Atomistic simulation of shear-coupled motion of [1 1 0] symmetric tilt grain boundary in α-iron

Computational Materials Science 148 (2018) 141–148

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Atomistic simulation of shear-coupled motion of [1 1 0] symmetric tilt grain boundary in a-iron Jian Yin, Yi Wang, Xiaohan Yan, Huaiyu Hou ⇑, Jing Tao Wang School of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

a r t i c l e

i n f o

Article history: Received 11 January 2018 Received in revised form 15 February 2018 Accepted 16 February 2018

Keywords: Shear-coupled grain boundary motion Molecular dynamic simulation a-iron Coupling factor Geometric model

a b s t r a c t Shear-coupled grain boundary (GB) motion (SCGBM) is an important and efficacious plasticity mechanism in the deformation of metals, especially nanocrystalline metals. In this work, molecular dynamic (MD) simulation has been performed to investigate the SCGBM of two [1 1 0] symmetric tilt GBs, R9 [1 1 0](2 2 1) and R17[1 1 0](2 2 3), in a-iron, and the effects of temperature and strain rate on SCGBM have been studied. The coupling factor b which is defined as the ratio of the velocities of GB lateral translation and migration was calculated, and a geometric model of b depending on the misorientation angle was constructed in [1 1 0] symmetric tilt GBs of BCC metals. The model was branched into two modes (h1 0 0i and h1 1 1i) corresponding to the perfect dislocation Burgers vectors in BCC metals. The b values calculated in the h1 1 1i mode were in good agreement with the MD simulation results for both the GBs. Further, the atomistic mechanisms of the SCGBM processes were also investigated. A same structural unit transformation was observed for the two GBs, which confirmed that both R9[1 1 0](2 2 1) and R17[1 1 0] (2 2 3) GBs moved in the h1 1 1i mode during the SCGBM process. Ó 2018 Elsevier B.V. All rights reserved.

1. Introduction Grain boundaries (GBs) play an important role in the mechanical properties of materials, especially for nanocrystalline materials. GB motion properties have a significant influence on recrystallization, grain growth and plastic deformation. The GB motion includes normal motion (migration), relative translation parallel to the GB plane coupled to migration, GB sliding, and grain rotation [1]. Under applied shear stress, the coupling of GB migration with lateral translation is observed in experiments [2–4]. This behavior is called shear-coupled grain boundary motion (SCGBM), and it is an important and efficacious plasticity mechanism in the deformation of metals, especially grain growth and recrystallization [5]. The coupling factor, b, is defined as the ratio of the velocity of GB translation, vp, to GB migration, vn [1], and it is indicated that its value depends on the GB structure (including tilt axis and misorientation angle) and perfect dislocation slipping direction. A geometric model of b was proposed by Cahn et al. [1,6] for FCC metals. The model is branched into two modes (h1 0 0i and h1 1 0i) corresponding to the perfect dislocation burgers vectors in FCC metals (h1 0 0i and 1/2 h1 1 0i).

⇑ Corresponding author. E-mail address: [email protected] (H. Hou). https://doi.org/10.1016/j.commatsci.2018.02.039 0927-0256/Ó 2018 Elsevier B.V. All rights reserved.

Due to the challenge and limitations in the direct measurements of GB mobility via experiments, molecular dynamic (MD) simulation has been widely used for researching GB migrations [7,8]. The process of SCGBM is convenient to carry out using MD simulation and the elementary atomic mechanisms can be investigated and analyzed from the MD results. Zhang et al. [9] examined the SCGBM of symmetric and asymmetric tilt GBs in copper using MD simulation and found that the SCGBM could be regarded as sliding of GB dislocations along the boundary plane. Frolov [10] and Zhang et al. [11] demonstrated the influence of temperature and local boundary structural transition [12] on SCGBM, and found that the motion and atomic mechanisms are sensitive to these factors. Most MD simulation researches for SCGBM focused on FCC metals such as copper, aluminum, and nickel [1,9–11,13–20] and confirmed that the model proposed by Cahn [1] was appropriate for both [1 0 0] and [1 1 0] symmetrical tilt GBs in FCC metals. In recent years, SCGBM has been investigated for [1 0 0] symmetrical tilt GBs of BCC metals such as Nb [21] and W [22,23] as well. It is suggested that the SCGBM plasticity mechanism exists in the plastic deformation of both FCC and BCC metals. Moreover Niu et al. [23] improved the geometric model of b in [1 0 0] symmetrical tilt GBs of BCC metals according to the perfect dislocation Burgers vectors in BCC (1/2 h1 1 1i and h1 0 0i) which is different from FCC metals. The geometric model for [1 0 0] symmetrical tilt GBs in

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BCC metals has the same expression as that of FCC metals. However, the SCGBM mechanism and coupling factor model of [1 1 0] symmetrical tilt GBs of BCC metals are still unclear. In this work, the SCGBM of two [1 1 0] symmetrical tilt GBs in BCC Fe is investigated using MD simulation. In Section 2, the general atomistic simulation method is described, and the simulation results are presented in Section 3. In Section 4, structural units of equilibrium GB structures and unit transformation in SCGBM are discussed, and a model for coupling factor b is constructed. Finally, the conclusion of this work is presented in Section 5.

2. Computational model In this work, MD simulation was carried out using the largescale atomic/molecular massively parallel simulator (LAMMPS) code [24]. The embedded atom method (EAM) interatomic potential of iron [25] was used to describe the interaction between atoms, and the atomic configurations were visualized using the Ovito software [26]. Two [1 1 0] symmetric tilt GBs, R9[1 1 0](2 2 1) with a misorientation angle of 141.06° and R17[1 1 0](2 2 3) with a misorientation angle of 86.63° in a-iron (in this paper the misorientation angle is the angle between [1 0 0] directions in both the crystals in the GB model), were constructed [27,28]. The simulation model with three dimension of 230.16 Å  291.57 Å  20.19 Å for R9(2 2 1) GB and 233.09 Å  306.60 Å  20.19 Å for R17(2 2 3) GB approximately to 80a0  100a0  7a0 (a0 is the lattice constant of iron) is shown in Fig. 1(a). The initial GB structure was energy minimized using conjugate gradient (CG) algorithm and relaxed sufficiently for calculating the GB energy, while periodic boundary conditions were applied along all the three directions. The GB ener-

gies calculated by Eq. (1) are 1289.3 mJ/m2 and 1623.1 mJ/m2 for R9[1 1 0](2 2 1) and R17[1 1 0](2 2 3) GBs, respectively.

EGB ¼

Etot  Ne ; 2SGB

ð1Þ

where Etot is the total system energy after minimization and relaxation, N is the total atom number, e is the cohesive energy and SGB is the area of the GB interface. Periodic boundary conditions were applied along x and z directions and free-surface boundary condition was applied for y direction in the SCGBM simulation. Two thin slabs of atoms (y direction) with 4–5 atom layers (the black blocks in Fig. 1(a)) were fixed, and another slab of atoms (red atoms in Fig. 1(a)) perpendicular to GB interface was selected for marking the movement of atoms in SCGBM. The simulations were performed in isothermal-isobaric NPT ensemble at five different temperatures (10, 100, 300, 600, 900 K), while pressures along the x and z directions were controlled to be around 0 GPa. We first relaxed the initial GB model for 40 ps to obtain the equilibrium structures of GBs at a given temperature. Then, a constant velocity (10 m/s) was applied to the top slab along the positive x direction and the bottom slab was fixed invariably. The equilibrium structures of both GBs at 10 K are presented in Fig. 1(b) and (c). The position of the GB plane is represented by the average coordinates of the defect atoms. 3. Results 3.1. SCGBM at 10 K The simulations of SCGBM of two [1 1 0] symmetric tilt GBs were carried out at 10 K to avoid thermal effects. Fig. 2(a) and (b) are the MD snapshots at about 400 ps for R9(2 2 1) and R17

Fig. 1. (a) Model of MD simulation. Top and bottom slabs of atoms are fixed, and top slab is used for applying constant strain. Red atoms initially perpendicular to GB interface in the middle of the box along x direction are selected for marking SCGBM. (b) and (c) are equilibrium structures of R9(2 2 1) and R17(2 2 3), respectively. Atoms are colored according to potential energy, and the color bar is presented for reference. (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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Fig. 2. MD snapshots of SCGBM of both the GBs: (a) R9(2 2 1) and (b) R17(2 2 3). S and H are the displacements of GB translation and migration, respectively. Black arrow indicates initial GB location and black dashed line indicates the final GB location.

(2 2 3) GBs, respectively. The horizontal black dashed line presents the final GB location after migration. The movement of atoms between the initial and final GB locations is distinct from other areas. The marked atoms in this area constitute an obvious slope. This suggests that the atoms in this area move along the x direction with displacement gradient. The atoms above this area along the y direction translate with the top boundary, whereas other atoms below this area do not move due to the fixed atoms in the bottom boundary. The displacements of GB translation (S) and migration (H) are equal to the width and height of the slope. As described in the introduction, the coupling factor, b, can be calculated by the displacements of GB translation (S) and migration (H), as shown in Eq. (2) [1].

b ¼ v p =v n ¼ S=H

ð2Þ

It is obvious from the MD snapshots that the two GBs migrate along opposite directions. Since velocity was applied in the positive x direction, we define that b is positive when the GB migrates along the positive y direction. Then, the values of b for the two GBs are: b ¼ 0:591 for R9(2 2 1) GB and b ¼ 0:412 for R17(2 2 3) GB. The xy component of shear stress (rxy)-time curves and the GB location-time curves for both the GBs are presented in Fig. 3. The sawtooth-shaped stress curve corresponds to the continuous platform-shaped GB location curve. In one sawtooth, the stress increases linearly until critical stress is reached. During increase in stress, the GBs remain stationary as the platform shown in the GB location-time curves. When the stress drops, the GBs start to move and rapidly reach a new position forming a step in the curves.

3.2. Effects of temperature and strain rate We chose five different temperatures (10, 100, 300, 600 and 900 K) for both GBs and a velocity 1 m/s for R9(2 2 1) GB at 10 K to investigate the effects of temperature and strain rate. For R9 (2 2 1) and R17(2 2 3) GBs, the GB location-time curves are presented in Fig. 4(a) and (b) and the relationship between SCGBM critical stress and temperature is presented in Fig. 4(c) and (d), respectively. The thermal effect decreases the critical stress and contributes to the activation of the GB migration. As shown in Fig. 4(a), the curves are similar at 10 K and 100 K, and it takes less time or a small strain to active GB migration at 100 K. The critical stress is about 1.70 GPa at 10 K and 1.50 GPa at 100 K as shown in Fig. 4(c). This shows that the SCGBM in R9(2 2 1) GB is different at low and high temperatures. The critical stresses for activating migration are about 3.84 GPa at 300 K, 2.71 GPa at 600 K and 2.14 GPa at 900 K which are much higher than those at low temperatures. For R17(2 2 3) GB, the slopes of the five GB locationtime curves at different temperatures are almost the same, as shown in Fig. 4(b). The time or strain of GB migration activation decreases with increasing temperature. Correspondingly, the critical stress decreases with temperature invariably, as we can see from the relationship of critical stress and temperature in Fig. 4 (d), and is caused by thermal effects as in the case of R9(2 2 1) GB. A different mechanism exists in SCGBM for R9(2 2 1) GB at high temperatures. As we can see from Fig. 5(a) (MD snapshot at 400 ps for R9(2 2 1) GB at 300 K), the displacement of GB horizontal translation is much larger at 300 K than at 10 K. As stated in the introduction, the GB motion includes normal motion (migration),

Fig. 3. Shear stress (rxy)-time curve and GB location-time curve at 10 K, (a) R9(2 2 1) and (b) R17(2 2 3) GBs.

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Fig. 4. (a) and (b) GB location-time curve at five different temperatures, and (c) and (d) the critical stress for GB migration at each temperature for R9(2 2 1) and R17(2 2 3) GBs.

relative translation parallel to the GB plane coupled to migration, GB sliding and grain rotation. GB sliding [29] is defined as the grain translation events that are not coupled to normal GB migration [1] and mostly occurs at relatively high temperatures. A higher critical stress and less migration displacement for the same strain indicate that GB sliding hinders migration. The relationship of S/H and temperature is presented in Fig. 5(b) with coupling factor calculated in Eq. (2). For R9(2 2 1) GB, the large value of S/H suggests that the GB sliding events occured at temperatures higher than 300 K. Moreover, it is indicated that no GB sliding events occured during SCGBM for R17(2 2 3) GB at temperatures lower than 900 K since the S/H value almost does not change. If GB sliding events occur, Eq. (2) should be changed as follows [1,6]:

b ¼ ðvp  vs Þ=vn

ð3Þ

where vs is the velocity of GB sliding. Cahn [1] reported that for SCGBM of [1 0 0] symmetrical tilt GBs in copper, low-angle GBs do not slide even when the temperature approaches melting point. For high-angle GBs, GB sliding events will happen if the temperature is higher than a critical temperature. In this simulation, although both the GBs have high misorientation angles, the critical temperature of R9(2 2 1) GB is lower than 300 K, while no GB sliding events occur even at 900 K for R17(2 2 3) GB. It is confirmed that the critical temperature of GB sliding for R9(2 2 1) GB is much lower than that of R17(2 2 3) GB. In our future work, we plan to determine the relationship between GB sliding and high temperature for [1 1 0] symmetrical tilt GBs in a-iron.

Fig. 5. (a) MD snapshot at 400 ps for R9(2 2 1) GB at 300 K, GB sliding occurs in SCGBM. (b) Ratio of the displacements of GB translation (S) and migration (H) for R9(2 2 1) and R17(2 2 3) GBs at five different temperatures.

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The relationship between GB location strain and the two strain rates applied in this simulation is shown in Fig. 6 for R9(2 2 1) GB at 10 K. Despite one order of magnitude difference between the two strain rates, the slope of the curve is almost the same; this implies that the strain rate has almost no effect on the SCGBM mechanisms. This has been confirmed in previous works of other researchers [1,23]. This is the reason we chose the velocity of 10 m/s applied in the SCGBM simulations of this work. 4. Discussion 4.1. Atomic mechanisms of SCGBM The GB energy and property depend on its structure. For lowangle GBs, the GBs can be regarded as an array of dislocations. On the other hand, the structures of high-angle GBs are complex and several models have been proposed for their characterization

Fig. 6. GB location-time curve for R9(2 2 1) GB with two applied velocities or strain rates.

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such as the coincident-site-lattice (CSL) model [29,30], structural unit (SU) model [22,31,32], and the O-lattice model. The SU model was proposed by Sutton and Vitek [33–35] to analyze the GB structure. In the SU model, GBs with certain misorientation angles correspond to ‘‘favored” structural units. All other GBs can be characterized by one or several neighboring favored boundaries [22,36,37]. We have identified several structural units in Fig. 7(a) and (b), where the Black and white atoms represent two atom planes along the [1 1 0] direction, and the [1 0 0] directions of the two grains are labeled with blue arrows. Structural unit A and A0 correspond to typical (1 0 0) and (1 1 1) components of perfect BCC lattice. The unit B in R9(2 2 1) GB is kite-shaped consisting of six atoms and symmetrical with the GB interface. The units B and B0 in R17(2 2 3) GB consist of six atoms as well and have two common atoms in the GB interface. Because of the misorientation angle of 86.63° which is close to 90° in R17(2 2 3) GB, the [1 0 0] direction in lower grain is close to the [1 1 0] direction in upper grain; this renders the GB structure a little more irregular and the units B and B0 unsymmetrical. The six atoms of approximate perfect lattice forming a rectangular box with edges in [1 0 0] and [1 1 0] directions in one grain are referred as unit C. Two adjacent MD snapshots of GB migration were selected for investigating the difference between the atom positions in both the GBs, which are shown in Fig. 7(c) and (d) where the atoms are colored according to atomic energy. The atoms are labeled with numbers to investigate their relative movements and the structural unit transformations. The transformation of unit C is the important atomic mechanism for the migration of GBs [11,38]. For R9(2 2 1), the GB structure is composed of adjacent structural units B and A and can be labeled as |BA.BA|, as indicated by the black dotted rectangle shown in Fig. 7(a). As we can see, structural unit C is adjacent to the GB unit B and they have two common atoms. In the process of SCGBM, the two units transform into each other by the relative displacement of partial atoms. The rectangular unit C transforms into the kite-shaped unit B and unit B becomes a perfect lattice of lower grain. Unit A adjacent to unit C, indicated by the green line in Fig. 7(a), is along the [1 0 0] direc-

Fig. 7. Structural unit analysis of (a) R9(2 2 1) and (b) R17(2 2 3) GBs and position variation of partial atoms of (c) R9(2 2 1) and (d) R17(2 2 3) GBs in SCGBM. Black and white atoms in (a) and (b) represent two atom planes along [1 1 0] direction. Atoms in (c) and (d) are colored according to potential energy, and the color bar is presented in the right of the figure. (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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tion in upper grain, and rotates to become parallel to the GB plane due to the relative displacements of atoms. As can be seen from Fig. 7(c), unit B transforms into C and unit A in the upper grain transforms into A of the GB interface through the relative movements of atoms in the units. While unit transformation occurs in the process of SCGBM, the upper grain translates along the x direction to accommodate the deformation. After the transformation, the structural units along the positive y direction turns into |CBC| from |BCC|, as shown in Fig. 7(c), and the GB migrates to a new position through the above-mentioned mechanism. The structural units of R17(2 2 3) GB are more complex than that of R9(2 2 1) GB. For description of unit transformation, the GB units are composed of two six-atom unsymmetrical units (B and B0 ) and four-atoms (1 1 1) components of perfect lattice unit (A0 ). We consider that the plane composed of units A0 is the GB plane, with unit B0 lying in the upper grain and B in the lower grain relative to the GB plane. The black dotted rectangle in Fig. 7(b) shows the R17(2 2 3) GB unit, which can be labeled as |BB0 A0 .BB0 A0 |. A similar mechanism was observed in the process of GB migration. The unit C close to perfect lattice transforms into unit B and unit B transforms into unit B0 . The common two atoms of units B and C displace relatively to become the new GB plane. The relative movements of atoms during unit transformation in R17(2 2 3) GB can be seen in Fig. 7(d). The structural units along the positive y direction turns into |CBB0 C| from |CCBB0 | and the position of unit A0 indicates that the GB migrates along the negative y direction to a new location. Overall, the structural unit analysis revealed that unit transformation from C to B is the main mechanism of SCGBM for both R9 (2 2 1) and R17(2 2 3) GBs. The structural unit transformation of [1 0 0] symmetric tilt GBs in copper has been discussed by Cahn et al. [1]. The atomic mechanisms of SCGBM depends on the modes of GB motion, which are h1 0 0i and h1 1 0i for FCC metals. Different unit transformations occur in the two modes in FCC metals, which may also be appropriate for BCC metals. In this simulation of BCC iron, even though the migration directions in R9(2 2 1) and R17(2 2 3) GBs are opposite, the SCGBM could be in the same mode which will be discussed in Section 4.2.

4.2. Geometric model of b The value of b is determined by GB geometry, including lattice structure and GB dislocation. In a previous work on shearcoupled [1 0 0] symmetric tilt GB motion in FCC Cu, Cahn et al. [39] developed a b model with two branches of misorientation dependence according to different GB dislocation contents. As is known, 1/2h1 1 0i and h1 0 0i are the Burgers vectors of two representative perfect dislocations in FCC crystal. The two branches are termed as h1 1 0i and h1 0 0i due to the perfect dislocation slip directions, as shown below:

bFCC h100i½100 ¼ 2 tan


 h 2

ð4Þ

and


bFCC h110i½100 ¼ 2 tan

p h 4





2

ð5Þ

where h is the GB misorientation angle. It is confirmed from dislocation theory and experiment results that compared to FCC crystal, the perfect dislocations in BCC crystal has the Burgers vectors of 1/2h1 1 1i and h1 0 0i. For [1 0 0] symmetric tilt GB in BCC crystal, Niu et al. [23] improved the model using the MD results of shearcoupled GB motion in tungsten. The model form is similar to that h p h of FCC crystal (bBCC and bBCC h100i½100 ¼ 2 tan 2 h111i½100 ¼ 2 tan 4  2 ). The model for [1 1 0] symmetric GB in BCC crystal can be derived from geometric considerations  tilt Consider that two grains form a symmetric GB with [0 1 1] axis and misorientation angle h, as shown in Fig. 8. Suppose the lower grain moves up in mode h1 0 0i. After the GB migrates by a length L, the [1 0 0] direction in grain 1 rotates along the [1 0 0] direction of grain 2. The atoms of this direction move parallel to  the GB plane by 2L tan 2h , as shown in Fig. 8(a). Then, the b in this mode can be written as the same as that of FCC crystal  (b ¼ 2 tan 2h ). Similarly, If the upper grain moves down in the h1 1 1i mode, the distance by which the atoms in grain 2 move paral  lel to the GB plane is 2L tan u2 , as shown in Fig. 8(b), where

Fig. 8. Coupling factor model for [1 1 0] symmetric GB in BCC crystal in two modes, (a) h1 0 0i mode and (b) h1 1 1i mode.

J. Yin et al. / Computational Materials Science 148 (2018) 141–148

pffiffiffi 2  2h. Considering the direction of applied velocity, the   pffiffiffi b value in this mode is b ¼ 2 tan tan1 2  2h . The two

u 2

¼ tan1

branches of b model for [1 1 0] symmetric tilt GBs in BCC metals are described as

bBCC h100i½110 ¼ 2 tan


 h 2

ð6Þ

and


pffiffiffi h  1 : bBCC 2 h111i½110 ¼ 2 tan tan 2

ð7Þ

For R9(2 2 1) and R17(2 2 3) GBs, the misorientation angles are 141.06° and 86.63°, respectively. Considering the direction of GB migration, R17(2 2 3) GB is attributed to the h1 1 1i mode. However, we cannot decide the mode for R9(2 2 1) GB only from the migration direction. As the misorientation angle ranges from 0° to 180° for [1 1 0] symmetric tilt GBs, the value of b in h1 1 1i mode may be positive, which implies that the direction of GB migration is probably along the positive y direction similar to the case of the h1 0 0i mode. We calculated the values of b in the two modes from the model for R9(2 2 1) GB: bh1 0 0i = 5.657 and bh1 1 1i = 0.566. The coupling factors obtained from the MD simulation results were 0.591 for R9(2 2 1) GB and 0.412 for R17(2 2 3) GB which were in agreement with the model calculated values of 0.566 and 0.404. It is suggested that the b model is correct and applicable to [1 1 0] symmetric tilt GBs in BCC metals. Thus, it is confirmed that R9(2 2 1) GB moves in the h1 1 1i mode because of the model value and the same unit transformation as that of R17(2 2 3) GB. 5. Conclusion In this work, the MD simulations of SCGBM for two [1 1 0] symmetrical tilt GBs, R9(2 2 1) and R17(2 2 3) GBs, in a-iron have been performed. Migration directions were opposite for the two GBs. GB sliding occurred in R9(2 2 1) GB at high temperatures (higher than 300 K), whereas no GB sliding was observed in R17 (2 2 3) GB. The critical temperature of GB sliding for R9(2 2 1) GB was much lower than R17(2 2 3) GB. Moreover, the strain rate had almost no effect on the SCGBM mechanisms. Unit transformation that a six-atoms rectangular unit C, which is close to perfect lattice transforms into a kite-shaped six-atoms unit B in the GB structure was observed for both the GBs. A geometric model of b for [1 1 0] symmetric tilt GBs in BCC metals was constructed depending on the misorientation angle. The b values calculated in the h1 1 1i mode were in good agreement with the MD simulation results for both the GBs. The same unit transformation and comparison of b theoretical value and simulation results indicated that both the GBs in our simulation moved in the h1 1 1i mode. Acknowledgements This work was supported by the National Key Research and Development Program of China under Grant No. 2017YFB0702201, the Chinese Ministry of Science and Technology of China under Grant No. 2017YFA0204400 and 2017YFA0204403 of the National Key Basic Research Program, the Natural Science Foundation of China under Grant No. 51520105001, and Fund of Key Laboratory of Advanced Materials of Ministry of Education No. 2017AML06.

Conflict of interest None.

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