved critical stress in ½001 is 1.94 GPa, which is much smaller than the stress needed to activate the ½001 pure sliding (about 3.50 GPa), at 100 K. On the other hand, the resolved critical stress in ½130 is 1.02 GPa, which is comparable to the stress needed
to activate the ½130 coupled motion (about 1.22 GPa). The small difference could be compensated by atomic shuffling. For further investigations, we have performed MD simulations at 100, 300, 600 and 900 K. Table 2 lists the resolved critical stresses and the critical stresses of deformation in the ½001; ½130 and ½136 directions. The MD results are consistent with experiment in that the resolved stress in the ½001 is much smaller than the critical stress of pure sliding. The resolved critical stress in the ½1 30 direction is comparable to the critical stress of coupled motion. Moreover, Fig. 13 shows that the GB is not displaced in the ½001 direction. Combining the coupling factor and the atomic structure evolution, we conclude that shearing along the ½136 direction shares the same mechanism as that along the ½130 direction. It can also be seen from Table 2 that the shear along the ½395 direction shows behavior similar to the deformation in the ½136 direction. The resolved critical stress in the ½001 direction is always much smaller than the stress needed to activate pure sliding, while the resolved critical stress in the ½130 direction and the critical stress along the ½130 direction are comparable. Thus we conclude that the deformation along the ½395 direction shares the same mechanism as the coupled motion in the ½130 direction.
Table 2 The critical stress Fc, stress F and resolved stress f (in GPa) at different temperatures. Temperature
Fc½001
Fc½130
F ½136
f½001
f½130
F ½395
f½001
f½130
100 K 300 K 600 K 900 K
3.50 4.20 3.20 2.50
1.22 0.58 0.31 0.18
2.19 1.07 0.61 0.43
1.94 0.95 0.54 0.38
1.02 0.50 0.29 0.20
1.18 0.58 0.32 0.22
0.55 0.27 0.15 0.10
1.04 0.52 0.28 0.19
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at various temperatures. Fig. 13. The stress–time relations as well as the GB displacements in different directions for shearing along ½136
½1315; ½1320 and ½1330 directions at 100 K. Fig. 14. The stress–time relation for shearing along the ½1310;
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Fig. 15. The stress and the GB displacement as functions of time for direction at 900 K. Arrow 1 is the ½130-like shearing along the ½1 330 coupled motion.
By investigating the deformation mechanisms in the first group at 100 K, we find that the simulations exhibit a ½130like GB coupled motion once the angle between the shear direction and the ½00 1 tilt axis is larger than the critical angle. In the second group, the angles between the shear directions and the tilt axis are smaller than the critical value, and the stress–time curves of the four shear directions at 100 K are shown in Fig. 14. The strain–stress curves display a non-periodic behavior, which indicates that the shear deformations belong neither to the ½1 30-like GB coupled motion, nor to the ½00 1-like GB pure sliding. shear direction (Fig. 14(a)), whose For the ½1 330 included angle h is 6:02 , there are three different stages of deformation. After the critical stress is reached, atomic shuffle appears first to partially release the stress. A large number of dislocations then nucleate on the GB. As the
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deformation proceeds, finally, there is a GB distortion which makes the stress drop rapidly. direction (Fig. 14(b)), For the shear along the ½1320 which has an included angle of h = 8:99 , there are two stages of deformation, which correspond to stages one direction. Dislocation nucleations and three of the ½1330 are not observed in our simulations. and ½1310 directions For the shears along the ½1315 (Fig. 14(c) and (d)), the included angles h are 11:90 and 17:54 , respectively. The GB coupled motions are observed to release the stress. However, the GB coupled motions have limited effects on the deformation, and atomic shuffle and GB distortion still dominate the shearing process. Based on the above analysis, we conclude that a transition region exists between the GB pure sliding and the GB coupled motion, which has different shear deformation mechanisms. With a larger included angle, the proportion of GB coupled motion increases. Nevertheless, we have not observed pure GB sliding, even for an extremely small angle. This is similar to the simulation along the ½00 1 direction at 1 K. A possible reason is that a higher temperature is needed to activate the GB pure sliding in those small-angle simulations, as shown in the following discussion. The stress and the GB position are shown as functions direction of time in Fig. 15 for shearing along the ½1330 and ½391 (perpendicular to at 900 K. Sliding along ½1330 as well as GB migration are observed in the simula½1330) and ½ tion. The displacements along the ½1330 39 1 directions are decomposed to the ½001 and ½130 directions. Based on the analysis of the GB position, we find that the deformation contains ½130-like GB coupled motion and ½001-like GB pure sliding. The stress–time curve displays a sawtooth behavior. The first small peak corresponds to GB sliding on the ½130 direction and migration in the vertical direction. For the second and
Fig. 16. Different types of GB motions with respect to temperatures and shear directions.
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fourth peaks, the GB slides along the ½1 30 and ½00 1 directions and simultaneously migrates; the ½00 1-like GB pure sliding dominates and releases most of the stress. The third and the fifth peaks indicate that the GB displays a pure sliding along the ½00 1 direction. Since the barrier of the ½130 direction coupled motion is much smaller than the ½00 1 direction GB pure sliding, it is reasonable that a handful of ½1 30 direction coupled motions take place at such high temperature. Thus, we consider that this simulation is similar to the ½00 1 direction GB pure sliding. We have summarized the types of the GB motions for different temperatures and shear directions as shown in Fig. 16. The MD simulations with ½1 5; ½1 3 6; ½ 39 3 1 shear directions exhibit ½130-like GB coupled motion. The defor ½1 ½1315 mations with shear directions along the ½1 330; 320; and ½1310 are similar to that along ½1330 at T = 900 K. In direction is a ½00 addition, shear along the ½1 330 1-like pure sliding at 300 K. It is found that the ½00 1 pure sliding is dominating in small-angle simulations at high temperatures, while GB dislocations, distortion and atomic shuffle are the main deformation mechanisms at low temperatures. 4. Conclusion We have performed MD simulations to study the shear responses of the R5h001if310g symmetric tilt GB in bcc Nb over a wide range of temperatures. Nine shear directions parallel to the GB plane have been studied, namely ½1 ½1315; ½1 ½1 ½001; ½1 5; ½1 330; 320; 310; 3 6; ½ 39 3 1 and ½130. For shear deformation along the ½1 30 direction, the GB always shows a coupled motion regardless of the temperature.The critical stress decreases exponentially with increasing temperature. The coupled motion displays a stick–slip behavior, i.e. the system is trapped in an energy minimum until the GB becomes unstable and jumps to a new minimum. At very low temperatures, we observe multiple jumps, which are different from the single jump at high temperatures. The GB coupling factor is found to be independent of temperature and can be predicted from geometric calculations. Therefore, we propose that shearing along the direction perpendicular to the tilt axis shares a mechanism with the geometric theory proposed by Cahn. For the shear deformation along the ½00 1 direction, the GB movement is pure sliding at most of the temperatures studied. The sliding is more difficult to activate than the coupled motion because its critical stress is much larger. Pure sliding is not observed at T = 1 K, and the GB exhibits an unusual behavior caused by the dislocation nucleating and extending. The shear deformations between the ½1 30 and ½001 directions are more complicated, and we have divided them into two groups according to their deformation mechanisms. The first group has large angle between the shear directions and the tilt axis, which exhibits a GB coupled motion and a geometrically predictable coupling factor.
We find that this type of deformation shares the same atomic structural evolution with the ½130 shear direction. The second group has smaller angles between the shear directions and tilt axis. Dislocation, atomic shuffle and GB distortions are dominant at low temperatures, while the ½001-like GB pure sliding is the principal mechanism at high temperatures. Acknowledgment This work was financially supported by the National Science Foundation of China (Grant Nos. 51071011 and 51371017). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
Sutton AP, Vitek V. Philos Trans R Soc A 1983;309:1. Mishin Y, Asta M, Li J. Acta Mater 2010;58:1117. Xu GQ, Demkowicz MJ. Phys Rev Lett 2013;111:145501. Bishop GH, Harrison RJ, Kwok T, Yip S. J Appl Phys 1982;53:5596. Mishin Y, Suzuki A, Uberuaga BP, Voter AF. Phys Rev B 2007;75:224101. Ivanov VA, Mishin Y. Phys Rev B 2008;78:064106. Caillard D, Mompiou F, Legros M. Acta Mater 2009;57:2390. Trautt ZT, Adland A, Karma A, Mishin Y. Acta Mater 2012;60:6528. Spearot DE, Jacob KI, Mcdowell DL. Acta Mater 2005;53:3579. Spearot DE, Tschopp MA, Jacob KI, Mcdowell DL. Acta Mater 2007;55:705. Pan Z, Li Y, Wei Q. Acta Mater 2008;56:3470. Fr£seth A, Swygenhoven HV, Derlet PM. Acta Mater 2004;52:2259. Farkas D, Fr£seth A, Swygenhoven HV. Scr Mater 2006;55:695. Cahn JW, Taylor Jean E. Acta Mater 2004;52:4887. Rajabzadeh A, Legros M, Combe N, Mompiou F, Molodov DA. Philos Mag 2013;93:1299. Sansoz F, Molinari JF. Acta Mater 2005;53:1931. Li Choh Hsien, Edwards EH, Washburn J, Parker ER. Acta Metall 1953;1:223. Molteni C, Marzari N, Payne MC, Heine V. Phys Rev Lett 1997;79:869. Shiga M, Shinoda W. Phys Rev B 2004;70:054102. Sutton AP, Balluffi RW. Interfaces in crystalline materials. Oxford: Clarendon Press; 1995. Cahn JW, Mishin Y, Suzuki A. Acta Mater 2006;54:4953. Zhang H, Du D, Srolovitz DJ. Philos Mag 2008;2:243. Cahn JW, Mishin Y, Suzuki A. Philos Mag 2006;86:3965. Molodov DA, Ivanov VA, Gottstein G. Acta Mater 2007;55:1843. Molodov DA, Gorkaya T, Gottstein G. J Mater Sci 2011;46:4318. Winning M, Gottstein G, Shvindlerman LS. Acta Mater 2001;49:211. Mompiou F, Legros M, Caillard D. J Mater Sci 2011;46:4308. Homer ER, Foiles SM, Holm EA, Olmsted DL. Acta Mater 2013;61:1048. Olmsted DL, Holm EA, Foiles SM. Acta Mater 2009;57:3704. Wan L, Wang S. Phys Rev B 2010;82:214112. Hyde B, Farkas D, Caturla MJ. Philos Mag 2005;85:3795. Campbell GH, Foiles SM, Gumbsch P, Ru¨hle M. Phys Rev Lett 1993;70:449. Melchionna S. Phys Rev E 2000;61:6165. Fellinger MR, Hyoungki P, Wilkins JW. Phys Rev B 2010;81:144119. Wang LG, van de Walle A. Phys Chem Chem Phys 2012;14:1529. Zhang RF, Wang J, Beyerlein IJ, Germann TC. Philos Mag Lett 2011;91:731. LAMMPS code homepage:
ScienceDirect Acta Materialia 77 (2014) 258–268 www.elsevier.com/locate/actamat
Atomic simulation of bcc niobium R5h001if310g grain boundary under shear deformation Bo-Wen Huang a, Jia-Xiang Shang a,⇑, Zeng-Hui Liu a, Yue Chen b a
School of Materials Science and Engineering, Beihang University, Beijing 100191, People’s Republic of China Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA
b
Received 8 March 2014; received in revised form 19 May 2014; accepted 19 May 2014 Available online 1 July 2014
Abstract The shear behaviors of grain boundaries are investigated using molecular dynamics simulations. The R5h001if310g symmetric tilt grain boundary (GB) of body-centered cubic (bcc) Nb is investigated and the simulations are conducted under a series of shear directions at a wide range of temperatures. The results show that the GB shearing along ½1 30, which is perpendicular to the tilt axis, has a coupled motion behavior. The coupling factor is predicted using Cahn’s model. The critical stress of the coupling motion is found to decrease 1 direction, which is parallel to the tilt axis, exponentially with increasing temperature. The GB under shear deformation along the ½00 has a pure sliding behavior at most of the temperatures investigated. The critical stress of sliding is found to be much larger than that of the coupled motion at the same temperature. At very low temperatures, pure sliding is not observed, and dislocation nucleating and extending is found on GBs. We observed mixed behaviors when the shear direction is between ½1 30 and ½00 1. The transition region between GB coupled motion and pure sliding is determined. If the shear angles between the shear direction and the tilt axis are larger than a certain value, the GB has a coupled motion behavior similar to the ½1 30 direction. A GB with a shear angle smaller than the critical angle exhibits mixed mechanisms at low temperatures, such as dislocation, atomic shuffle and GB distortion, whereas for the ½00 1-like GB pure sliding is the dominating mechanism at high temperatures. The stresses to activate the coupling and gliding motions are analyzed for shear deformations along different directions at various temperatures. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Grain boundary; Shear deformation; Molecular dynamics; Bcc
1. Introduction Grain boundaries (GBs) play a significant role in determining the mechanical properties of polycrystalline materials[1,2], e.g. nanocrack healing by migrating GBs [3]. Numerous investigations have been conducted in order to understand GB structures, energies and thermodynamics. Molecular dynamics (MD) simulation is a powerful method, providing a comprehensive understanding of the microscopic mechanisms of GB movements [4–8], plastic ⇑ Corresponding author. Tel./fax: +86 10 8231 6500.
deformations [9,10] and other atomic behaviors [11–13]. One of a GB’s most important features is its mobility, which depends on the GB crystallography and external conditions. It has been found that the normal GB motion is often coupled to the tangential translation of grains (referred to as coupled GB motion). Stress-induced GB motions have been studied in great detail, both in simulations and experimentally. GB motions can be classified into different groups based on analysis of the atomic structure evolution [14–16], which includes GB coupled motion, GB sliding, grain rotation and dislocation emission. GB coupled motion was first observed in experiments by Li et al. [17] in small-angle Zn GBs. Molteni and
E-mail address: [email protected] (J.-X. Shang). http://dx.doi.org/10.1016/j.actamat.2014.05.047 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
B.-W. Huang et al. / Acta Materialia 77 (2014) 258–268
Shiga et al. investigated the tilt and twist GB motions in Ge, Al and Ni using density functional theory and found that the sliding and migration motions exhibit a stick–slip behavior [18,19]. The Frank–Bilby equation defines the dislocation of an interface between two crystals. The solutions of this equation are meaningful only for low-angle tilt GBs [20], though recently the equation has been extended to high-angle tilt GBs [21,22]. Cahn et al. [23] proposed a geometric model to predict the GB coupled motion. The model considers the upper grain as a parallelogram. In particular, one face of the parallelogram is along the GB plane, while another one is normal to the tilt axis, and the third one is parallel to the slip plane. To calculate the coupling factor b, which is defined as the ratio of the normal to the lateral motion, the parallelogram is first sheared along the direction parallel to the slip plane by a magnitude of B (the sum of the Burgers vectors). Then the parallelogram is rotated clockwise by h (tilt angle) to align the upper grain with the lower grain. The model gives two opposite signs for the coupling factor, which refer to the different slip mechanisms. For the slip plane of GB dislocation along the f100g plane: b ¼ 2tanðh=2Þ
ð1Þ
For the slip plane of GB dislocation along the f110g plane: b ¼ 2tanðp=4 h=2Þ
ð2Þ
It can be seen that the coupling factor only depends on the misorientations. Caillard et al. [7] have further proposed a model of shear-migration coupling for general GBs. A series of MD simulations of the h100i tilt GBs in Cu have been performed by Cahn et al. [5,6,21], confirming the above models. The coupling factors predicted by the geometric theory show excellent agreements with MD simulations and experimental data [24–27]. In addition, three typical tilt GBs (h100i; h110i and h111i) in Ni have been investigated [28,29] using a synthetic driving force method. It was found that the GBs exhibit excellent agreements with the theory for most of the h100i boundaries, whereas the h110i and h111i boundaries did not obey the previous models. Wan and Wang [30] studied the R9h001if221g GB in Cu, Al and Ni with different shear directions at room temperature. They observed various GB motions including coupling and sliding, while the motion types that are activated depended on the shear directions and materials. In another report, occasional sliding has been observed during coupled motions as the strain rate increases [6]. In other words, coupling and sliding take place once the stress has accumulated to a critical level. If the critical stress for coupling is larger than that needed to activate sliding, sliding motion will be dominant; otherwise, coupled motion will take place. In certain conditions, e.g. low temperatures or along some particular shear directions, if the critical stresses for coupling or sliding exceed the yield stress, dislocations accompanied by atomic shuffles will take place.
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Most of the investigations of GB motions focus on materials with face-centered cubic (fcc) structure such as Ni, Al and Cu. The GB motions of body-centered cubic (bcc) metals have rarely been studied except for the bccFe R5h001if310g symmetric tilt GB, which has been investigated using MD under shear deformation from 1 to 600 K. The nucleation and gliding of partial GB dislocations were found in GB migration [31]. Do the GBs in bcc materials exhibit a behavior similar to that seen in fcc materials? Does the geometric theory hold valid in metals with bcc structure? What is the relation between GB motions, temperatures and shear directions? To answer these questions, we choose the R5h001if310g symmetric tilt GB of niobium (Nb) with bcc structure as a model in this paper. Nb is a refractory metal, and one of the most important elements in superalloys with promising applications. The GB is constructed using the coincidence site lattice (CSL) model. By applying shear loads parallel to the GB plane at a wide range of temperatures and different directions, we observe that there are various kinds of GB motion which depend on the shear conditions. We have shown that the shear deformation along the ½1 30 axis is consistent with the prediction of geometric theory. Shear deformation along the ½001 axis displays a GB pure sliding behavior. Other simulations whose directions are between the ½130 and ½001 axes show a mixed behavior. A transition region between the coupling and sliding motions, which depends on the temperatures and directions, is described. The paper is organized as follows. In Section 2, the general atomistic simulation method is described. In Section 3, the results and discussions are presented. Finally, a summary of the present work and conclusions are given in Section 4. 2. Simulation methodology The shear deformation of the Nb R5h001if310g GB is investigated in this paper. Simulation models are constructed using the CSL method. We create the GB model by concatenating two separate grains with specific crystallographic orientations. The orientation of the lower grain is shown in Fig. 1. The upper grain is built by rotating the lower grain around the [310] axis for 180°. The size of the simulation model is approximately 10 nm 10 nm 20 nm, which contains about 120,000 atoms. The periodic boundary condition (PBC) is applied along three dimensions to mimic the bulk material conditions. In order to avoid the interference of the second GB which is caused by the PBC, we fix the atoms on the top and bottom layers of the model. The fixed atoms are located in their perfect lattice positions. We define the thickness of each fixed region as twice of the cut-off distance. Two grains undergo a rigid-body translation within the GB plane to find the position with the lowest energy. A conjugate gradient algorithm is applied for energy minimization in this work. Once the optimized structure is obtained, we then run MD simulations for sufficiently long times to ensure
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Fig. 1. Schematic illustration of the GB simulation model that used in this work. The crystallographic directions are defined with reference to the lower grain. The ½00 1 axis is the tilt axis.
the stability of the model. The initial equilibrium GB structure is shown in Fig. 2, which is viewed along the tilt axis. According to their CSL notation and the GB normal direction, this GB is termed R5h001if310g; its structure was observed experimentally in Ref. [32]. In Fig. 2, the structural unit, which contains six atoms located on the GB plane, is outlined and the orientation vectors for both the GB period and normal directions are given. The black and white atoms define two adjacent atomic planes along the tilt axis. It should be noted that each kite-like structure unit is composed of atoms belonging to two neighboring atomic planes. MD simulations are then performed to deform the GB model at a constant shear strain rate of 1108 s1 . The
shear direction is parallel to the GB plane. Table 1 lists the shear directions and the included angles between the ½001 tilt axis and the shear directions. The shear process deforms the simulation box as a whole. Stresses on all directions except the shear direction are allowed to relax during the simulations. A canonical ensemble (NPT) with the Nose–Hoover thermostat is applied [33]. Stress is calculated using the standard viral expression. We adopt the Nb embedded atom method potential [34] which has been used in other studies [35,36]. The accuracy of this potential is confirmed by testing the elastic constants, melting point, lattice constant and thermal expansion. Common neighbor analysis (CNA) is used to display the atomic structure. MD simulations are realized using the LAMMPS code [37].
Fig. 2. The R5h001if310g symmetric tilt GB structure in Nb at 0 K. Filled and empty circles represent the Nb atoms in two adjacent atomic planes. The structural units are outlined in solid blue lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Table 1 The shear directions, critical stresses and GB motion types at 100 K. Direction ½001 ½1330 ½1320 ½1315
½1310
½136
½395
½131
½130
Angle h Critical stress (GPa) Motion type
17.55° 2.80 Mix
27.79° 2.20 Couple
62.21° 1.18 Couple
72.45° 1.10 Couple
90° 1.22 Couple
0° 3.50 Slide
6.02° 5.00 Mix
8.99° 4.80 Mix
11.90° 4.10 Mix
3. Result and discussion 3.1. Shear along the ½1 30 direction Shear deformation is applied along the ½1 30 direction with a constant strain rate. The ½1 30 direction is located on the GB plane and is perpendicular to the ½00 1 tilt axis. We perform MD simulations at different temperatures to analyze the mechanisms of GB motion. Fig. 3 shows the stress–time relations at temperatures between 1 and 2400 K (the bulk melting point 2750 K). It is obvious that the curves display a strong temperature dependence. The yield stress decreases exponentially with increasing temperature as shown in the inset of Fig. 3. At temperatures between 1 and 2100 K, the stress–time curves display a sawtooth behavior. The stress drops when the critical value is reached. The magnitude of the dropping and the period of each curve are temperature independent between 300 and 2100 K. Ivanov [6] reported that the grain size in the normal direction affects the stress behavior during shear deformation. At temperatures below 100 K, the stress–time curves display larger stress drops and longer periods. The atoms at temperatures below 100 K are difficult to move; thus the GBs can accumulate and release more elastic energy in one period. When the temperature is above 2400 K, the stress is close to zero. After checking the GB structure at 2400 K using the radial distribution function (RDF), we find that the GB has already premelted, and this explains the abnormal behavior of the stress–time relation at 2400 K. In order to understand the relation between GB motion and stress, we plotted the time dependency of the shear
stress and GB displacement at 300 K (see Fig. 4). The GB position is tracked by the CNA computation, which gives values of 3 and 5 for atoms in the bcc lattice and the GB region, respectively [38]. It is obvious that the GB migration curve displays a regular serrated profile, which is the so-called “stick–slip” behavior. While the “stick” stages correspond to elastic straining, the “slip” stages relate to certain structural transformations. During the elastic deformation stage, the shear stress increases almost linearly. After the critical stress is reached, the GB rapidly moves to a new position. As the deformation continues, the stress drops after GB migration; this is then followed by a new increase in the stress until the next peak. Each peak of the stress correlates exactly with an increment of the GB motion. One possible explanation is that the GB becomes mechanically unstable as the deformation continues and requires some mechanism to release the excess energy. GB migration is one of the mechanisms that is likely to occur in this condition. The accompanying grain translation produces a permanent shear deformation. This type of GB motion is trapped in one energy minimum until it loses stability and jumps to a new minimum. Fig. 5 shows the GB migrations at different temperatures. It can be seen that the GB migration is similar for most of the temperatures considered here. An obvious stick–slip behavior is seen in our simulations at low temperatures. The displacement steps firstly decrease and then maintain at a certain value as the temperature increases. Thermal fluctuation tends to smooth out the discontinuous stick–slip behavior. The smallest displacement step is confirmed to be the distance between two neighboring atomic planes in the GB region.
Fig. 3. Stress–time curves for shear along the ½130 direction at temperatures between 1 and 2400 K. The simulations are performed with a constant strain rate of 1 108 s1 . The inset of the plot shows the exponential fitting of the critical stress.
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Fig. 4. The shear stress and the GB migration as functions of time for shearing along ½1 30 at 300 K. The process of migration exhibits a stick– slip behavior. The blue line is the stress–time curve and the red line represents the GB displacement which normal to the GB plane. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Migrations of the GB as functions of time at different temperatures for shearing along ½1 30. Migrations exhibit stick–slip behavior at most temperatures. The GB exhibits a triple jumps and double jumps at 1 and 100 K, respectively. At 2400 K, the GB shows random motions.
At 100 K, the GB jumps by twice the smallest height and the drop in stress is also doubled. The atoms cannot move easily at this low temperature; therefore, the accumulated elastic strain energy is so large that one single jump cannot release it completely. It is expected that at temperatures below 100 K, higher multiple jumps may be observed. Simulation at 1 K shows a triple-distance migration which confirms this expectation. At 2400 K, the migration is interrupted because of the GB pre-melting. These phenomena are consistent with a recent publication that reports the transition from stick–slip to driven Brownian dynamics for fcc materials [5]. We measure the GB displacement on the GB plane to study whether sliding is also occurring.The magnitude of sliding is calculated using relative positions between two
predefined atoms located on different sides of the GB. The relative distance between this two marked atoms is equal to zero at time ðtÞ ¼ 0. In all of our simulations, GB sliding along the ½1 30 direction is found to accompany GB migration. The temperature dependence of sliding (see Fig. 6) has a behavior similar to that of migration, i.e. stick–slip behavior. The GB migration and sliding velocities can be obtained from Figs. 5 and 6 (displacement–time curves), respectively. Our results indicate that the GB velocity is independent of temperature, and is determined by the shear rate and GB structure. The shear rate is also referred to as the sliding velocity by other authors [5,6,21]. Considering that the velocities of migration and sliding remain constant, we assume that a coupled relation exists in these two motions. Fig. 7(a) shows the GB motions in both of the ½130 and ½310 directions at 100 K. It can be seen that the migration and the sliding motions take place simultaneously, which means they couple perfectly. The coupled motion is similar to other stick–slip motions, e.g. the tip movements in atomic force microscopy. The coupling factor b for GB introduced by Cahn et al. [21] is defined as the ratio between the GB sliding velocity (v//) and the GB migration velocity (vn). From the geometric model of coupling, the perfect coupling factor of our GB is b ¼ 2tanðp=4 h=2Þ ¼ 0:99. Our simulations give a consistent result that b ¼ v== =vn 1. In the range of 1–2100 K, b is found to be practically independent of temperature and in good agreement with its geometrical value (see Fig. 7(b)). This indicates that the coupling is nearly perfect for this temperature range. When the temperature is above 2400 K, the coupling behavior is destroyed by GB pre-melting. It was mentioned in a previous publication for fcc Cu [21] that the GB coupling behavior is determined solely by the GB geometric structure. In other words, the coupling factor is fixed once the GB structure is determined.
Fig. 6. Sliding of the GB as functions of time at different temperatures for shearing along ½130. Sliding exhibits stick–slip behavior at most temperatures. The stick–slip behavior is weakened as temperature increases.
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deformation of the units. This process can be viewed as the glide of the parallel arrays of the GB dislocations along ½001. The coupling factor can be obtained, which is the ratio of the unit displacement length on plane to that in the vertical direction. 3.2. Shear along the ½001 direction
Fig. 7. The GB coupled motion for shearing along ½130. (a) The GB migration and sliding as functions of time at 100 K. The blue line describes migration and the red line represents the sliding process. (b) The coupling factor b as a function of temperature. The dashed line is the b predicted from the geometric model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
As our simulations reveal similar behaviors in Nb, the same rule may also apply to other bcc materials. However, more GBs in other bcc materials need to be further tested in order to confirm this. To study the structure evolution of the GB, snapshots of the atomic positions are recorded at fixed time-step intervals. Fig. 8 shows a plot of the vector field of the R5h001if310g GB, which is viewed along the ½001 direction. It records the position variation before and after one migration. The green unit (abcdef) belongs to the lower grain and is interlocked with the kite-like blue unit (ABCDEF). The green unit is a slightly distorted version of the perfect lattice. Importantly, the green and blue units are topologically identical and can transform to each other by relatively small atomic displacements. At each step of boundary motion, the green unit changes its shape and transforms to a kite-like blue unit, whereas the blue unit simultaneously transforms to an orange unit. Note that the latter is a mirror reflection of the green unit. As a result, the GB position shifts one step down while the upper grain translates to the right to accommodate the
The ½001 direction is parallel to the tilt axis. We apply shear deformation along this direction as was done for the ½130 direction. Several phenomena that are different from those we have discussed in the ½130 direction are observed. It is determined that the GB motion is pure sliding along the ½001 direction. Stress–time curves at different temperatures are shown in Fig. 9. The critical stress is much larger than that of the ½130 direction, which means pure sliding along the ½001 direction is harder to activate than coupled motion. The critical stress does not strongly depend on temperature in the range 100–300 K. A periodic sliding behavior similar to the GB coupled motions is observed at most temperatures except 1 K. Based on the analysis on the variance of the GB position, it is found that there is no displacement perpendicular to the GB plane. The GB can only slide along the ½001 direction within the GB plane. Similar to the ½130 direction, the GB sliding velocities display temperature independence and remain constant. The shear stress and the GB in-plane displacement at 900 K are shown as functions of time in Fig. 10. The sliding along ½001 can also be regarded as a stick–slip process. The GB initially stays in a stick stage, and it accumulates strain throughout the crystal. The slip process takes place when strain reaches the critical level. Simulation at T = 1 K shows an abnormal behavior. In order to understand the mechanisms in detail, we monitor the crystal structure evolution at this temperature. Fig. 11 shows the deformation along the ½001 direction at 1 K. It can be seen that a partial dislocation nucleates and is emitted from the GB, due to its lower activation energy compared to pure GB sliding at this temperature. Once the partial dislocation moves toward the fixed boundary, the region occupied by the stacking fault will extend. Meanwhile, detailed analysis shows that limited GB sliding and local atomic shuffle also take place simultaneously. The nucleation and gliding of partial GB dislocations has previously been observed in bcc Fe at low temperature [31]. 3.3. Shear between the ½130 and ½001 directions In this section, we conduct a series of simulations with shear directions between the ½130 and ½001. The specific information about the shear directions are summarized in Table 1. We have divided the simulations into two groups based on the results at T = 100 K. The first group represents shearing along the ½136; ½395 and ½131 directions. These simulations behave similarly to the ½1 30 direction (coupled motion). The second group consists of simulations
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Fig. 8. Vector field of atomic displacements for shearing along ½130 at 300 K. The black dots represent the original positions of the atoms, while the red arrows show the atomic trajectories. The blue and green lines represent the GB structural units before and after migration takes place, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. The stress–time relations for shearing along the ½001 direction at temperatures between 1 and 2100 K. The simulations are performed with a constant strain rate of 1 108 s1 . The inset shows the exponential fitting of the critical stress.
direction as Fig. 10. The shear stress and the GB sliding in the ½001 functions of time for shearing along ½001 direction at 900 K.
½1 ½1 and ½1 directions; these along the ½1 330; 320; 315 310 show complex mechanisms at low temperatures and ½001like pure sliding at high temperatures.
In the first group, we have studied the ½136 and ½ 5 39 directions in detail. The ½136 direction is about 27:79 clockwise to the ½001 direction and the ½395 direction is 62:21 anticlockwise to the ½001 direction. Two kinds of simulations behave similarity. The GB migrates and slides with stick–slip behavior which is similar to the deformation along the ½130 direction. Furthermore, the GB in-plane movements do not only take place in the shear direction, but also in the vertical direction. The GB can move along the ½136 and ½395 directions simultaneously. As we have discussed previously, the GB exhibits a coupled motion and a pure sliding when it is sheared along the ½1 30 and ½001 directions, respectively. Thus, we have decomposed the velocities of sliding into the ½001 and ½130 directions. Fig. 12 shows how we have decomposed the velocity and stress. For shear along ½136, the resolved velocity in ½00 1 is close to zero and the resolved velocity in ½130 is 3.81 m s1. The GB migration velocity is 3.85 m s1. Therefore, the coupling factor is 0.99, which is consistent with that of the ½130 direction. A similar decomposition is applied to
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Fig. 11. Selected snapshots of dislocation nucleation, motion and atomic shuffle on the GB at 1 K. Atoms are colored according to the centrosymmetry parameters.
Fig. 12. Decomposition of the ½136 displacement into the ½001 and ½130 directions.
the critical stress. The resolved critical stress in ½001 is 1.94 GPa, which is much smaller than the stress needed to activate the ½001 pure sliding (about 3.50 GPa), at 100 K. On the other hand, the resolved critical stress in ½130 is 1.02 GPa, which is comparable to the stress needed
to activate the ½130 coupled motion (about 1.22 GPa). The small difference could be compensated by atomic shuffling. For further investigations, we have performed MD simulations at 100, 300, 600 and 900 K. Table 2 lists the resolved critical stresses and the critical stresses of deformation in the ½001; ½130 and ½136 directions. The MD results are consistent with experiment in that the resolved stress in the ½001 is much smaller than the critical stress of pure sliding. The resolved critical stress in the ½1 30 direction is comparable to the critical stress of coupled motion. Moreover, Fig. 13 shows that the GB is not displaced in the ½001 direction. Combining the coupling factor and the atomic structure evolution, we conclude that shearing along the ½136 direction shares the same mechanism as that along the ½130 direction. It can also be seen from Table 2 that the shear along the ½395 direction shows behavior similar to the deformation in the ½136 direction. The resolved critical stress in the ½001 direction is always much smaller than the stress needed to activate pure sliding, while the resolved critical stress in the ½130 direction and the critical stress along the ½130 direction are comparable. Thus we conclude that the deformation along the ½395 direction shares the same mechanism as the coupled motion in the ½130 direction.
Table 2 The critical stress Fc, stress F and resolved stress f (in GPa) at different temperatures. Temperature
Fc½001
Fc½130
F ½136
f½001
f½130
F ½395
f½001
f½130
100 K 300 K 600 K 900 K
3.50 4.20 3.20 2.50
1.22 0.58 0.31 0.18
2.19 1.07 0.61 0.43
1.94 0.95 0.54 0.38
1.02 0.50 0.29 0.20
1.18 0.58 0.32 0.22
0.55 0.27 0.15 0.10
1.04 0.52 0.28 0.19
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at various temperatures. Fig. 13. The stress–time relations as well as the GB displacements in different directions for shearing along ½136
½1315; ½1320 and ½1330 directions at 100 K. Fig. 14. The stress–time relation for shearing along the ½1310;
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Fig. 15. The stress and the GB displacement as functions of time for direction at 900 K. Arrow 1 is the ½130-like shearing along the ½1 330 coupled motion.
By investigating the deformation mechanisms in the first group at 100 K, we find that the simulations exhibit a ½130like GB coupled motion once the angle between the shear direction and the ½00 1 tilt axis is larger than the critical angle. In the second group, the angles between the shear directions and the tilt axis are smaller than the critical value, and the stress–time curves of the four shear directions at 100 K are shown in Fig. 14. The strain–stress curves display a non-periodic behavior, which indicates that the shear deformations belong neither to the ½1 30-like GB coupled motion, nor to the ½00 1-like GB pure sliding. shear direction (Fig. 14(a)), whose For the ½1 330 included angle h is 6:02 , there are three different stages of deformation. After the critical stress is reached, atomic shuffle appears first to partially release the stress. A large number of dislocations then nucleate on the GB. As the
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deformation proceeds, finally, there is a GB distortion which makes the stress drop rapidly. direction (Fig. 14(b)), For the shear along the ½1320 which has an included angle of h = 8:99 , there are two stages of deformation, which correspond to stages one direction. Dislocation nucleations and three of the ½1330 are not observed in our simulations. and ½1310 directions For the shears along the ½1315 (Fig. 14(c) and (d)), the included angles h are 11:90 and 17:54 , respectively. The GB coupled motions are observed to release the stress. However, the GB coupled motions have limited effects on the deformation, and atomic shuffle and GB distortion still dominate the shearing process. Based on the above analysis, we conclude that a transition region exists between the GB pure sliding and the GB coupled motion, which has different shear deformation mechanisms. With a larger included angle, the proportion of GB coupled motion increases. Nevertheless, we have not observed pure GB sliding, even for an extremely small angle. This is similar to the simulation along the ½00 1 direction at 1 K. A possible reason is that a higher temperature is needed to activate the GB pure sliding in those small-angle simulations, as shown in the following discussion. The stress and the GB position are shown as functions direction of time in Fig. 15 for shearing along the ½1330 and ½391 (perpendicular to at 900 K. Sliding along ½1330 as well as GB migration are observed in the simula½1330) and ½ tion. The displacements along the ½1330 39 1 directions are decomposed to the ½001 and ½130 directions. Based on the analysis of the GB position, we find that the deformation contains ½130-like GB coupled motion and ½001-like GB pure sliding. The stress–time curve displays a sawtooth behavior. The first small peak corresponds to GB sliding on the ½130 direction and migration in the vertical direction. For the second and
Fig. 16. Different types of GB motions with respect to temperatures and shear directions.
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fourth peaks, the GB slides along the ½1 30 and ½00 1 directions and simultaneously migrates; the ½00 1-like GB pure sliding dominates and releases most of the stress. The third and the fifth peaks indicate that the GB displays a pure sliding along the ½00 1 direction. Since the barrier of the ½130 direction coupled motion is much smaller than the ½00 1 direction GB pure sliding, it is reasonable that a handful of ½1 30 direction coupled motions take place at such high temperature. Thus, we consider that this simulation is similar to the ½00 1 direction GB pure sliding. We have summarized the types of the GB motions for different temperatures and shear directions as shown in Fig. 16. The MD simulations with ½1 5; ½1 3 6; ½ 39 3 1 shear directions exhibit ½130-like GB coupled motion. The defor ½1 ½1315 mations with shear directions along the ½1 330; 320; and ½1310 are similar to that along ½1330 at T = 900 K. In direction is a ½00 addition, shear along the ½1 330 1-like pure sliding at 300 K. It is found that the ½00 1 pure sliding is dominating in small-angle simulations at high temperatures, while GB dislocations, distortion and atomic shuffle are the main deformation mechanisms at low temperatures. 4. Conclusion We have performed MD simulations to study the shear responses of the R5h001if310g symmetric tilt GB in bcc Nb over a wide range of temperatures. Nine shear directions parallel to the GB plane have been studied, namely ½1 ½1315; ½1 ½1 ½001; ½1 5; ½1 330; 320; 310; 3 6; ½ 39 3 1 and ½130. For shear deformation along the ½1 30 direction, the GB always shows a coupled motion regardless of the temperature.The critical stress decreases exponentially with increasing temperature. The coupled motion displays a stick–slip behavior, i.e. the system is trapped in an energy minimum until the GB becomes unstable and jumps to a new minimum. At very low temperatures, we observe multiple jumps, which are different from the single jump at high temperatures. The GB coupling factor is found to be independent of temperature and can be predicted from geometric calculations. Therefore, we propose that shearing along the direction perpendicular to the tilt axis shares a mechanism with the geometric theory proposed by Cahn. For the shear deformation along the ½00 1 direction, the GB movement is pure sliding at most of the temperatures studied. The sliding is more difficult to activate than the coupled motion because its critical stress is much larger. Pure sliding is not observed at T = 1 K, and the GB exhibits an unusual behavior caused by the dislocation nucleating and extending. The shear deformations between the ½1 30 and ½001 directions are more complicated, and we have divided them into two groups according to their deformation mechanisms. The first group has large angle between the shear directions and the tilt axis, which exhibits a GB coupled motion and a geometrically predictable coupling factor.
We find that this type of deformation shares the same atomic structural evolution with the ½130 shear direction. The second group has smaller angles between the shear directions and tilt axis. Dislocation, atomic shuffle and GB distortions are dominant at low temperatures, while the ½001-like GB pure sliding is the principal mechanism at high temperatures. Acknowledgment This work was financially supported by the National Science Foundation of China (Grant Nos. 51071011 and 51371017). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
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