A modified synthetic driving force method for molecular dynamics simulation of grain boundary migration

Acta Materialia 100 (2015) 107–117

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A modified synthetic driving force method for molecular dynamics simulation of grain boundary migration Liang Yang a, Saiyi Li a,b,⇑ a b

School of Materials Science and Engineering, Central South University, Changsha 410083, China Key Laboratory of Nonferrous Metal Materials Science and Engineering of the Ministry of Education, Changsha 410012, China

a r t i c l e

i n f o

Article history: Received 5 May 2015 Revised 22 August 2015 Accepted 23 August 2015

Keywords: Grain boundary migration Mobility Molecular dynamics Driving force Driving pressure

a b s t r a c t The synthetic driving force (SDF) molecular dynamics method, which imposes crystalline orientationdependent driving forces for grain boundary (GB) migration, has been considered deficient in many cases. In this work, we revealed the cause of the deficiency and proposed a modified method by introducing a new technique to distinguish atoms in grains and GB such that the driving forces can be imposed properly. This technique utilizes cross-reference order parameter (CROP) to characterize local lattice orientations in a bicrystal and introduces a CROP-based definition of interface region to minimize interference from thermal fluctuations in distinguishing atoms. A validation of the modified method was conducted by applying it to simulate the migration behavior of Ni h1 0 0i and Al h1 1 2i symmetrical tilt GBs, in comparison with the original method. The discrepancies between the migration velocities predicted by the two methods are found to be proportional to their differences in distinguishing atoms. For the Al h1 1 2i GBs, the modified method predicts a negative misorientation dependency for both the driving pressure threshold for initiating GB movement and the mobility, which agree with experimental findings and other molecular dynamics computations but contradict those predicted using the original method. Last, the modified method was applied to evaluate the mobility of Ni R5 h1 0 0i symmetrical tilt GB under different driving pressure and temperature conditions. The results reveal a strong driving pressure dependency of the mobility at relatively low temperatures and suggest that the driving pressure should be as low as possible but large enough to trigger continuous migration. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

1. Introduction Grain boundary (GB) migration plays a key role in governing the microstructure evolution in polycrystalline materials during recrystallization and grain growth [1]. While experimental studies have provided remarkable insights into various migration behavior and mechanisms [2–8], they are often inadequate for a comprehensive evaluation under various conditions. This is essentially related to difficulties in preparing pure bicrystal specimens with specific crystallographic orientation and boundary geometry, and difficulties in exploring migration kinetics under controlled driving forces [1]. Atomistic computer simulation based on molecular dynamics (MD) can overcome such deficiencies and it has been well received as an effective means to complement experiments in understanding GB migration behavior.

⇑ Corresponding author at: School of Materials Science and Engineering, Central South University, Changsha 410083, China. E-mail address: [email protected] (S. Li). http://dx.doi.org/10.1016/j.actamat.2015.08.051 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Existing MD techniques for GB migration simulations can be sorted into two categories: ‘‘fluctuation” methods and ‘‘driving force” methods. In the first category, investigation into GB migration is realized by analyzing thermal fluctuations of a boundary, in the absence of an external driving force [9,10]. This kind of method may not be applied to low temperature and is not suitable for investigating migration kinetics [11,12]. In the second category, a boundary is driven to move under controlled driving forces, which can arise from GB curvature [3,13,14] and elastic anisotropy [15–17] or be related to crystallographic orientations [18–20]. With a curvature-driven technique, a U-shaped GB is often constructed to provide the driving force for its motion. This technique accounts for only three out of the five crystallographic parameters (i.e. misorientation) of a GB and extracts a reduced GB mobility. By contrast, the other two driving force-based techniques consider all five crystallographic parameters of a GB. Simulation of GB migration due to elastic anisotropy imitates a real physical process, but the applied elastic strain cannot provide driving force for symmetric boundaries [12,21]. In simulations utilizing crystallographic orientationcorrelated driving force (OCDF) [18–20], the driving force arises

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from artificial potential energies added to atoms according to their local orientations in a bicrystal system. It has been shown that OCDF-based methods can be conveniently applied to simulate GB migration under desired temperature and driving force conditions [22]. The implementation of OCDF by Janssens et al. [20], which is often known as the synthetic driving force (SDF) method, has been applied to investigate various aspects of GB migration, e.g. boundary roughening [23], shear coupled migration [22,24] and grain growth stagnation [25]. In this implementation, the driving forces for boundary motion are imposed after distinguishing atoms in grains and GB, according to their local orientations characterized by order parameters in conjunction with two adjustable cutoff values [20]. Despite the revealing results obtained with this method, cautions have been raised about its limitation in applications and defects of the technique to distinguish atoms. Olmsted et al. [22] warned that this method is effective only when the boundary misorientation is large enough such that the local structural difference between the two perfect crystals is greater than the typical differences due to thermal fluctuations. Similarly, Rahman et al. [26] advised that considerable cares should be taken in applying the SDF method to low-angle GBs, especially at high temperatures. More specifically, Zhou and Mohles [27] pointed out that the limitation of this method could be related to improper distinction of atoms in grains and GB, which may lead to flawed energy and force distributions in the bicrystal. Several attempts have been made to overcome the deficiencies of the SDF method stated above. First, Zhou and Mohles [27,28] suggested that using only one set of order parameters (as adopted in the original SDF method) is insufficient to distinguish thermal displacements from those caused by crystal reorientation and they proposed to use two sets of order parameters. Although their modified SDF method was criticized for its controversial definitions of atomic energy and forces [29,30], it seems to bring some improvement in avoiding misidentification of atoms for some cases. In a follow-up work by Ulomek and Mohles [31], it was argued that the original SDF method with only one set of order parameters results in an asymmetric transition of order parameter in the GB area and hence an asymmetric profile of the artificial energy in the bicrystal. They applied an order parameter analogous to that considered in the capillary fluctuation method [9] to generate the so-called symmetrical driving forces for GB migration. However, their method yielded the same mobility results as the original approach within the margin of error [31]. Moreover, similar to that of the original SDF method, the GB migration results obtained using these modified methods [27,28,31] depended strongly on the cutoff values that were arbitrarily chosen without necessary justifications. Furthermore, for cases that atoms in grains and GB have similar order parameter values, which are typical at high temperatures due to significant thermal fluctuations, it is impossible to distinguish these atoms by comparing only their order parameters with the cutoff values. A robust technique with sufficient generality to identify atoms in grains and GB under the influence of thermal fluctuations, which is critical in developing the SDF method as a dependable tool to investigate GB migration, is still lacking. Another major concern in applying the SDF method for GB migration simulations is the magnitude of applied artificial driving pressure (i.e. a global value of the individual driving orces). In experimental and force-driven computational studies [1,13,16,17], the GB mobility M is often extracted by assuming a linear relation between migration velocity V and driving pressure P in the low limit of P (i.e. M = dV/dP|P?0). However, due to constraints on the computational expense in typical MD simulations, the driving pressure generally applied in the SDF method is around 10–400 MPa [20,22,23,26,32,33], which are orders of magnitude

higher than experimentally applied values (104–1 MPa) [1,34]. Under such high driving pressures, the migration velocity can be as large as several hundreds meters per second (typical experimental value is 108–103 m/s [1,34]) and it often varies nonlinearly with the driving pressure. The resulting activation energies for GB migration are often significantly lower than those determined from experiments [22,23,26,32,33]. Moreover, an unrealistically high driving pressure has also been observed to alter GB migration mechanism [11] and lead to somewhat surprising results, e.g. zero mobility for R3 boundary with plane normal h1 1 1i while quite high mobility for other R3 boundaries [20,22]. Under these circumstances, it is no longer certain that the mobility obtained from the MD simulations is a reasonable approximation of the intrinsic mobility. It is necessary to investigate the proper range of the driving pressure to consider in practical applications of the SDF method. In the present work, a modified SDF method has been developed after evaluating the deficiency of the SDF method in distinguishing atoms. The modified method was applied to investigate the migration behavior of several Ni h1 0 0i symmetrical tilt GBs (STGBs) and Al h1 1 2i STGBs. The results are compared with those obtained using the original method to reveal the significance of the present modification. Principles for selection of driving pressure in extracting intrinsic mobility are discussed based on a detailed examination of the effect of driving pressure on the mobility of Ni R5 h1 0 0i STGB at different temperatures. 2. Method 2.1. Principles of SDF-based method for GB migration During recrystallization and grain growth, boundaries are driven to move under forces arising from the free energy difference between two neighboring grains [1]. In a bicrystal system structured with a MD technique, the free energy difference caused by the structure difference between adjoining grains and thermal noises are insufficient to drive a boundary to migrate continuously, if no other sources of free energy difference (e.g. chemical or mechanical ones) are available. In the SDF method [20], the free energy difference is enlarged by artificially adding an orientation-dependent potential energy difference to the bicrystal, e.g. adding a potential energy to one grain while not to another, and adding an intermediate energy to GB. A clear distinction of atoms in grains and GB indisputably is necessary before adding the artificial potential energy and thus providing driving forces for GB migration. To this end, Janssens et al. [20] defined an order parameter ni for each atom to characterize the deviation of an actual local structure from a reference structure. In a bicrystal consisting of grain A with orientation I and grain B with orientation J, the order parameter for atom i is defined as [20]

ni ¼

n X jrj  rIj j;

ð1Þ

j¼1

where rj is a position vector of nearest-neighbor atom j of atom i, and rIj denotes the ideal relative vector in the reference grain A. n is 12 for face-centered cubic materials. ni is expected to be small for atoms in the reference (or favored) grain, large for atoms in the non-reference (or unfavored) grain, and intermediate for atoms in the GB. Then, atoms in grains and GB are distinguished based on their order parameters by introducing a pair of cutoffs [20], nlow ¼ f nIJ and nhigh ¼ ð1  f ÞnIJ , where nIJ characterizes quantitatively the orientation difference between the two grains at 0 K P (nIJ ¼ nj¼1 jrJj  rIj j) and f is a parameter to adjust the cutoffs.

L. Yang, S. Li / Acta Materialia 100 (2015) 107–117

Following this distinction, an artificial potential energy uðri Þ with a maximum of U is added to each atom [20]: atoms with ni 6 nlow are considered to be inside the favored grain (uðri Þ ¼ 0), atoms with ni P nhigh inside the unfavored grain (uðri Þ ¼ U), and atoms with intermediate ni values in the GB (0 < uðri Þ < U). Then, the artificial driving force (F) exerted on each atom can be derived by calculating the derivative of uðri Þ. Under these driving forces, the boundary will migrate towards the grain with a higher potential energy to reduce the total free energy of the bicrystal system. When the artificial potential energy uðri Þ is added based on a clear distinction of atoms, the free energy difference DG between adjoining grains will be equal to the added energy difference U and thus the driving pressure P for GB migration can be deduced directly from U. As mentioned in Section 1, the current discussion on the SDF method in the literature concentrates on its technique used in distinguishing atoms in grains and GB. For a given bicrystal system (with a given ni distribution), the discrimination of atoms relies on the cutoffs nlow and nhigh , or essentially nIJ and f. As noted by Janssens et al. [20], the optimal value of f depends on the material, potential and temperature. This statement implies that the value of f has to be determined after examining the specific order parameter distribution in the system under investigation. In practice, however, f was set to be 0.25 in studying various Al h1 1 1i GBs at a temperature of 800 K [20] and 388 Ni GBs at 600–1400 K [22]. Since the order parameter distributions in bicrystals structured with these GBs could be very different and may change significantly with the temperature even for the same bicrystal, it is surprising that such a fixed f value would work well for all these cases. Actually, Olmsted et al. [22] noticed for some cases that there were atoms with order parameters above nlow in the

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reference grain and below nhigh in the other grain, indicating misidentification of atoms in grains and GB. These authors realized that under these circumstances the driving pressure P could not be deduced directly from the added energy difference U, and they attempted to determine P by other means. However, the misidentification would lead to incorrectly added artificial potential energy and hence incorrect driving forces for individual atoms, which should have an immediate effect on the GB migration. Therefore, the validity of simulated GB migration behavior for these cases requires further justification, though this was not discussed in their work. 2.2. Deficiency of existing implementations To evaluate the feasibility of using a fixed cutoff parameter f to distinguish atoms in grains and GB for different cases, we conducted MD calculations for a series of widely studied Ni h1 0 0i STGB using the SDF method implemented in the Lammps MD package [35,36]. Fig. 1 plots the order parameters of atoms as a function of their normalized coordinates, yi , along GB normal (GBN), for several representative cases with different misorientations and at different temperatures. The cutoffs (nlow and nhigh ) corresponding to f = 0.25 are indicated in the plots. These cutoffs have been used in studying the same GBs by Olmsted et al. [22]. As follows from Fig. 1a–d, the order parameter distribution changes significantly with the misorientation angle h at the same temperature (T = 1000 K). For the h = 42.1° bicrystal (Fig. 1c), atoms in grains and GB can be roughly distinguished with the chosen cutoffs. In this case, a considerable number of atoms apparently inside the favored grain (yi > 0) but with ni P nlow would be

Fig. 1. Distribution of order parameter ni in Ni h1 0 0i STGB bicrystals: (a) h = 8.8° at 1000 K; (b) h = 26.0° at 1000 K; (c) h = 42.1° at 1000 K; (d) h = 78.6° at 1000 K; (e) h = 42.1° at 1200 K. The two red lines represent the cutoff values defined with f = 0.25, as in Ref. [22], for distinguishing atoms in grains and GB. The two green lines in (b) represent another set of cutoffs better adapted for the specific GB. Note the value of nIJ quoted in the legend varies with h. (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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counted as atoms in GB and another group of atoms presumably in GB but with ni P nhigh would be counted as atoms in the unfavored grain. This sort of misidentification becomes much severer in the case of h = 26.0° (Fig. 1b). For the h = 8.8° (Fig. 1a) and 78.6° (Fig. 1d) bicrystals, which are featured by highly-disordered GB structures, the cutoffs completely fail in identifying the belongings of atoms. Comparing results for the 42.1° bicrystal at T = 1000 and 1200 K (Fig. 1c and e) further reveals that the misidentification becomes severer at the higher temperature due to increased thermal fluctuations. It is worthwhile to note that an attempt has also been made to choose the two cutoffs, independently, for each order parameter distribution. It turns out that the misidentification could be somewhat reduced by using other cutoff values, e.g. nlow ¼ 0:55nIJ and nhigh ¼ 0:85nIJ for the 26.0° bicrystal (Fig. 1b). For cases like the 8.8° and 78.6° bicrystals, however, it is impossible to distinguish atoms in grains and GB using any cutoffs, because many atoms in the interior of grains have order parameters similar to those of atoms in GB (Fig. 1a and d). The cutoff-based approach simply does not work in such cases. It is possible to show that similar problems also exist in the adapted implementations of the SDF method in Refs. [27,28,31]. Moreover, the cutoffs in the original and these modified methods were all chosen without a general criterion. In the SDF method, the artificial driving force imposed on single atom is derived directly from the added artificial potential energy uðri Þ [20]. To evaluate the consequences of misidentification of atoms in GB migration simulation, it is useful to look into the distribution of uðri Þ. Fig. 2 plots the normalized artificial potential energy uðri Þ/U as a function of the normalized coordinate of each atom along GBN, for two representative GBs. The artificial energy added to each atom was determined based on the distinction of atoms with f = 0.25 (see Fig. 1c and d). It can be seen by comparing Figs. 1c and 2a that for the h = 42.1° bicrystal, a considerable number of atoms apparently in the favored grain are added with artificial energies and thus suffer unexpected driving forces. The number of such atoms drastically increases in the case of the h = 78.6° bicrystal (compare Figs. 1d and 2b). Moreover, for the 78.6° bicrystal, a large number of atoms apparently in GB are not imposed with any artificial energy and they suffer no driving forces. It is reasonable to anticipate that the GB migration simulated in these cases will contradict normal GB migration and lead to somewhat unphysical consequences (e.g. disordered structure in the grain interior). The misidentification will result in a deviation of free energy difference DG from its deserved value, i.e. the added energy difference U. More specifically, DG will be smaller than U and the difference will increase with the extent of misidentification. It is then unfeasible to deduce directly the driving pressure P from U, as stated earlier. Olmsted et al. [22] did notice the effect of misidentification on DG and the

subsequent determination of P. They attempted to circumvent this problem by calculating DG through a thermodynamic integration of the potential energy. However, this post-processing amendment could not change the fact that the GB migration behavior itself was simulated under incorrectly exerted driving forces. It is clear that the deficiency in the stage of distinguishing atoms has to be solved in implementing the SDF concept for GB migration simulations. 2.3. A modified SDF method It has been demonstrated in the preceding section that in SDF-based simulations of GB migration, atoms in grains and GB cannot be properly distinguished based on the order parameter distribution and cutoffs defined in Ref. [20]. In the present work, another form of order parameter, which will be named as crossreferenced order parameter (CROP) nci , is utilized to quantify the local crystalline orientation for each atom. CROP is defined for each atom by subtracting order parameter defined referring to grain B (with unfavored orientation J) from that defined referring to grain A (with favored orientation I), i.e.

nci ¼

n   X jrj  rIj j  jrj  rJj j ;

ð2Þ

j¼1

where rIj and rJj denote the ideal vectors in grain A and B, respectively. Accordingly, the value of CROP will be an approximately constant negative value for atoms in the interior of grain A and a positive value of the same magnitude for those in grain B, with variations due to thermal fluctuations. This order parameter is analogous to the one considered in the capillary fluctuation method for tracing the position of an interface between two misoriented grains [9]. It has been applied in previous MD studies to help estimate the GB position during migration [22] and to quantify the local orientation for individual atom [31]. As an interface between two adjoining grains, GB is a structural transition region from one grain to another and the arrangement of atoms in this region is disordered with respect to the structures in both grains. To illustrate that, Fig. 3 shows the (a) atomic-scale structure and (b) distribution of nci for the Ni 78.6° h1 0 0i STGB at a finite temperature. The distributions of order parameter and artificial energy added to each atom according to the original SDF method can be found in Figs. 1d and 2b, respectively. As follows from Fig. 3b, the nci values are positive for atoms in the unfavored grain and negative in the favored grain. Meanwhile, the nci values for atoms in the transitional or interface region vary from positive to negative values, indicating a significant change in local orientations across this region. This change is consistent with that indicated by the centro-symmetry parameters in Fig. 3a. The

Fig. 2. Distribution of normalized artificial potential energy (uðri Þ/U) imposed on atoms in Ni h1 0 0i bicrystals at 1000 K: (a) h = 42.1°; (b) h = 78.6°. The artificial potential energies were added based on the distinction of atoms using the original SDF method with f = 0.25, as shown in Fig. 1c and d.

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Fig. 3. (a) Bicrystalline MD simulation cell of the Ni 78.6° h1 0 0i STGB at 1000 K, visualized by the centro-symmetry parameter of each atom using the Atomeye tool [37]; (b) Illustration of the present approach for distinguishing atoms based on the distribution of CROP; (c) Distribution of order parameter with data points for atoms in grains and GB identified in (b) shown in different colors; (d) Distribution of normalized artificial potential energy based on the identification of atoms in (b). The thick black lines in (a) indicate the interface region which needs to be defined according to the CROP distribution. Blue points a and b in (b) stand for the two intersection points between the ncpln curve and ncav e lines.

overall feature in the nci distribution stated above becomes more apparent in the plane-by-plane average CROP values (ncpln ), which are superimposed on the distribution of CROP (see black curve in Fig. 3b). It is possible to see that, nci < 0 (or nci > 0) is a necessary but not a sufficient condition for determining an atom as belonging to the favored (or unfavored) grain. This means that an additional measure is needed in order to avoid misidentifying atoms. More specifically, it calls for a more objective definition of the interface region according to the actual CROP distribution. For that, we

uðri Þ ¼

8 > <0 > :

yi > ya and nci < 0; or yi 6 ya and nci < ncav e ðfavored grainÞ U ð1  cos 2xi Þ yb 6 yi 6 ya and  ncav e 6 nci 6 ncav e ðGBÞ 2 U

N 1X jnc j; N i¼1 i

ð4Þ

yi < yb and nci > 0; or yi P yb and nci > ncav e ðunfavored grainÞ;

introduce first an average absolute CROP value (ncav e ) for all atoms in the bicrystal system defined by

ncav e ¼

from their coordinates along this direction by Dy ¼ ya  yb , gives a reasonable estimation of the dimension of the interface region along GBN or the width of GB. Though this dimension can be measured at every time step during GB migration, we expect it changes little during migration and it is thus reasonable to use the initially measured dimension for the whole migration process. With the CROP and other parameters defined above, it is possible to distinguish atoms in grains and GB and then add the artificial potential energy to each atom, as follows:

ð3Þ

where N denotes the total number of atoms in the system. It is possible to show that the ncpln curve intersects at least once with each of the two lines corresponding to ncav e and ncav e , respectively. Points a and b marked in Fig. 3b are such intersection points that are closest to the center of the interface region,1 and they indicate the borders of structure transition between grains and GB. Then, the distance between points a and b along GBN, which can be readily calculated 1 Under non-periodic boundary condition along the GBN direction, only the interface region corresponding to the GB in the middle of the simulation cell needs to be defined.

where xi is a normalized CROP defined as xi ¼ p2

nci þncav e 2ncav e

and U (pos-

itive) denotes the potential energy added to atoms inside the unfavored grain. It is clear that with this distinction approach, atoms apparently in grains but having nci values deviating significantly from ncav e or ncav e due to thermal fluctuations can be readily distinguished. Note also that atoms belonging to grains and GB are mixed in the interface region [yb , ya ]. Fig. 3c re-plots the distribution of the conventional order parameter ni for the 78.6° bicrystal, which is representative of cases that are impossible to deal with using the original technique (see Fig. 1d), but using different colors to discriminate atoms in grains and GB distinguished according to Eq. (4). Evidently, the new technique performs well in separating atoms in GB from other atoms. The corresponding normalized artificial potential energy distribution is plotted in Fig. 3d. In contrast to the erroneous distribution obtained using the original technique (see Fig. 2b), the distribution determined according to the present

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technique (Fig. 3d) follows the principles of the SDF approach, i.e. atoms in the two neighboring grains are respectively added with the maximum and minimum energies while those in GB with intermediate values. This ensures that the free energy difference DG equals the added energy difference U and that only atoms in boundary suffer driving forces. Similar to that in the original method [20], the potential energy function defined in Eq. (4) is differentiable with respect to the position of atom. The driving force imposed on an atom in GB can be calculated by n @uðri Þ U @ cos 2xi X U @ cos 2xj ¼  @ri 2 2 @ri @rj j¼1 ( ! J I n X dij dij pU ¼ c sin 2xi  J I 4nav e jd j jd ij j¼1 ij j " !#) J I n n X X djk djk þ  sin 2xj  J ; I jdjk j j¼1 k¼1 jdjk j

Fðri Þ ¼ 

ð5Þ

where dIij ¼ rIj  rj and dIjk ¼ rIk  rk . The indices j and k denote one of the nearest neighboring atoms of atom i and j, respectively. For atoms in grains, the additional energy will be a constant and thus no driving forces will be imposed. Under the artificial driving forces, the boundary will be forced to migrate towards the unfavored grain to reduce the total free energy of the system. A key difference between the present SDF modified method and the original method lies in the technique for distinguishing atoms in grains and GB. Our technique is featured by adopting CROP to characterize local orientations and introducing a CROP-based definition of the interface region. It is possible to see by comparing Fig. 3b and c that, the use of CROP has the effect of ‘‘enlarging” the local orientation differences between atoms in the favored and unfavored grains. In principle, for cases that the CROP or order parameter of atoms in GB smoothly transits from one grain to another (e.g. the CROP distribution in Fig. 3b), one may look for some cutoff values to separate the atoms in grains and GB, as in the original SDF method [20] and other modified versions [27,28,31]. However, there is no straightforward method to determine the cutoff values, despite of their significant impact on the subsequent GB migration simulations. Therefore, we limit the GB atoms within the interface region, whose dimension along the GBN is bounded by the intersection points of the ncpln curve with the ncav e lines (see Fig. 3b). Importantly, the ncpln and ncav e values used in this procedure are derived directly from the CROP distribution at the considered temperature, without any adjustment. These values are different from the cutoffs in previous developments [20,22,27,28,31], which were determined using an adjustable parameter (f), together with an average order parameter difference between the two grains at 0 K (nIJ in Refs. [20,22,28]) or at a chosen temperature (Dn in Ref. [27]), or were simply specified by an adjustable parameter (g in Ref. [31]). With our procedure, it is possible to distinguish atoms in grains and GB in a straightforward manner without any interference from the observer, while maintaining sufficient generality necessary for comparison of migration behaviors for GBs with various misorientations and at various temperatures. It has to be emphasized that the present technique of specifying the interface region is only applicable to flat boundaries, instead of curved or U-shaped boundaries. It is also worth to note that GB migration is discontinuous at a mesoscale (atomistic scale), as discussed with the help of an island-based mechanism in Ref. [33]. This discontinuity is obscure in a small simulation system and becomes more apparent in a larger system (i.e. a larger GB area) [33]. However, even in large systems as studied in Ref. [33] the island height is only about 1–2 atoms. It is then reasonable to assume that the boundary, as a whole, is still flat and migrates

continuously, at a macroscale. Such a hypothesis of the GB motion mode has been a choice in many other studies of GB mobility wherein the overall movement of a GB has to be tracked. Therefore, the present technique is expected to be still applicable in migration simulations with a large system showing apparent discontinuity in the motion of GB atoms. 3. Applications 3.1. Simulation set-up The modified SDF method proposed in this work was applied to simulate the migration of a series of STGBs in pure Ni and Al to evaluate its performance in comparison with the original SDF method. First, we simulated the migration of Ni h1 0 0i boundaries with four misorientations (h = 7.8°, 26.0°, 42.1° and 78.6°) at T = 1000 K and a driving pressure of 0.01 eV/atom. As mentioned earlier, these boundaries are representative of cases that the original method performs very differently in distinguishing atoms in grains and GB (see Fig. 1); they were shown to have quite different mobility according to the computation in Ref. [22,24]. The results obtained will be compared with those of the original method presented in Ref. [22,24] to assess the significance of atom distinction on the migration behavior. For the 36.9° (R5) h1 0 0i boundary, in particular, we extended the simulations by considering a large range of temperature (T = 500–1300 K) and driving pressure (P = 0.001–0.1 eV/atom) to study the effect of the magnitude of driving pressure on the GB mobility at different temperatures. This boundary is typical of low-period and high-angle CSL boundary, and its migration behavior and mechanism have been widely studied [1,11,22,23]. Second, we investigated the migration behavior of four Al h1 1 2i boundaries (h = 8.8°, 11.7°, 17.4° and 23.1°) at T = 300 K with P = 0.001–0.012 eV/atom. This set of simulations was aimed to explore the misorientation and driving pressure dependencies of GB migration. The results will be compared with those of the original SDF method and random walk method presented in Ref. [26]. The simulations stated above were carried out using the Lammps software package [35,36]. A bicrystal simulation cell was used to construct a flat GB with a given crystalline orientation. It contained one GB and two free surfaces along GBN, i.e. nonperiodic boundary condition along this direction, while the other two directions parallel to the GB plane were set to be periodic. We chose these boundary conditions for two main reasons: (i) the free surfaces can accommodate shear-coupled GB migration and avoid the build up of stresses during migration [26]; (ii) the current boundary conditions are consistent with those in studies in Refs. [22,26], to which the results obtained by the present method will be compared. To avoid interaction between free surfaces and GB, the normal dimension of each grain Ly was set to be the larger value between 30a0 (a0 is lattice constant) and two repeats of the periodic structure. The other two dimensions Lx and Lz were chosen to be the larger value of 15a0 and two repeat structures. The embedded atom method (EAM) potentials of pure Ni and Al, developed by Foiles and Hoyt [9] and Liu et al. [38], respectively, were used as the baseline potentials to describe interatomic energy and force. After the construction of a bicrystal system for a given GB, the energy of the system was minimized at 0 K using the scheme introduced in Ref. [39]. Subsequently, the bicrystal system was elevated to and then equilibrated at a chosen temperature. The temperature of the system was controlled by velocity scaling over a period of 10 ps (with a default time step of 1 fs). The equilibrium state of the system was obtained by performing NVE (constant number of atoms, volume and energy) ensemble, and the time of equilibration MD runs is generally about 0.1–1.5 ns, depending

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L. Yang, S. Li / Acta Materialia 100 (2015) 107–117 Table 1 Mobility and velocity results of Ni h1 0 0i STGBs at 1000 K simulated by the modified and original SDF methods with an added energy difference of 0.01 eV/atom.

a

h (°)

Mobility (105 Å/ps MPa)

Velocity (102 Å/ps)

Modified method

Original method [24]

Modified method

Original method [24]

Modified method

Original method

8.8 26.0 42.1 78.6

201.2 51.7 48.9 467.7

475.6 67.4 47.3 1018.6

146.3 146.3 146.3 146.3

4.4 132.8 140.5 28.6

29.4 7.6 7.2 68.4

2.1 9.0 6.6 29.1

Driving pressure (MPa)

a

These velocities were derived using the M and P values provided in Ref. [24] by V = MP.

on the temperature and the dimension of the simulation cell. The final equilibrium state was determined by the steady-state variations in system pressure and energy, e.g. the pressure along GBN should be quite close to zero in the equilibrium state. After adding the artificial driving forces, the GB was required to migrate for at least 50 Å. The specific time for a certain GB to migrate such a displacement depends on the orientation, temperature and applied driving pressure. It was about 0.01–0.1 ns for the Al h1 1 2i boundaries and 0.001–1 ns for the Ni h1 0 0i boundaries. The GB displacement during migration was determined by tracking the center position of the interface region, which was approximated by the average value of GBN coordinates of atoms having CROP values sufficiently close to zero (between 0.05 and 0.05). The GB migration velocity was extracted from the slope of displacement vs. time data.

3.2. Results and discussion 3.2.1. Effect of atom misidentification on GB migration In this section, a quantitative comparison between the GB migration behavior simulated by the original and modified SDF methods is conducted to assess the effect of the misidentification of atoms on boundary migration. Table 1 compares the mobility M, migration velocity V and driving pressure P obtained from the migration simulations of the Ni h1 0 0i boundaries (h = 8.8°, 26.0°, 42.1° and 78.6°) at 1000 K and with P = 0.01 eV/atom, using the two methods. The M and P results of the original method are taken from Table 1 in Ref. [24] and the corresponding V values were derived from these M and P data based on a V = MP relation assumed in Ref. [24]. To ensure a fair comparison, the same linear relation was considered in deriving our mobility results from the simulated velocities.2 Note, however, the different P values utilized in the two methods. As discussed in Section 2.2, with the modified method, P was derived directly from the added energy difference U (U = 0.01 eV/atom P = 146.3 MPa for Ni). With the original method, it was deduced from the free energy difference DG (6U) determined through a thermodynamic integration [22,24] after adding the artificial energy to atoms according to the cutoff-based technique. It is possible to see by comparing Table 1 and Fig. 1 that, when the misidentification of atoms is insignificant, the P values calculated by the original method are similar to ours, which are equal to the theoretical value. Table 1 shows that, in spite of their considerable differences in magnitude, the mobility results obtained using the original and modified methods indicate a similar trend in their variation with the misorientation, i.e. a much higher mobility for the 78.6° GB than that of the 8.8° GB and then the other two GBs. However, in consideration of the different driving pressures used in deriving the mobility results, this agreement in the misorientationdependency of mobility does not necessarily dictate a similar agreement in the migration behavior. It is more meaningful to 2 In principle, the mobility of a GB should be extracted through a linear V–P relation existed in the low limit of P, as discussed in ongoing Section 3.2.3.

compare the migration velocities, which reflect directly the migration behavior. As shown in columns 6 and 7 in Table 1, the migration velocities predicted by the two methods indeed vary differently with the misorientation. The discrepancies in magnitude are more significant for GBs that the two methods give rise to larger differences in the distinction of atoms in grains and GB. For the 26.0° or 42.1° GB, atoms are relatively less misidentified by the original method (Fig. 1b and c) and only minor differences are found in the velocities predicted by the two methods. For the 8.8° or 78.6° GB, the original method predicts a much lower velocity than that using the modified method. According to the original method, a large number of atoms apparently in the favored grain would be misidentified as atoms in GB and most of atoms apparently belonging to GB would be misidentified as atoms in the unfavored grain (see Fig. 1a and d). Consequently, the actual free energy difference driving the GB motion obtained by the original method would be significantly lower than that by the modified method (compare Figs. 2b and 3d for the 78.6° GB). It is then conceivable that the underestimated migration velocities by the original method are predominantly attributed to its flawed technique used in distinguishing atoms, though minor contributions to the discrepancies from unavoidable differences in simulation setups (e.g. dimension of the simulation cell and time of equilibrating a system) between the present work and those in Refs. [22,24] cannot be ruled out. Last, it should be emphasized that any errors in the migration simulations will be carried on in subsequent calculation of GB mobility and that this effect of atom misidentification cannot be corrected by adjusting the driving pressure (see discussion in Section 2.2).

3.2.2. Driving pressure and misorientation dependencies of GB migration In this section, a comparison between the original and modified SDF methods is conducted in terms of the driving pressure and misorientation dependences of migration simulated for Al h1 1 2i STGBs at T = 300 K. Fig. 4a shows the migration velocities of four h1 1 2i GBs simulated under different driving pressures (P = 0.001–0.012 eV/atom) using the modified SDF method. As shown, a linear V–P relation is evident for the 7.8° and 11.7° GBs when the driving pressure reaches a critical value, which can be estimated to be somewhere between 0.005 and 0.008 eV/atom for the 7.8° GB or between 0.002 and 0.005 eV/atom for the 11.7° GB. For the 17.4° and 23.1° GBs, no critical values observed and the velocities of these GBs increase linearly with the driving pressure in the entire range of driving pressure considered in these simulations. These differences indicate a negative misorientation dependence of the driving pressure threshold for initiating a boundary movement. In addition, comparison of the velocities for the four GBs under P > 0.008 eV/atom reveals a decreasing velocity with an increase of misorientation. To help understand the misorientation dependence of the driving pressure threshold, we visualized the atomic structures of the four h1 1 2i STGBs using the software Atomeye [37]. It was found that these boundaries are consisted of periodic structural units

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θ= o 7.8 11.7

17.4o 6

23.1o °

4

2

0 0.000

(b)

0.055

0.002

0.004

0.006

0.008

0.010

0.012

Driving pressure (eV/atom)

Mobility (A/ps MPa)

°

Velocity (A/ps)

8

(a) o

0.045

0.035

0.025

0.015 5

10

15

20

25

Misorientation angle (°)

Fig. 4. (a) Variation of migration velocity with driving pressure and (b) variation of mobility with misorientation simulated for Al h1 1 2i STGBs at 300 K, using the modified SDF method. The mobility was extracted by linear fitting of the V–P data in (a) over the range of the driving pressure that each boundary can migrate. The error bars in (b) display the standard deviation of the fitted mobility.

with the distance between adjoining units decreasing with the misorientation, in agreement with theoretical expectations about GB structures in literatures [34,40]. Since a shorter distance between the adjoining units signifies a higher structural disorder density and hence a higher energy, it is reasonable to anticipate that a GB with a higher misorientation is thermodynamically less favorable and less stable, and thus requires lower thermal fluctuations and driving forces to initiate its motion. This explains why a higher driving pressure threshold is predicted for a lower-angle h1 1 2i GB (Fig. 4a). Note also that this misorientation dependence agrees with the experimental observation by Winning et al. [7]. They found that the migration activation enthalpies of Al h1 1 2i STGBs with h < 13.6° are much higher than that of GBs with h > 13.6°. Contradictory to the modified SDF method, the original SDF method predicted a driving pressure threshold of 0.003 eV/ atom for the 23.1° GB and no driving pressure threshold for the 7.8° GB (see Fig. 8 in Ref. [26]), indicating increased activation energy at higher misorientation. The discrepancy between the results obtained using the original and modified SDF methods stated above should not be attributed to any physical causes, such as migration mechanism or thermal kinetics, but mainly to the technique for distinguishing atoms in grains and GB, which is the main aspect that differs in the two methods. Another remarkable difference between the results obtained for these h1 1 2i GBs using the original and modified SDF methods can also be found in the misorientation dependence of mobility. To reveal that, the mobility extracted using the modified method is plotted against the misorientation in Fig. 4b. For each GB, its mobility was extracted by a linear fitting of V–P data over the range of the driving pressures that the GB can migrate. These results indicate a negative misorientation dependence of mobility, which is consistent with that simulated by the random walk method in Ref. [26]. By contrast, the mobility results obtained by the original method showed a positive dependence on the misorientation at the same temperature (see Fig. 7 in Ref. [26]). Note that for the 23.1° GB, in which case atoms in grains and GB are least misidentified among the GBs considered (see ongoing Fig. 5d), the original and modified SDF methods predict nearly the same mobility (0.02 Å ps1 MPa1). To facilitate the discussion, Fig. 5 shows the order parameter distributions of the four h1 1 2i GBs at 300 K and the order parameter cutoff values specified with f = 0.25 (one of the f values considered in the simulations using the original SDF method in Ref. [26]). Evidently, atoms in grains and GB would be significantly misidentified by the original method in all cases and the extent of misidentification decreases with increasing misorientation. As

discussed in 3.2.1, the misidentification will lead to an unexpectedly low free energy difference DG (i.e. low effective driving pressure) and eventually an underestimated migration velocity. This means that the underestimations of effective driving pressure and migration velocity will be more significant for GB with a lower misorientation due to a more severe misidentification. Therefore, the misidentifications could potentially change a negative misorientation dependency of the migration velocity (as expected from the modified method, see Fig. 4a) into a positive one. Meanwhile, since the mobility is deduced from dV/dP and the misidentification tends to underestimate both the effective driving pressure and migration velocity, the effect of misidentification on the mobility becomes uncertain and it depends on the relative significance between the underestimations in the two aspects. It is reasonable to speculate from the positive misorientation dependency of mobility predicted by the original method (see Fig. 7 in Ref. [26]) that, the misidentification effect on the driving pressure was overshadowed by that on the velocity. All these comparisons between the original and modified methods demonstrate the importance of the technique for distinguishing atoms in grains and GB during SDF-based GB migration simulations. 3.2.3. Proper driving pressures to extract GB mobility The driving pressures applied in existing SDF-based GB migration simulations [20,22,23,27,28,32,33,41] are orders of magnitude higher than those considered in experiments (104–1 MPa) [1,34], and usually result in unrealistically high migration velocities. Moreover, the mobility results extracted from these simulations often vary significantly with the diving pressure [22,23,32,33,41], which contradicts the general assumption that the intrinsic mobility of a GB should show no dependence on the applied diving pressure [1]. A useful guidance to simulate the boundary migration by the SDF method would be to apply driving pressures as low as possible, such that the mobility results obtained are close to the intrinsic mobilities and are comparable to those from experimental studies. However, applying a too low driving pressure often requires huge and even unbearable computation expenses. It is thus worthwhile to explore a proper range of driving pressure to consider in extracting the intrinsic GB mobility with reasonable computation expenses. In the present work, we chose the Ni 36.9° (R5) h1 0 0i STGB as an example and applied the modified SDF method to simulate its migration behavior under different driving pressures (P = 0.001– 0.1 eV/atom) and at different temperatures (T = 500–1300 K), as described in detail in Section 3.1. Fig. 6 plots the (a) migration velocity and (b) nominal mobility obtained at various temperatures, as a

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Fig. 5. Distribution of order parameter in differently-misoriented Al h1 1 2i STGB bicrystals at 300 K: (a) h = 7.8°; (b) h = 11.7°; (c) h = 17.4°; (d) h = 23.1°. The solid and dash lines represent two cutoff values with f = 0.25.

Fig. 6. Variation of (a) migration velocity and (b) nominal mobility with applied driving pressure in migration simulation of Ni R5 h1 0 0i STGB at different temperatures using the modified SDF method. The nominal mobility was extracted by M = V/P at each V–P data point in (a).

function of the driving pressure. To help illustrate the sensitivity of mobility to the driving pressure, the nominal mobility was extracted directly by M = V/P at each data point without any fitting. Fig. 6a shows that, as expected, the migration velocity increases with the driving pressure. While an overall linear V–P relation can be found at relatively high temperatures (say T P 1000 K), significant nonlinearity is evident at low temperatures. This suggests that, at the high temperatures it is reasonable to extract mobility assuming a linear V–P relation and the results are not sensitive to the particular driving pressure value considered in the simulation. By contrast, the selection of driving pressure becomes important at the low temperatures due to the loss of a linear V–P relation. This temperature dependence of the V–P relation can be attributed to the variations of boundary structure [23,41] and migration mechanism [11] with the temperature. Meanwhile, an increasing trend of velocity with the temperature is observed under sufficiently low driving pressures (<0.025 eV/atom). Under higher driving pressures

(e.g. P = 0.05 eV/atom), the velocity often shows a negative dependency on the temperature, which is unlikely to happen according the existing understanding of migration mechanisms and thermal kinetics. The significance of driving pressure selection in GB migration simulations stated above becomes more apparent from the nominal mobility results shown in Fig. 6b. It can be seen that at high temperatures (T = 1000–1300 K), the mobility does not change significantly with the driving pressure. At other temperatures, the mobility changes significantly with the driving pressure, especially in the range of P = 0.025–0.075 eV/atom. The ratio of maximum to minimum mobility values in the entire driving pressures reaches about 3 at T = 900 K and 6 at 500–800 K. The selection of driving pressure is clearly critical at these low temperatures. Meanwhile, being consistent with the negative temperature dependency of the velocity, the mobility shows also a negative or ill dependency on the temperature under high driving pressures.

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Fig. 7. Variation of mobility with temperature for Ni 36.9° (R5) h1 0 0i STGB. The mobility at each temperature was extracted by linear fitting of the V–P data in Fig. 6a with P = 0.001–0.01 eV/atom.

It becomes clear that in order to obtain mobility results complying with a positive temperature-dependency, the driving pressure should not be higher than 0.01 eV/atom. Interestingly, at this level of driving pressures, an approximately linear V–P relation is held also for the low temperatures (see Fig. 6a). Therefore, a driving pressure within the range of 0.001–0.01 eV/atom should be appropriate to extract the mobility of the R5 GB by the modified SDF method, though these driving pressures are still orders of magnitude higher than the experimental values. Fig. 7 shows the corresponding mobility results extracted by a linear fitting of the V–P data within this range, as a function of temperature. The results indicate an approximately constant mobility at 500–800 K and an increasing trend in the mobility at higher temperatures. It is worth noting that the authors have performed additional simulations using driving pressures lower than the lowest P value discussed above. Under such low driving pressures (e.g. 0.00075 eV/atom), the GB was found to migrate in a mode of forward and backward hops, without showing any continuous motion along a certain direction, especially at higher temperatures (probably due to higher thermal fluctuations). This kind of migration mode is known to be a typical mode under pure thermal fluctuations [9,10]. In our trials the artificial driving forces imposed on atoms are simply too low to alter the mode. Therefore, the applied driving pressure should be as low as possible, but large enough to maintain a continuous GB migration. 4. Conclusions In this work, we analyzed the deficiency of the cutoff-based technique for distinguishing atoms in grains and GB in the original SDF method and proposed a modified SDF method by introducing a new technique to distinguish the atoms. This new technique utilizes CROP parameters to characterize local lattice orientations in a bicrystal system, and introduces a CROP-based definition of the interface region. The modified method has been applied to investigate the migration behavior of the Ni h1 0 0i and Al h1 1 2i STGBs, and the corresponding results have been compared with those obtained using the original method. The following conclusions can be drawn from the results: 1. The cutoff-based techniques for distinguishing atoms in grains and GB as applied in the original SDF method and several adapted versions all rely on adjustable parameters that are chosen without a general criterion. They often fail to perform

when atoms in grains have similar order parameters as those in GB due to thermal fluctuations. The resulting misidentification of atoms will lead to an unexpectedly low free energy difference between adjoining grains and hence erroneously exerted driving forces on atoms. 2. With the modified SDF method, atoms in grains and GB can be successfully distinguished in a straightforward and sufficiently general manner without any interference from the observer. Comparison of migration simulations for the Ni h1 0 0i STGBs reveals that, the discrepancies between migration velocity results predicted by the original and modified methods are proportional to the differences in distinguishing atoms in grains and GB. This correlation reveals the importance of using a reliable atom distinguishing technique in SDF-based simulations. 3. The modified SDF method predicts a higher driving pressure threshold for initiating the movement of a GB with a lower misorientation and a negative misorientation dependency of mobility for the Al h1 1 2i GBs, which agree with experimental findings and other molecular dynamics computations but contradict those predicted using the original method. 4. The driving pressure in SDF simulations of GB migration should be chosen based on a detailed investigation of the V–P relation considering a relatively large range of driving pressure for the concerned GB and temperature. It should be as low as possible, on the condition that a continuous GB migration is ensured. When extraction of the intrinsic mobility is desired, a linear V–P relation and a positive temperature dependence of velocity should exist in the vicinity of the chosen driving pressure.

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