The Formation and Dynamic Evolution of Antiphase Domain Boundary in FeAl Alloy: Computational Simulation in Atomic Scale

The main objective of the present work is to build a model and analyze the dynamic evolution process of antiphase domain boundary (APDB) in FeAl alloy. The formation, evolution of APDB, long range order (LRO), the crystal structure transition, impact of temperature on LRO, are investigated. Comparisons with experiments proves that the model is competent for the dynamic investigation of APDB in microscopic scale and able to predict different boundary-types and their corresponding atoms distribution. The dynamic process shows that the initial distribution of premier micro domains determine the morphology of APDB. The morphological evolution of APDB significantly affects the quantity of APDB. The in situ observation shows that the crystal structure of a micro domain is altered by the APDB movement. The calculated LRO revealed that the atoms in Al-sublattice mainly contribute to the low order degree of FeAl at different temperatures.


Introduction
In recent years, due to the development of modern industry the demand for structural materials is growing 1 . The intermetallic compounds have attracted a lot of attention due to their peculiar properties. Among various compounds, the B2-FeAl has been considered as an important high-temperature structural candidate. Which has increasingly attracted the interest of materials scientists. In addition to its excellent moderate to high-temperature property, FeAl compound also has other extraordinary properties 2-7 , including: high specific strength/weight ratio, excellent corrosion and oxidation resistance as well as sulfidation resistance, low raw material costs, retaining LRO up to critical temperature of A2 ⇌ B2 phase transformation, stress anomaly, wear resistance.
In spite of numerous studies, researches referring to APDB in FeAl are seldom paid attention 33,34 . To our knowledge, the APDB has a share in influencing the magnetic 35 , mechanical 36 and plastic 37 properties of the alloy. The microscopic structures determine the macroscopic properties. To get a better control on the microscopic structures, it is important to understand the specific evolution process of microscopic structures. So investigations on APD and APDB are needful and significative. However, for the solid to solid phase transformation, due to the spatial and temporal restrictions it is difficult and expensive to accomplish an instantaneous tracking on the whole transformation process. Fortunately, the phase field method which is a powerful tool for materials research, make it available for the investigation on the whole phase transformation process [38][39][40] . Therefore, the main objective of this work is to build a model for the investigation of the APDB in FeAl alloy. And make it possible to clearly capture the formation process of APDB in atomic scale, predict the motion as well as evolution of APDB during the order-disorder solid phase transformation. It is believed that the model is helpful in having a further and comprehensive understanding about the FeAl-based alloy and meaningful for the material design.

The MPF model
Based on Ginzburg-Landau theory, Khachaturyan firstly introduced the microscopic phase field (MPF) theory 41  c 0 is solute composition of the alloy, k B is Boltzmann constant, Tis temperature, L 0 (r-r')denotes the matrix of kinetic coefficients related to probabilities of elementary diffusional jumps from site r to r' of a Bravais lattice during a unit time. φ is the total energy of the system. It can be written as (2) w(r-r') is the effective interchange energy between two atoms at lattice sites r and r'. It consists of two parts: the chemical interaction energy w ch (r-r') and the strain-induced elastic interaction energy w el (r-r'). The more detailed descriptions of the MPF can be referred to references 42 to 46 .

The application of MPF model to the bcc alloy
The FeAl's structure is CsCl-type (  Figure 1). Thus, the FeAl unit cell can be described as two interpenetrating simple cubic sublattices Fe and Al. In the structure, each atom has a nearest coordination sphere containing eight atoms of the other type. So, for an atom in the perfect B2 unit cell, its first-nearest vector is r a b c V and the second-nearest is r 2 =a.
In the Fourier space, w ch (r-r') and L 0 (r-r') are indicated as w ch (k) and L 0 (k), respectively. And the two parameters are correlated with the crystal lattice site The V i (i=1,2,3) is the nearest neighbor effective interchange energy from the first coordination sphere to the last. L is a constant. k=2π(ha * , kb * , lc * ) is the vector in reciprocal lattice space of FeAl. (a * , b * , a * ) are the reciprocal unit vectors corresponding to (a, b, c). In Figure 1, a random r is written as r=(ma, nb) in a (x, y) Cartesian coordinate, and  With vector (5), the expressions of w ch (r-r') and L 0 (r-r') in reciprocal space are expressed as The strain-induced elastic interaction energy w ch (r-r') in reciprocal space is in form of (8) In Fourier space, , u x and u y are constitutive vector of u along x and y directions. ε results from crystal lattice misfit between solid solution lattice and precipitated phase lattice. c 11 , c 12 , c 44 are the three independent elastic anisotropic constants of b.c.c alloys.

The parameters for the model
In this paper, the parameters are: ∆t=0.00001; ε=0.00692; 1 . The simulation work was performed in a matrix (square lattice) containing 512×512lattice points, where the periodic boundary conditions were implemented along both directions. In the temperatural and compositional ranges, a B2-FeAl single phase exists 1 .

The formation of FeAl APDB
The micrographs of formation and dynamic evolution of APDB reproduced by MPF simulation for Fe-35at. % Al at T = 973K are presented in Figure 2. In the initial stage, the alloy system is a disordered solid solution which is generated by assigning the average compositions of Fe and Al to the respective distribution functions at each lattice site. As visually depicted by pure color existed in Fig. 2(a), the solute Al occupies each sublattice site with a probability equal to Al solute concentration 0.35. When t = 950, the micro domains form in the solid solution. Because of the random thermal noise, the micro domains possess different LRO degrees (the thermal noise is removed at t = 200). It is found that the formation process is very fast, , , , which can be illustrated by the comparison of Figure 2b, 2c, 2d. Figure 2d shows the complete formation of B2 APDB. Before this time, accompany with the precipitations of new ordered micro domains, the existed micro domains hold their growing. There are two evolutional possibilities for all micro domains during the formation process: coalescence and collision. Micro domains increasingly grow into one larger domain when they coalesce with each other. In this case, APDB does not form. In the second circumstance, the conflict of micro domains generates an APDB in the contact area. The micro domains of smallest bulk firstly disappear. Examples can be given by the marked part in Figure 2e. For the coalescence, only the micro domains having the same sublattice configurations in their junction crystal lattice sites can coalesce to produce a larger domain. In the right bottom of Figure 2, an instance of coalescence is clearly displayed. The left image is extracted from Figure 2b. Its location in Figure 2b is realizingly identified by its coordinate numbers.
The right is the in situ photograph extracted at time t = 1150.
In the selected region, six ordered micro domains precipitate from the parent phase, it is clearly seen that they are separated by disordered solid solution when the ordering starts. During the ordering process, the micro domains marked with 1, 2 and 3 coalesce with each other; meanwhile, the 5 th and 6 th micro domains also merge into one domain. After a check in atomic scale, it is certified that the 1 st , 2 nd and 3 rd micro domains completely overlap their same elementary sublattice in the connecting lattice sites. However, an APDB generates when the 4th micro domain encounters with the 1 st and 2 nd micro domains. Which can be observed in the right imagine, another longer APDB is the conflicting result of a coalesced domain (6 th and 5 th ) against the coalesced domain (1 st , 2 nd and 3 rd ). As illustrated in the Figure 2. The face-to-face contact of colliding micro domain sublattices produces an area that the thickness is several atomic layers. The atomic ordered arrangement of this area is not consistent with the periodic arrangement of any micro domains. This misfit accounts for the reason of APDB formation: a result of collision between competitive growths of micro domains.

The dynamic evolution of FeAl APDB
Competitive growths of micro domains lead to the morphological change of B2 APDB. Initially, numerous micro domains occur randomly with various irregular shapes in the matrix, which determines that the APDB have multifarious winding forms and distribute in a high density. As is shown in Figure 2d-2h, when the time t increases, the density of APDB decreases. The APDB is one type of defects that have a higher free energy, which is less stable than the perfect structure and makes the system more unstable. Thus, the decrease of APDB volume derives from the improvement of systematic stability. For the very same reason, the APDB likewise adjusts its shape toward reducing the total system energy. Hence, the closed APDB gradually modulates its shape from an anomalous closed shape to an approximately circular pattern and, simultaneously, shrinks its boundary length. However, as for the dynamic evolution, unlike the closed APDB, the unclosed APDB regulates its form by transforming a straight shape, such as the marked APDB: h1, h2 and h3 in Figure 2h. Deleting windings is the common feature for different typed APDB though they present different dynamic evolution routes. The dynamic evolution of APDB can be expressed as a function of time. In view of numerous production of micro APDB in the early stage, the dynamic evolution of APDB can impact the total APDB quantity.
It is clearly presented that in the Figure 2, lots of closed APDB appear during the ordering transformation process. The APDB is an important factor affecting the material properties. Therefore, it is needful to study its formation process. In order to explicitly depict the formation of closed APDB, its early formation process is listed in Figure 3. To date, there is no work related to the formation process of B2-FeAl APDB in atomic scale. The MPF provides the formation process in the present work. Based on the theoretical results, it is found that the formation mechanism of closed APDB includes two types. The first type is presented in Figure 3a1 to 3a3. As displayed, the island micro domain (in the white circle) generates in a disordered region of the ordered domain. In fact, this case is rare in contrast with the second. According to the Ostwald Ripening, the island micro domain tends to dissolve. However, its wrapping domain mainly grows at expense of consuming the Al solute of the outside solid solution. The island micro domain can compete in depleting solute with the wrapping domain until the solid solution completely disappears. The second type is indicated in Figure 3 b1 to 3b3, the island micro domain (in white box) is separated with its neighbor island micro domains by solid solution. In the next competitive growth stage, its neighbor micro domains merge into a larger domain that completely wraps the island micro domain. This case is the main formation route for the closed APDB.

Impact of moving APDB on the crystal sublattice configuration
The moving APDB has an impact on the crystal sublattice. The influence of moving APDB on the sublattice is exhibited in Figure 4.
To explicitly interpret the effect of moving APDB on the crystal structure. The in situ cell structure transformation and atomic distribution function of sublattice are exhibited in Figure 4. Three conditions are investigated. Firstly, a region where no APDB passes is checked. Figure 4a shows the Feand Al-distribution functions versus time in a B2 lattice of the above region. X Y denotes the X atom locates at Y sublattice site. About at t = 1000, the curves obviously start to change, which indicates that Al and Fe atoms separate fast in the solid solution and selectively occupy different sublattices. After t = 1150, all curves reach their equilibrium and do not The Formation and Dynamic Evolution of Antiphase Domain Boundary in FeAl Alloy: Computational Simulation in Atomic Scale  have any changes, this means no structural transformation occurs. Figure 4d1 is extracted from the morphological evolution figures. It is only existed cell structure before t = 1000. And the structure showed in Figure 4d2 is the solely existed structure after t = 1150. Associated with Figure 4a, it is concluded that without the APDB movement, only one structural transformation occurs. Secondly, a region where only one APDB goes through is studied (the APDB reaches the investigated sublattice at t = 4000). Figure 4e2 to 4e3 shows the existed structures, respectively, corresponding to the second (t = 1150-4000) and third (t = 6000-100000) equilibrium period of distribution functions in Figure 4b. In this condition, comparison of Figure 4e2 and 4e3 demonstrates that after the APDB leaves, the left region has an inverse sublattice arrangement. Also, the phenomenon appears in the third condition. Two APDB successively pass the investigated area during the whole ordering process. Structures presented in Figure 4f1 to Figure 4f4 and the corresponding changes of curves exhibited in Figure 4c explicitly illustrate that APDB movement alters the crystal lattice for two times. In a word, during the moving process once the APDB passes a position, the crystal lattice of this area is influenced to alter once.

Impact of temperature on the LRO
In the case of intermetallic compound alloys, as a fundamental parameter reflecting the materials' properties, the LRO has an important contribution to its characteristics, such as the mechanical properties. Aiming at obtaining a further comprehension of the ordering behaviors under various temperatures, it is necessary to consider the LRO changes under different temperatures and know its variation rule. This information is profound for the design, improvement, doping and control of related materials, which motivates the present interest in the influence of temperature on the LRO of B2-FeAl. Figure 5(a) exhibits the LRO degree variation as a function of different sublattices at different temperatures for the Fe-35 at. % Al alloy. In Figure 5(a) the Fe-and Al-sublattice are particularly differentiated which is very different from the traditional method. To some extent, this distinction can characterize the antisite defects of different sublattices. Graphically, it manifests that both the LRO degrees of Fe-and Al-sublattice decrease with temperature rise. The reason for this variation tendency is that as the temperature increase all atoms simultaneously vibrate more intensively and frequently. So it becomes more difficult to sustain its perfect structure. However, it is found that the Fe-sublattice is consistently in a higher order than Al-sublattice. This is because the formation of Fe-antisite needs much higher energy than the Al-antisite. Thus, the Fe-sublattice retains a better LRO degree than the Al-sublattice, which is consistent with other investigations 1,15,49 . So, the current work can act as a prediction for future researches. Figure 5(b) depicts the starting time of ordering as a function of temperature. For the control of phase transformation, the information about the relationship between time and process is necessary. What showed in Figure 5(b) reveals that the B2 structure domain precipitates earlier at low temperatures than at high temperatures. According to equation (2), compared with high temperature, the reduction of total free energy Δφ, which is the phase change driving force, has a larger value. This means that the ordering transformation is more likely to occur despite the fact that the atoms diffusion are more likely to be restricted duo to the slowed thermodynamic crystal vibration at low temperature. Figure 6 is the comparison between the simulation results and the experiment results. Figure 6a is the theoretical micro APD which is in the same size as Figure 6c. Figure 6c is the atomic micrograph obtained through the HTEM. Here, it should be pointed out that the Figure 6d is the show of displays an atomic observation in nano-sized level.) But it does not hamper us to give a visual comparison between the simulation and experiment. It is proper because both the FeAl and Fe 3 Al compound are both based on the body centered cubic structure. From the comparison, we can find that the simulation results and experiment results are very similar, which proves that the simulation results are true.

Conclusion
Through the comparisons between the computational results and experimental results, it is found that the computational results agree well with the experimental results. In contrast to the previous phase field models based on a function of concentration-field, in the present work results in atomic level resolution are obtained. The present model provides a method for the investigations on the formation process, dynamic evolution routes of APDB in FeAl-based alloy. The theoretical formation process can be used as a supplement and reference for the experimental investigations of APDB in early stage, because the instantaneous observation in experimental solid phase transitions is a hard field. Meanwhile, dynamic evolution routes provide explicit information on the moving direction especially clear atoms configurations of different boundary-types which is one of most interesting topics for material researchers. In this present work, it is found that: (i) the formation process shows that two ways account for the formation of closed APDB; the ordered domain grows through depleting micro APDB. (ii) The in situ research on crystal lattice shows that the movement of APDB thoroughly changes the crystal lattice configurations of the area where it passes by; and the change frequency of crystal lattice is determined by APDB migration times. (iii) The obtained LRO exhibits an inverse relationship with temperature increasing. With increasing temperature, the LRO degrees at Fe-sublattice and Al-sublattice decrease. However, differentiation in LRO exists in Fe-and Al-sublattice, which demonstrates that the Al-sublattice has a major contribution to the low order of FeAl domain.