A Molecular Dynamics Study on the Effects of Lattice Defects on the Phase Transformation from BCC to FCC Structures the

Molecular dynamics simulations of the phase transformation from body-centered-cubic (bcc) to face-centered-cubic (fcc) structures were performed. A Morse-type function was applied, and the parameters were determined so that both fcc and bcc structures were stable for the perfectcrystal model. When the fcc structure was superior to the bcc structure, the bcc model transformed to fcc. Two mechanisms, based on the Bain and Nishiyama-Wasserman (NW) relationships, were considered. Then, point or linear lattice defects, i.e., randomly scattered or regularly aligned vacancies, were introduced. Consequently, bcc models tended to transform to an fcc structure, whereas fcc models remained stable. The transformation process was also investigated in detail. BCC-to-FCC transformation is often considered as a homogeneous process based on changes in the axis lengths, and such a process was observed for the perfectcrystal model. Conversely, for the defect models, local heterogeneous deformation patterns, including cylindrical domain and planar interface formation, were observed. These behaviors are considered to be related to plastic deformation during phase transformation, and the validity of the presented model for further investigation was confirmed.


Introduction
The microstructure of materials strongly affects the macroscopic mechanical properties, and controlling the microstructure is important for achieving re-How to cite this paper: Uehara pering, and annealing are the most effective routes that are indispensable for producing useful materials [1]. This process involves intrinsic phase transformations between austenite, ferrite, and martensite; austenite phase is stable at high temperatures, with a face-centered-cubic (fcc) structure, while ferrite and martensite phases are stable or quasi-stable at room temperature with body-centered-cubic (bcc) and body-centered-tetragonal (bct) structures, respectively.
Heat treatment processes are classified into many detailed subdivisions, and the obtained phases are named as pearlite, bainite, troostite, and so on, while the fundamental crystal structures are considered to be fcc and bcc/bct. The development of heat treatment processes has a long history, but advancement has been achieved through trial and error based on enormous experiments. Additionally, the precise control of not only the microstructure but also the deformation and stress distribution has been required. In addition to distortion due to phase transformation, plastic deformation plays a substantial role in deformation and is termed transformation plasticity. However, even the fundamental mechanism remains unclear, and intensive investigations involving modeling the phenomena and clarifying the mechanism have been conducted [2]- [7].
Alternatively, computer simulation is an indispensable tool in engineering processes. In heat treatment processes of steel, it has become possible to simulate macroscopic deformation and stress distribution, where the phase transformation is usually represented by an empirical kinetic law such as the Johnson-Mehl-Avrami-Kolmogorov equation, and the state of a phase is normally represented using the volume fraction of eachphase [8] [9]. From a smaller-scale viewpoint, it has become possible to simulate the microstructure using a phase field model [10] [11], in which two phases are represented by a single variable with a value of 0 or 1, and phase transformation is expressed by the domain boundary motion. However, these computational simulations are phenomenological, and the essential phenomena, i.e., change in the crystal structure or atomic configuration, are not directly represented. In order to achieve an intrinsic understanding and application through more detailed and precise modeling and simulation, atomistic investigations are necessary.
Experimental technology on a nanometer scale has achieved rapid progress using electron and atomic-force microscopy [12] [13], however, direct observation of dynamic behavior is not yet possible. Molecular dynamics (MD) simulation is useful for exploring the atomic behavior, and various researches including on the structural phase transformation have been carried out [14]- [19]. However, few studies on the mechanism of the transformation plasticity are reported.
In this study, we modeled the phase transformation on an atomic scale to clarify the deformation mechanism during phase transformation, including induced plastic behavior. In MD modeling, a choice of potential function is crucial, and this should be carefully demonstrated when a specific material is studied. In this study, we concentrated on capturing a general aspect of the phase  [20], was applied. The parameter was determined initially so that both fcc and bcc structures are stable, and then the effects of lattice defects on the transformation process were investigated. Three types of defects, i.e., point, partial line, and full-line defects, were considered, though these are all based on vacancies, and their influence on the stable phase was investigated. Generation of plastic behavior during phase transformation was also investigated by observing the dynamic process.

Crystallographic Relationship between Bcc and Fcc
The crystallographic relationship between the fcc and bcc structures is usually explained as the Bain relationship. Figure 1 This structure corresponds to a face-centered-tetragonal (fct). If the length of the vertical axis (z in Figure 1) is increased by 2 , or the two perpendicular axes are shortened by 1 2 , the cell becomes cubic. Accordingly, the phase transformation from bcc to fcc can occur only by changing the edge length without changing the atomic arrangement.
Another relationship between fcc and bcc structures is Nishiyama-Wasserman (NW) relationship, as shown in Figure 1(b). This is usually used to explain martensitic (bct) phase transformation, which is interpreted as a bcc-fcc relationship when combined with a change in the edge length [21]. The crystal plane and orientation are related through fcc {111}//bcc {011} and fcc <112> // bcc <011>. In Figure 1(b)(i), two bcc unit cells are drawn using green lines, two parallel (011) planes are drawn using black and red lines, and the 01 1     direction is indicated by an arrow in magenta. Figure 1  direction, and if the edge length in the corresponding direction is slightly elongated so that the hexagon becomes regular, then this plane becomes the (111) plane with appropriate stacking, and the bcc-to-fcc transformation is complete. This mechanism requires both the change in edge length and relative motion of the atomic layer, but the required edge-length ratio is much smaller than that for the Bain relationship. Figure 1  x-y and z-y planes when the bcc-to-fcc transformation occurs. The original (001) plane becomes the (011) plane of the fcc structure, following the Bain relationship, and a rectangular cell is prominent. However, relative motion of alternate layers, represented by black lines, occurs in the NW relationship, giving a somewhat disordered impression. The regular unit can be found by selecting a 45˚ rotated square, depicted in magenta, in which the center atom is shifted to one side. In the perpendicular (z-y) plane, the long and short edged rectangular cells are alternatively accumulated along the z direction.

Molecular Dynamics Method
A classical MD method with two-body interatomic potential function was used in this study, and the following Newton equation of motion was then solved numerically.
Here, r i and m i are the position vector and mass of the i-th atom, F i is the force acting on the i-th atom, f ij is the interatomic force between the i-th and j-th atoms, and φ is the interatomic potential energy function. In this study, the following Morse-type function is applied.
Here, D, R, and α are the parameters, and the values for most major metallic elements are listed in the literature [20]. Equation (2) is standardized by dividing φ by D and r by R as In this paper, all variables are expressed as the standardized values, except for temperature, and the superscript * is omitted. Numerical integration was performed using the conventional velocity Verlet algorithm.

Simulation Model and Conditions
As a preliminary study, the potential parameter α was varied, and the stable structures were explored using the models illustrated in Figure 2. Using these models, MD simulations were demonstrated at a constant temperature under a constant pressure, p = 0. The velocity scaling method was applied for temperature control with T = 10, 400, and 800 K. Total time step was set as 5000 steps, which may not be long enough to distinguish an actual stable structure, but this does not matter in this study because the transformation is the major objective.    Therefore, it is confirmed that phase transformation from bcc to fcc has occurred according to the NW relationship.

Results
Similarly, an fcc transformation occurred for the bcc-(011) model. The initially rectangular calculation cell became cubic by stretching the shorter edge (y in this case), as shown in Figure 3(b)(ii) and Figure 4(b). The relative configuration of atoms did not change; therefore, it is concluded that an fcc transformation occurred in accordance with the Bain relationship. In both cases, the potential energy was decreased; hence, the fcc structure is more stable for α = 1.60.
Note that the difference in the potential energy between Figure 4(a) and Figure   4(b) is due to the use of a different value in α.
The simulated results are summarized in a phase diagram shown in Figure   5(a), in which, for example, the notation "fcc-> fcc" indicates that the fcc model maintained its structure, and "bcc-> fcc" indicates that phase transformation from bcc to fcc occurred. The fcc structure was stable for the whole parameter range shown in the diagram, and no plots for "fcc->bcc" exist. It is concluded that both the fcc and bcc structures are stable or quasi-stable in the range of α≤ 1.55, while the bcc structures transform to fcc for α ≥ 1.60. Figure 5(b) represents the potential energy obtained for fcc and bcc structures at T = 10 K. Note that the potential energies of bcc for α ≥ 1.60 are measured before the transformation to fcc occurred. The potential energy for fcc phase is slightly lower than that for bcc; consequently, the fcc structure is superior to bcc, and the bcc is quasi-stable for small values in α. The difference in the potential energy becomes larger as the value in α becomes larger. Then the bcc phase is unable to retain the structure, and transformation occurs for α ≥1.60.

Simulation Models and Conditions
Based on the results for perfect crystal models, the models with lattice defects were applied to the MD simulations. Models based on a bcc-(001) model are shown in Figure 6. The defect line is set in the center of the x-y plane, as shown in Figure 6(b) and   According to the phase diagram in Figure 5(a), both fcc and bcc structures are stable for α ≤ 1.55. In this section, similar simulations are performed using the defect models, and the effects on the stable structure are examined. The simulation conditions are the same as in the previous section; α is set between 1.20 and 1.80, T = 10, 400, and 800 K, and the initial configuration is based on fcc, bcc-(001), and bcc-(011) models. Three trials are demonstrated for each parameter set by varying the random-number series, which is used for defining the initial velocity of every atom and is considered to influence the results to a certain degree.   These results are summarized in a phase diagram in Figure 8(a), in which the small marks represent the results for a perfect crystal. The results for the defect models were only plotted when a bcc-to-fcc transformation occurred. Three trials were executed for each case, and the case was plotted if transformation occurred one or more times. The results for Point-5 and Point-10 models, and Line-2 and Line-3 models showed similar trend, and their results are plotted together. This figure clearly shows that the domain where the fcc structure is stable is extended, and the threshold value of α is decreased from 1.60 to 1.45. That is, the defects accelerate phase transformation from bcc to fcc structures. Figure   8(b) represents the potential energy in the defect models for α =1.40 and T = 10 K, in which the value is the average of all atoms in the calculation cell. Blue and red lines represent the value for the perfect bcc and fcc crystal models, respectively. The potential energy for defect models is increased, but the increment is small compared to the difference between fcc and bcc. Point-defect models show  higher values than that for line-defect models, but no specific effects on the phase transformation was found. Therefore, the initiation of the transformation is considered to originate in a local behavior, which is discussed in the next section.

Plastic Behavior during Phase Transformation
According to both the Bain and NW relationships, the bcc-to-fcc transformation is an overall phenomenon, involving changes in the edge length and/or simultaneous sliding of the atomic layer. Therefore, if no local irregularity exists, phase transformation progresses homogeneously. However, defects may induce local irregular behavior; in some of the simulations, notable patterns, which may be related to plastic deformation, were observed. Another notable pattern is shown in Figure 10, which also shows results for the bcc-(001) model, Line-3 defect, and T = 10 K, with α = 1.50. Phase transformation   is initiated at about the 1500th timestep, which can be observed from the differences in the atomic configuration between the 1000th and 1500th time steps.
Proceeding to the transformation, faint oblique lines are observed at the 1000th time step. These become sharper with the transformation at the 1500th time step and finally disappear when the transformation is completed by the 5000th time Similar behavior was observed for the full line models. Figure 11(a) represents the result for the same sets of parameters as for Figure 9, for which a circular pattern was observed. In this case, an arc can be seen at the 2000th timestep, as shown in Figure 11(a)(ii), but a closed circle is not formed, and two parallel boundaries are formed at the 3000th timestep. The origin of the generated plane can be found at the original defect, which is located at the center of the side edge in this figure. Similarly, the result of a comparative calculation using the same parameters as in Figure 10 is shown in Figure 11(b). In this case, the same pattern consisting of oblique sharp lines is observed. A minor difference is that the line does not disappear completely until the end, and the lines are partially retained.

Discussion
Overall, no specific patterns as shown in Figs 9 and 10 were observed during phase transformation for the scattered point defect models, although the total number of vacancies was much higher in a Point-10 model than that in a Line-3 model, and the increase in the potential energy was larger as shown in Figure   8(b). This is because heterogeneity is higher for aligned vacancies, or a line-defect model, than a randomly scattered case. Also, these patterns were not observed very frequently for the bcc-(011) models. This is because the Bain relationship induced often in the (011) model is rather straight forward, and local irregularities did not tend to be formed. In MD simulation, however, random inhomogeneity is introduced intrinsically, and different results were obtained occasionally. Figure 11(c) shows a result for the bcc-(011) Line-full model with α = 1.50 and T = 10 K. A unique pattern with crossing lines is generated on the transformation between the 3000th and 4000 th time steps. Some other patterns were also observed, and detailed investigation of the universal features will continue.

Conclusion
Molecular dynamics simulations of the phase transformation from bcc to fcc structures were performed using a Morse-type potential, and the effects of lattice defects on the phase transformation were investigated. Consequently, it was revealed that the defects influenced the stable structure and accelerated transformation from bcc to fcc. Different types of lattice defects, such as cylindrical domain formation and planar defects, were also generated during phase transformation, which might result in the initiation of plastic deformation. Further investigation is necessary to clarify the relationship between the phase transformation and induced plastic deformation, and it is concluded that the validity of the present model is verified. In addition, we have also reported microstructural  Very recently, an interesting study on the transformation mechanism between the bcc and fcc structures for a binary-alloy system was reported [24], where the transformation is initiated at the two-phase boundary and progresses based on the NW relationship. Further modeling combined with these models will lead to complete clarification of the transformation and induced plastic behavior.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.