Dislocation-mediated migration of nterphase boundaries

1. Introduction

Various industrial alloys obtain their final properties via solid-state phase transformations that occur during heat treatment processes. Transformations in many technically important alloys often yield product phases with a lath morphology, usually surrounded by faceted interphase boundaries (IPBs) with irrational orientations (i.e., not low-index crystal planes) in both phases, particularly when the phases have fcc, bcc, or hcp structures in metallic systems [1]. These faceted IPBs often incorporate one or two sets of interfacial dislocations parallel to the long axis of the lath [[2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]] that are also found along irrational orientations (i.e., not low-index crystal directions) in both phases. For an in-depth understanding of the mechanism of the phase transformation, it is crucial to obtain a clear physical picture of the migration of the faceted IPBs. For decades, many theoretical models have been proposed to interpret the observed facets and interfacial structures based on the geometrical matching between two phases at faceted interfaces [8,9,[13], [14], [15], [16], [17]]. However, limited progress has made in the study of the migration of the IPBs together with these interfacial dislocations. Direct observation of the atomic-scale mechanism of IPB migration through high-resolution transmission electron microscope (HRTEM) is a challenging task, because the observation is usually limited by the irrational geometrical characteristics of the interfaces and interfacial dislocations. An alternative method, i.e., molecular dynamics (MD) simulation, offers atomic resolution and was successfully used to study interface migration in both homophase [[18], [19], [20], [21], [22], [23], [24], [25], [26]] and heterophase [[27], [28], [29], [30], [31], [32], [33], [34]] systems. These MD simulations of heterophase systems mainly addressed the interfaces parallel to the close-packed planes or with artificially introduced interfacial defects that were not observed experimentally. Until recently, the migration of IPBs containing two sets of crossed dislocations in an austenite/martensite system was simulated following the HRTEM observations [35]. However, as mentioned above, the faceted interfaces in many precipitation systems usually contain parallel dislocations [[2], [3], [4], [5], [6],36] that were developed in a manner different from that of the martensitic transformation with specific constrain described in the previous simulation [35]. In the present work, MD simulations are employed to investigate the migration mechanism of IPBs containing one or two sets of parallel interfacial dislocations in an fcc/bcc system, with the crystallographic features close to the experimental results.

To reveal the details of IPB migration by MD simulations, appropriate interfacial dislocation structures must be constructed first, and this construction should be guided by the experimental observations. Faceted IPBs usually have various interfacial structures in different alloy systems or under different heat treatment conditions, as characterized by the transmission electron microscope (TEM) experiments in various alloy systems [5,7,9,12,[36], [37], [38]]. There are some common features associated with the IPBs surrounding a lath product phase: (1) reproducible orientation relationships (ORs) between the two phases separated by the faceted IPBs, near (but not exactly defined by) a rational OR (i.e., described as parallelism of low-index crystal directions), e.g. K-S ({11$\bar{1}$}_f∥{01$\bar{1}$}_b, <101>_f∥<111>_b) or N-W ({11$\bar{1}$}_f∥{01$\bar{1}$}_b, <1$\bar{1}$0>_f∥<100>_b) in an alloy system containing fcc and bcc phases; (2) the parallel interfacial dislocations lie along an invariant line [13] that is misfit-free and is usually in an irrational direction in a general system. However, the condition of the existence of an invariant line itself does not fully constrain the OR because there are numerous possible invariant line directions in the unextended cone of a Bain strain [39]. The observed irrational OR can often be explained with the O-line model [15,40], which requires the complete cancellation of the interfacial misfit by a single set of dislocations (see Appendix A for mathematical definition and calculation details). Further criteria incorporating the O-line model to constrain the OR have been established based on a variety of experimental observations. This approach has been verified by observations in the Zr-Nb alloy [38], Ti-Cr alloy [36] and the duplex stainless steel [5], showing consistence in the specific dislocation direction, irrational orientation of the faceted IPBs and associated irrational ORs. The O-line condition for the OR will be adopted for generating hypothesized interfaces containing either a single set or two sets of dislocations in the present simulation.

Closely related to IPB migration, the surface relief effect associated with the growth of some lath shaped precipitates is a long-standing unsettled issue [41]. The phenomenological theory of martensite crystallography (PTMC) [42,43] has proved to be successful in explaining the surface relief effect associated with the martensitic transformation. In this theory, the overall transformation deformation, described by the Bain deformation plus a rigid rotation, is divided into two parts: macroscopic deformation leading to the surface relief effect and lattice invariant deformation of martensite. However, the surface relief effect associated with the growth of the lath-shaped precipitates remains controversial [[44], [45], [46], [47]]. A study of the migration of the IPBs at the atomic scale may shed some light on this controversial issue.

In this work, different migration modes are analyzed at three hierarchical scales: the macroscopic deformation associated with IPB migration, interfacial dislocation motion and the atomic displacement. The simulation results reveal the relationship between the migration behaviours and interfacial structures at the atomic scale, and elucidate the connections between macroscopic deformation, dislocation motion and atomic displacement. The insights obtained from the simulation results are helpful for understanding the growth mechanism of the lath-shaped product phase, together with the surface relief effect associated with the interface migration.

The rest of this work is organized as follows. Section 2 provides the details of the MD simulations, including the geometry of the IPBs used in this work, the validation of the potential function and the simulation conditions. Section 3 provides information related to interface migration, including the calculation of the driving force and migration rate. Detailed simulation results and discussions are presented in Section 4. In this section, the two migration modes observed in the present simulation are described in detail, including the corresponding interfacial dislocation motion and atomic displacements. For IPBs containing one set of dislocations, the macroscopic deformation is calculated and compared with that of the PTMC. All of the results are summarized in Section 5.

2. Simulation details

An fcc/bcc bi-crystal system in pure iron was selected for the present MD simulations, as illustrated in Fig. 1. To mimic the migration of faceted IPBs in a real system, two types of irrational interfaces were selected for the simulations. The first type is IPBs containing a single set of dislocations, which usually act as the habit planes in many alloys [[2], [3], [4],6,34]. They are referred to as “O-line interfaces (OIFs)” because the dislocations must lie alternately with the O-lines that are the periodic linear misfit-free positions in the interface [35,36]. The interfaces of the second type contain two sets of parallel dislocations. These are called “side interfaces (SIFs)” because this kind of dislocation structure is usually present in a side facet of a lath precipitate [2,4,6,9,34]. An IPB containing one or two sets of dislocations is a common feature in any precipitate system consisting of two of the fcc, bcc, or hcp phases. Thus, our results will contribute to the general understanding of the relationship between the interfacial dislocation structure and the kinetic properties of IPBs.

Seven representative interfaces, namely, five OIFs (OIF1-OIF5) and two SIFs (SIF1 and SIF2), were constructed to simulate the details of interface migration. Initially, OIF1-OIF5 have only one set of dislocations with correlated Burgers vectors of the 1/2<101>_f|1/2<111>_btype for OIF1-OIF3 and of the 1/2<1$\bar{1}$0>_f|<100>_btype for OIF4-OIF5, where subscripts ‘f’ and ‘b’ denote the crystal directions or planes in fcc and bcc, respectively. Both of the two SIFs contain two sets of dislocations with the 1/2<101>_f|1/2<111>_btype. The crystallographic features of these IPBs are listed in Table 1. The input for the MD simulations, including the ORs and the interface orientations constrained by the selected dislocation structure and corresponding slip planes [48], were determined based on the O-lattice theory [14] with the O-line criterion [15] (see Appendix A for the calculation details). The resulting ORs for all of the above IPBs were close to the K-S or N-W, which are most frequently observed in fcc/bcc systems. It should be noted that all of the calculated IPBs lie in irrational orientations. In addition, all of the interfacial dislocations in these IPBs are also along irrational crystal directions and are of a mixed nature.

The x direction of the simulation cell is along the dislocation line and the z-axis is along the normal direction of the interface. Periodic boundary conditions were applied in x and y directions in order to minimize the influence of the free surface during interface migration, and therefore, the x and y dimensions should be integer multiples of the shortest crystal vectors along these two directions. However, as observed from the data presented in Table 1, the dislocation line direction defined by an invariant line usually has an irrational index in both crystals, so a small strain (<1%) must be applied on both phases to obtain x and y directions with approximately rational indexes. A detailed description of the determination of the small strain is presented in Appendix A. Here, we define a₁ and a₂ as the smallest lattice vector along the x and y directions after adding the small strain, and then 2a₁ and 2a₂ were selected as the dimensions of the x and y. Since the free surface perpendicular to the z direction has little influence on the interface migration before the interface approaches the surface, the z direction was allowed to relax freely and the dimension along this direction is labelled in Fig. 1. The sizes of simulation cell and the numbers of atoms for each IPB are listed in Appendix A.

A modified Finnis-Sinclair potential [28] was employed to describe the interatomic interactions. In this potential, an extra cutoff function is added in the atomic charge density term of the original Finnis-Sinclair potential [49]. Since a bi-directional phase transformation will be studied in the present work, the transition temperature of the fcc and bcc phase is the most important property. To make the transition temperature close to that in the phase diagram (1185 K), we set the cutoff radius in the extra cutoff function as R_c = 3.03 Å, giving rise to a transition temperature of approximately 1100 K. Appendix B reports the detailed formulation and test of the potential.

The simulation systems were relaxed to obtain an equilibrium initial structure of IPBs. Then, the MD simulations were performed in an NPT ensemble at 500 K for an overall simulation time of 200 ps. The temperature was controlled by a Nosé-Hoover thermostat, while the pressure was held at zero by a barostat. All of the simulations were carried out using the open-source LAMMPS package [50]. In the simulation box, the crystal structures were identified by adaptive common neighbor analysis (a-CNA) [51] using OVITO [52], while the interfacial dislocations were identified by applying a method [53] based on the singular-value decomposition of the Nye tensor [54].

3. Calculation of the driving force and the migration rate

For the NPT ensemble applied in this work, the driving force of the interface migration is the Gibbs free energy difference ΔG(T)between the fcc and bcc phases, which can be calculated by the Gibbs-Helmholtz equation:

(1)

where H_f(T)and H_b(T)are the enthalpies of the fcc and bcc phases respectively as a function of the temperature T, and Tc is the transition temperature of the two phases. H_f(T) and H_b(T) can be obtained by the method used in Ref. [30]. The potential used in the present work has the advantage over most other Fe potentials in that it can give a transition temperature of two solid phases below the melting point. Therefore, it is more convenient to use the transition temperature T_c, instead of the melting temperature T_m used in Ref. [30], as the reference point for thermodynamic integration. Fig. 2 compares the driving forces from the present potential to that from the formula of the Gibbs free energy difference of alpha ferrite and austenite in pure iron [55], which is consistent with the Fe-C phase diagram. The driving force of the present potential is close to that determined according to the phase diagram near the transition temperature T_c, though a larger discrepancy is observed for the temperature far away from T_c.

The average interface migration rate vIPB was extracted from MD simulations by applying the formula [30]

v_IPB=(1/SΩL)(dE/dt) (2)

where S, Ω and L are the interface area, number of atoms per volume and the latent heat per atom that remains constant during the transition from the fcc phase to the bcc phase. dE/dt is the rate of the total potential energy change with respect to the simulation time. This formula shows a linear relationship between vIPB and the rate of potential energy change for a certain interface. Therefore, by acquiring the total potential energy before and after interface migration in MD simulation, we can extract the average interface migration rate via Eq. (2).

4. Results and discussion

4.1. Migration rate

The potential energy vs. time plots for different IPBs are shown in Fig. 3, where SIF2 is absent because it is immobile within the simulation time. The slopes of the curves represent the rates of change of the potential energy that can reflect the interface migration rate and the interface migration mode of each IPB. For all interfaces presented in Fig. 3(a), i.e., OIF1-OIF4, the slope of the interface has a constant value in the full range of the interface migration (the deviation at end of OIF3 is due to approaching the z limit of the simulation box), implying that the interfaces of this type have a constant migration rate and exhibit a uniform migration behavior. By contrast to the interfaces that show uniform migration, both the OIF5 and SIF1 do not have a constant migration rate. The migration rate is relatively high at the beginning and then slows down before it increases again. This process repeats until the interface approaches the free surface, indicating a typical “stick-slip” migration behavior similar to grain boundary migration [20], as shown in Fig. 3(b). The “stick” stage of SIF1 is shorter than that of OIF5 due to different dislocation interactions, as elucidated in the next section. It should be noted that an energy jump is observed for OIF3 and SIF1 when the migrating interface approaches the top free surface. This is due to the attractive interactions between the interfaces and the free surface. The interaction only becomes prominent as the distance between the surface and an interface is close to half of the dislocation spacing, which is the effective influencing distance of the stress field of a semi-coherent interface [56]. Therefore, the energy jump occurs at a distance of approximate 1 nm from the surface, since the spacing of dislocations in either OIF3 or SIF1 is approximately 2 nm.

Table 2 lists the values of the average migration rate vIPB determined from Eq. (2). These values diverge significantly, implying that the migration rate is very sensitive to the interfacial structure. This divergence results from the different modes of dislocation motion as mentioned above and will be discussed in the following sections, together with their influence on the migration rate.

4.2. Migration modes

4.2.1. Uniform migration

The uniform migration of a semi-coherent IPB involves its movement at a constant rate that is observed for OIF1-OIF4. Uniform migration of OIF1-OIF4 is illustrated by the overlapped MD snapshots shown in Fig. 4(a)-(d), where only the atoms near the dislocation cores are shown for clarity. As observed from these figures, migration of each interface for this mode is accompanied by the continuous gliding of the interfacial dislocations in their slip planes. The migration behaviors of OIF1-3 are similar because their correlated Burgers vectors are of the same type, i.e., 1/2<101>_f|1/2<111>_b, though the geometries of their slip planes with respect to the interfaces are different. However, the dislocations in OIF4 spontaneously decompose into two sets immediately after relaxation and remain decomposed during the migration, as shown in Fig. 5(d). The initial Burgers vector is 1/2[1$\bar{1}$0]_f|[100]_b, and the decomposition in each phase can be expressed as:

[100]_b→1/2[111]_b+1/2[1$\bar{1} \bar{1}$]_b (3a)

and

1/2[1$\bar{1}$0]_f→1/2[[101]]_f+1/2[0$\bar{1} \bar{1}$]f (3b)

The two decomposed Burgers vectors still lie in the slip plane for OIF4, so that they can glide synchronously in (11$\bar{1}$)_f|(01$\bar{1}$)_b during interface migration. That is the reason why OIF4 can migrate uniformly though the migration process involves the motion of two sets of interfacial dislocations.

The association of migration of OIF1-OIF4 with dislocation gliding can be used to interpret the variation of their velocities v_IPB listed in Table 2. Dislocation gliding velocity can be decomposed with a component v_dis⊥, which is perpendicular to the interface. The interface also migrates with the speed of v_dis⊥ because the interface moves together with dislocations. Provided that the dislocations in different interfaces migrate at similar velocities, it is found that v_dis⊥ increases with the increased angle between the interface and the slip plane. Consequently, interface migration rate is high when the angle between the interface and the slip plane is larger. This explains why the v_IPB values for OIF1, OIF2, and OIF4 in Table 2 increase with the angle between the interface and slip plane (θ₁). OIF3 is an exceptional case, where migration at the highest rate is observed because of its extended core structure of the dislocations, as shown in Fig. 5. As identified by using the a-CNA method, the core structure of the dislocations in OIF3 was found to contain a narrow stacking fault extending along the slip plane, in contrast to the dislocations in OIF1 and OIF2, where decomposition was not observed. The high interface migration rate indicates that the partial dislocations at the border of the narrow stacking fault glide with a small energy barrier, compared to a full dislocation, making interfacial dislocations have a high v_dis⊥ and hence a high v_IPB. Therefore, the interface migration rate can be accounted by both the geometric factors of the IPB (the angle between the interface and the slip plane) and the core structure of the interfacial dislocations. The geometry of the slip plane with respect to the IPB decides the major limit for the interface migration rate.

Macroscopic shear deformation is detected during the migration of OIF1-OIF4, indicating clearly a shear-coupled migration character. An example of the shear displacement associated with the migration of OIF1 is given in Fig. 6. To directly display the shear effect, atoms that enter the opposite side of the simulation box due to the periodic boundary conditions have been moved to their true positions, as shown in Fig. 6(b). The initial configurations of OIF1-OIF4 were selected intentionally to satisfy the condition of the phenomenological theory of martensite crystallography (PTMC), for convenience of comparison. The magnitude and direction of the macroscopic shears obtained from the MD data were calculated by averaging the displacements in the untransformed zone per unit migration distance. The simulated results are compared with the theoretical results based on the PTMC (details are given in Appendix C) in Table 3. It is found that both the direction and the magnitude of the macroscopic shear deformation in our simulation are in reasonably good agreement with that of PTMC. Note that the shear deformation associated with IPB migration is not a simple shear (no volume change) because there is a volume change due to the phase transformation. This consistency indicates that the macroscopic shear deformation associated with the growth of the lath product phase can possibly yield surface relief.

The displacement $d_i$$d_i$ associated with each atom at the final position of $x_i$$x_i$ clearly differs from the result caused by the homogeneous phase transformation, $T{x}_i$$T{x}_i$, where $T$$T$ is the transformation displacement matrix [14]. The difference vectors, $T{x}_i-d_i$$T{x}_i-d_i$, plotted in Fig. 7(b), lie approximately along the Burgers vector $b_{\text{f}}$$b_{\text{f}}$ of the dislocations, defined on the fcc basis. According to their lengths, the difference vectors can be arranged in zones specified by a zone sequence number $n$$n$. Namely, $n=0$$n=0$ represents the zone containing the origin, while $n<0$$n<0$ and $n>0$$n>0$ representthe zones to the left and right of the $n=0$$n=0$ zone, respectively. The difference vectors in zone n can be approximately described by $n{b}_{\text{f}}$$n{b}_{\text{f}}$, with the zone borders parallel to the slip planes of the dislocations, as observed from Fig. 7(b). Therefore, the real atomic displacement $d_i$$d_i$ can be expressed as:

$d_i=T{x}_i-n{b}_f$$d_i=T{x}_i-n{b}_f$(4)

And it is also shown in Fig. 7(c). Eq. (4) indicates that the collective atomic displacement caused by the interface migration, which contributes to the macroscopic shear deformation, is the remaining part of the displacement generated by the homogenous phase transformation, after the displacements due to dislocation gliding are subtracted.

While in principle, this conclusion should agree with the PTMC, Eq. (4) is not directly comparable with the PTMC. The relationships between them can be elucidated via the extended O-lattice theory [57]. Accordingly, when the interface migrates with the gliding of a set of dislocations, the displacement $T{x}_i$$T{x}_i$ can be decomposed into the sum of the displacement associated with a long-range strain (the first term) and that cancelled out by the interfacial dislocations (the second term), as follows:

Tx_i=(Δg_u'x_i/Δg_u'b_b)Δb+(g_bu'x_i/g_bu'$x^{O}_{\perp}$)b_f (5)

Here, the subscript $\text{'}u\text{'}$$\text{'}u\text{'}$ refers to the unit vector, while $b_{\text{b}}$$b_{\text{b}}$ is the Burgers vector of the interfacial dislocations defined on the bcc base, correlated to $b_{\text{f}}$$b_{\text{f}}$, and $\Delta b=b_{\text{b}}-b_{\text{f}}$$\Delta b=b_{\text{b}}-b_{\text{f}}$. Vectors $g_{}\text{f}\text{'}$$g_{}\text{f}\text{'}$ and $g_{}\text{b}\text{'}$$g_{}\text{b}\text{'}$ are reciprocal vectors of the slip planes of the dislocations in the fcc and bcc phases, respectively (a vector with superscript “ $\text{'}$$\text{'}$” denotes a row vector while that without “ $\text{'}$$\text{'}$” represents a column vector). $\Delta {\text{g}}_u\text{'}$$\Delta {\text{g}}_u\text{'}$ is the unit vector along $g_{}\text{f}\text{'}-g_{}\text{b}\text{'}$$g_{}\text{f}\text{'}-g_{}\text{b}\text{'}$, which must be normal to the interface according to the property of the O-lines [33]. $x_{\perp }^O$$x_{\perp }^O$ is the vector lying in an OIF and perpendicular to the dislocation line, while its length is equal to the dislocation spacing. If we let $x_i$$x_i$ define a point in the migrating interface, then $\Delta g_{\text{u}}\text{'}{x}_i$$\Delta g_{\text{u}}\text{'}{x}_i$ is the migration distance. When the point lies in zone $n$$n$, we obtain $g_{\text{bu}}\text{'}{x}_i/g_{\text{bu}}\text{'}{x}_{\perp }^O=n+k$$g_{\text{bu}}\text{'}{x}_i/g_{\text{bu}}\text{'}{x}_{\perp }^O=n+k$, where $k\in \left(-1/2,1/2\right)$$k\in \left(-1/2,1/2\right)$. By substituting this result and the expression for $\mathbf{T}{x}_i$$\mathbf{T}{x}_i$ from Eq. (5) into Eq. (4), it is found that:

d_i=(Δg_u'x_i/Δg_u'b_b)Δb+kb_f (6)

All of the vectors in Eqs. (5) and (6) and the relationship between them are schematically illustrated in Fig. 7(b) and (c). As the migration proceeds, the value of $\Delta g_{\text{u}}\text{'}{x}_i/\Delta g_{\text{u}}\text{'}{b}_{\text{b}}$$\Delta g_{\text{u}}\text{'}{x}_i/\Delta g_{\text{u}}\text{'}{b}_{\text{b}}$ becomes considerably larger than $k$$k$ because $\Delta g_{\text{u}}\text{'}{x}_i\gg \Delta g_{\text{u}}\text{'}{b}_{\text{b}}$$\Delta g_{\text{u}}\text{'}{x}_i\gg \Delta g_{\text{u}}\text{'}{b}_{\text{b}}$. Consequently, $d_i$$d_i$ will approach $(\Delta g_{\text{u}}\text{'}{x}_i/\Delta g_{\text{u}}\text{'}{b}_{\text{b}})\Delta b$$(\Delta g_{\text{u}}\text{'}{x}_i/\Delta g_{\text{u}}\text{'}{b}_{\text{b}})\Delta b$ for large values of $x_i$$x_i$. This explains the overall trend of the atomic displacements along one direction near the top region in Fig. 7(a). The direction of $\Delta b$$\Delta b$ is consistent with that of the shear direction in the PTMC [57,58], although this was not stated explicitly in the original theory. In addition, the magnitude based on Eq. (6) is also consistent with the PTMC, since the formulation of Eq. (5) based on the O-line is equivalent to that of the PTMC [59]. While only OIF1 was taken as an example to illustrate the atomic displacement in migration in Fig. 7, similar displacement behaviors were observed from all four OIFs that exhibit a uniform migration mode, which is in good agreement with the comparison results listed in Table 3.

4.2.2. Stick-slip migration

In addition to the uniform migration of interfaces, some interfaces show a “stick-slip” migration mode, which refers to the scenario in which an interface migrates by repeating the “stick-slip” cycle [60]. This migration mode occurs in IPBs that involve the gliding of two sets of dislocations in different slip planes, as illustrated by the examples of OIF5 and SIF1 in Fig. 8. In the “slip” stage, dislocations glide smoothly in their slip planes. At a certain migration distance, the dislocations of different sets meet and interact. The interaction clearly retards the migration process, but it does not stop the migration of either OIF5 or SIF1 in the simulation condition, and hence the “slip” stage can resume after an apparent “stick” stage.

Despite the similar stick-slip migration behavior, the migration rate of SIF1 is much higher than that of OIF5 (see Table 2). This difference is due to the difference in the interfacial dislocation structures. The single set of dislocations in OIF5 spontaneously decomposes in the same manner as that for OIF4 (Eq. 3 (a, b)). In contrast to the case of OIF4, in which the dislocations can glide in the same slip plane, the two sets of dislocations in OIF5 have different slip planes, with the slip planes ${(10\overline{1})}_{\text{f}}|{(11\overline{2})}_{\text{b}}$${(10\overline{1})}_{\text{f}}|{(11\overline{2})}_{\text{b}}$ for the $1/2{[101]}_{\text{f}}|1/2{[111]}_{\text{b}}$$1/2{[101]}_{\text{f}}|1/2{[111]}_{\text{b}}$ dislocations and an irrational plane close to ${(0\overline{1}1)}_{\text{f}}|{(1\overline{1}2)}_{\text{b}}$${(0\overline{1}1)}_{\text{f}}|{(1\overline{1}2)}_{\text{b}}$ for the $1/2{[011]}_{\text{f}}|1/2{[\overline{1}11]}_{\text{b}}$$1/2{[011]}_{\text{f}}|1/2{[\overline{1}11]}_{\text{b}}$ dislocations. The overall migration process of OIF5 consists of the repeated decomposition and recombination of the dislocations, as shown in Fig. 8(a) and (b). To the best of the authors’ knowledge, this type of migration behavior of a semi-coherent interphase boundary has not been reported previously, though the recombination of zonal dislocations was observed during twin boundary migration [61]. However, these dislocation reactions are not observed for SIF1. The stick-slip behavior of SIF1 stems only from the tangling of dislocations of different sets, as shown in Fig. 8(c). The dislocation reaction process appears to require much longer time than dislocation tangling, such that SIF1 migrates faster than OIF5. In addition, following the same rule as the uniform migration mode, during the “slip” stage, SIF1 also moves faster than OIF5, due to the larger angles between the interface and slip planes for both sets of dislocations (by the comparison of Fig. 8(a) and (c)). It is noteworthy that the slip planes for the dislocation gliding can be high index, as seen in Fig. 8(c), where the Burgers vector of the major dislocations is $1/2{[011]}_{\text{f}}|1/2{[\overline{1}11]}_{\text{b}}$$1/2{[011]}_{\text{f}}|1/2{[\overline{1}11]}_{\text{b}}$ and that of the minor dislocations is $1/2{[10\overline{1}]}_{\text{f}}|1/2{[11\overline{1}]}_{\text{b}}$$1/2{[10\overline{1}]}_{\text{f}}|1/2{[11\overline{1}]}_{\text{b}}$. Their correlated slip planes are ${(12\overline{2})}_{\text{f}}|{(\overline{1}3\overline{2})}_{\text{b}}$${(12\overline{2})}_{\text{f}}|{(\overline{1}3\overline{2})}_{\text{b}}$ and ${(\overline{1}7\overline{1})}_{\text{f}}|{(\overline{4}3\overline{1})}_{\text{b}}$${(\overline{1}7\overline{1})}_{\text{f}}|{(\overline{4}3\overline{1})}_{\text{b}}$, respectively. This implies that the condition of a low index plane being the dislocation slip plane, as commonly assumed in the PTMC and used for OIF in Table 1, is not a necessary condition for a glissile semi-coherent interface.

The interfaces may not remain flat during the migration because the difference in $v_{\text{dis}\perp }$$v_{\text{dis}\perp }$ leads to varying migration rates at different locations. For SIF1, there is a large difference in $v_{\text{dis}\perp }$$v_{\text{dis}\perp }$ of the two sets of dislocations most likely due to the difference in their Peierls-Nabarro stress. Specifically, the slip planes of the minor dislocations ( ${(\overline{1}7\overline{1})}_{\text{f}}|{(\overline{4}3\overline{1})}_{\text{b}}$${(\overline{1}7\overline{1})}_{\text{f}}|{(\overline{4}3\overline{1})}_{\text{b}}$) show greater deviation (43.3°) from the nearest low index slip planes ( ${\{11\overline{1}\}}_{\text{f}}|{\{01\overline{1}\}}_{\text{b}}$${\{11\overline{1}\}}_{\text{f}}|{\{01\overline{1}\}}_{\text{b}}$) than the slip planes of the major dislocations ( ${(12\overline{2})}_{\text{f}}|{(\overline{1}3\overline{2})}_{\text{b}}$${(12\overline{2})}_{\text{f}}|{(\overline{1}3\overline{2})}_{\text{b}}$) (15.8°). Therefore, the minor dislocations may experience a high Peierls-Nabarro stress, which can significantly reduce the gliding velocity. As a result, the minor dislocations lag behind the major ones, even though the angle between the interface and their slip planes is large. The unevenness is enhanced by the dislocation tangling, leading to a wavy growth front, as shown in Fig. 8(d). In contrast, OIF5 remains flat during interface migration, since the slip planes of the two sets of dislocations after the decomposition have similar crystal indexes and similar angles with respect to the interface, yielding similar glide velocities.

The atomic displacements during interface migration become more complicated when the dislocations glide in different slip planes. In these cases, the expression for atomic displacements can be generalized from Eq. (4) as

d_i=Tx_i- $\sum\limits_{j}$ n_jb_fj (7)

where $n_j$$n_j$ denotes the zones separated by the jth set of dislocations for which the Burgers vector is $b_{fj}$$b_{fj}$. Fig. 9(a) shows a typical schematic plot of atomic displacements during the SIF1 interface migration based on Eq. (7). Fig. 9(b) gives the MD simulation result of the distribution of $T{x}_i-d_i$$T{x}_i-d_i$ corresponding to the migration of SIF1. These two figures match each other well, validating the simple expression in Eq. (7).

While all IPBs discussed above are mobile, SIF2 does not migrate within the simulation time. This is because the inclination angle of the slip planes of one set of dislocations is very small with respect to the interface, though the slip planes of the other set is nearly perpendicular to the interface, as shown in Table 2. The $v_{\text{dis}\perp }$$v_{\text{dis}\perp }$ for the former set of dislocations is so small that it severely delays the overall interface migration, leading to the immobile nature of SIF2.

A comparison between the present and previous work shows that the atomic mechanism of IPB migration may be substantially different when the IPBs are associated with different interfacial structures. The present simulation concerns about the migration of IPBs with parallel interfacial dislocations. All these IPBs migrate associated with dislocation gliding, i.e., these IPBs exhibits a dislocation-mediated IPB migration behavior. In contrast, the previous simulations mainly focus on the migration of IPBs with crossed interfacial dislocations [28,29,33] or artificially introduced disconnections [30,31,62]. When the IPB with N-W or K-S OR is parallel to the close-packed plane, the interface migration will start from the low-energy region in the interface and develops into the high-energy areas [29,33]; when the IPB contains disconnections, the interface migrates faster near the disconnections, while at other sites the interface migration proceeds by the nucleation and lateral growth of the secondary disconnections [31,62]. Correspondingly, the velocity of IPB migration may vary associated with different atomic mechanisms. For example, Tateyama et al. [28] found that the velocity of the parallel close-packed plane with an N-W OR is 15-25 m/s, lower than most of the IPBs studied in the present simulation. Other researchers have also reported the velocity range for different IPBs, e.g., 200-700 m/s from Bos et al. [29], 0.7-3.4 m/s from Song et al. [30], and 1.19-4.67 m/s from Tripathi et al. [62]. However, these results come from simulations with different potentials from the present one. Therefore, it is not evidential to draw a conclusion based on the comparison among them. It is also noteworthy that the interfacial structure not only affects the atomic mechanism of interface migration, but may also lead to different mechanical behaviors. For example, interface with steps will enhance the shear resistance of the interface [63], while serrated interfaces contribute to both higher barriers for dislocation nucleation and higher resistances to interfacial shear [64].

Based on the present simulation of IPB migration, we found that the movable interfacial dislocations in any IPB always prefer pure slip motion, even in high-index slip planes. This indicates that, under the present simulation conditions, dislocation climbing does not occur due to the involvement of atomic diffusion, which is limited within the present simulation duration. Therefore, the present results can be used for understanding the phase transformation with the migration of semi-coherent interfaces in cooperation with dislocation gliding. Apparently, these interfaces are glissile. While the presence of glissile interfaces is a necessary condition for athermal growth, as is found in a martensitic transformation involving cooperative atomic motion [39], slip of dislocations may also occurs when the interface migration is thermally activated. Though diffusion is limited, the present simulation permits thermal-activated motion of individual atoms, and their mobility was found temperature dependent (details to be published elsewhere), similar to a massive transformation or grain growth process. The macroscopic shear generated during the migration of an O-line habit plane may give rise to a surface relief effect. The surface relief effect associated with diffusional phase transformation has been a long-standing issue [41]. Admittedly, in a diffusional phase transformation, dislocations motion should not be limited to pure slip, since dislocation climb is facilitated by atomic diffusions. It is still possible to quantitatively determine the amount of residual macroscopic displacement according to the actual path (p2) of the dislocations, as described in Appendix C. If dislocation gliding remains the dominant event during growth of a precipitate, then a surface relief effect may be associated with the precipitate. In this sense, the present simulation result provides a useful insight into the long-standing issue of surface relief associated with lath-shaped precipitates for the habit plane containing a set of glissile dislocations.

5. Conclusions

MD simulations were employed to investigate the migration behaviors of irrational IPBs containing parallel dislocations of one set (OIFs) and two sets (SIFs), including macroscopic shear deformation, dislocation motion and atom displacements associated with the migration of these IPBs. The geometries of the semi-coherent interfaces were constructed according to the experimental observations of typical lath-shaped precipitates. The results provide the links between the kinetic properties of IPBs and the interfacial structures, which will be useful for understanding the lath morphology of the product phase and the associated surface relief effect. The major conclusions of this study are summarized as follows.

(1) The migration of the IPBs is accompanied with the slip of interfacial dislocations, even in high-index slip planes. Two interface migration modes were observed in the present simulation: uniform migration of the interfaces occurs when one or two sets of dislocations glide in common slip planes, while stick-slip migration occurs when two sets of dislocations glide in different slip planes. The simulation reveals a new type of mode, i.e., the decomposition-gliding-recombining dislocation motion mode.

(2) The migration rate of an interface is controlled by the motion of interfacial dislocations. The migration rate is high when the dislocation cores extend in their slip planes and the inclination angle of the slip plane with respect to the interface is large. Reactions (decomposition, combination or tangling) between two sets of dislocations will substantially retard the interface migration. An interface can be immobile when the slip planes of the interfacial dislocations are nearly parallel to the interface.

(3) Uniform motion of all OIFs exhibit a shear-coupled migration feature. The macroscopic shear displacement is in a good agreement with the result obtained from PTMC theory. The macroscopic shear is caused by the collective atomic displacements during the migration of these OIFs. An analytical formula was established to calculate the collective displacements based on the dislocation structure, slip plane, and interface normal. The connection between the PTMC and collective atomic displacements was elucidated by applying the decomposition of the extended O-lattice theory.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Nos. 51471097 and 51671111) and the National Key Research and Development Program of China (No. 2016YFB0701304).The authors thank Andy Godfrey for fruitful discussions. We are grateful to Jian Wang for detailed discussions on simulation techniques.

Appendix A. Interface geometry and construction of the simulation cells

The interface geometry in the present study was constructed with the O-line condition, as explained in the introduction. The O-lines, defined in the O-lattice theory [14], are the periodically distributed linear misfit-free positions in the interface. To allow the existence of the O-line, the OR must be special so that the phase transformation strain is characterized with an invariant line, which defines the direction of the dislocations in the habit plane, and an invariant line in reciprocal space, which must be perpendicular to the Burgers vector of interfacial dislocations [15]. With these conditions and known lattice parameters of two phases, the invariant line in reciprocal space can be solved following a method suggested in [5]. To constrain the remaining degree of freedom in the OR, i.e., the rotation around the invariant line in reciprocal space, we assume that the interfacial dislocations lie in low-index crystal planes. This constraint overlaps partially with the hypothesis in the phenomenological theory of martensite crystallography (PTMC), but it differs from the PTMC in two aspects: 1. The slip plane may not be a typical slip plane in either phase. 2. While the initial slip plane always contains the invariant line, the Burgers vector of interfacial dislocations in an interface may not lie in the selected initial slip plane. The final slip plane of a set of dislocations in our simulation is determined by the cross product of the dislocation line direction and the Burgers vector. Therefore, in the simulation results, some dislocations may not glide in the initial slip planes. Combining the constraint of selected Burgers vectors (b) and slip planes (p), we can solve for the various orientations of the interfaces containing specific dislocations structures together with the OR, using the following two groups of equations:

(A-1)

and

(A-2)

where ${\mathbf{x}}_{\text{in}}$${\mathbf{x}}_{\text{in}}$ and ${\mathbf{x}}_{\text{in}}^{*}$${\mathbf{x}}_{\text{in}}^{*}$ are the unit vectors along the invariant line in real and reciprocal space, respectively, ‘ $\text{'}$$\text{'}$’ is the transpose symbol of a vector or matrix. $\mathbf{B}=[\sqrt{2}\lambda 000\sqrt{2}\lambda 000\lambda ]$$\mathbf{B}=[\sqrt{2}\lambda 000\sqrt{2}\lambda 000\lambda ]$ ( $\lambda =a_b/a_f$$\lambda =a_b/a_f$ is the lattice parameter ratio of the two phases) is the Bain deformation matrix. For known values of $\mathbf{B}$$\mathbf{B}$, $\mathbf{p}$$\mathbf{p}$ and $\mathbf{b}$$\mathbf{b}$, ${\mathbf{x}}_{\text{in}}$${\mathbf{x}}_{\text{in}}$ and ${\mathbf{x}}_{\text{in}}^{*}$${\mathbf{x}}_{\text{in}}^{*}$ can be solved using the above groups of equations. The rotation between two phases can be determined by [5].

where ${\mathbf{x}}_{\text{n}}$${\mathbf{x}}_{\text{n}}$ is the unit vector along the cross product of ${\mathbf{x}}_{\text{in}}$${\mathbf{x}}_{\text{in}}$ and ${\mathbf{x}}_{\text{in}}^{*}$${\mathbf{x}}_{\text{in}}^{*}$, while ${\mathbf{x}}_{\text{np}}$${\mathbf{x}}_{\text{np}}$ is the unit vector along the cross product of $\mathbf{B}{\mathbf{x}}_{\text{in}}$$\mathbf{B}{\mathbf{x}}_{\text{in}}$ and ${(\mathbf{B}\text{'})}^{-1}{\mathbf{x}}_{\text{in}}^{*}$${(\mathbf{B}\text{'})}^{-1}{\mathbf{x}}_{\text{in}}^{*}$. Therefore, the transformation matrix $\mathbf{A}$$\mathbf{A}$, which connects the corresponding lattice vectors of two phases ${\mathbf{x}}_{\beta }=A{\mathbf{x}}_{\alpha }$${\mathbf{x}}_{\beta }=A{\mathbf{x}}_{\alpha }$, can be determined by

A=RB (A-4)

Correspondingly, the vectors in reciprocal space are related by ${\mathbf{g}}_{\beta }\text{'}={\mathbf{g}}_{\alpha }\text{'}{A}^{-1}$${\mathbf{g}}_{\beta }\text{'}={\mathbf{g}}_{\alpha }\text{'}{A}^{-1}$ (By default, vectors in the direct space are represented by column vectors, while the vectors in the reciprocal space are row vectors).

The next step is to determine the orientations of possibly faceted interfaces. These can be obtained using the displacement vectors in reciprocal space [40]

${\Delta \mathbf{g}}^{\prime }={{{\mathbf{g}}_{\alpha }}^{\prime }}_{}-{{{\mathbf{g}}_{\beta }}^{\prime }}_{}={{{\mathbf{g}}_{\alpha }}^{\prime }}_{}-{{\mathbf{g}}_{\alpha }}^{\prime }{A}^{-1}$${\Delta \mathbf{g}}^{\prime }={{{\mathbf{g}}_{\alpha }}^{\prime }}_{}-{{{\mathbf{g}}_{\beta }}^{\prime }}_{}={{{\mathbf{g}}_{\alpha }}^{\prime }}_{}-{{\mathbf{g}}_{\alpha }}^{\prime }{A}^{-1}$ (A-5)

In this work, we only construct the interfaces corresponding to $\Delta {\mathbf{g}}_{\{111\}}\text{'}$$\Delta {\mathbf{g}}_{\{111\}}\text{'}$ because they are frequently observed in experiments [3,5]. Based on the extended O-lattice theory, an interface must be normal to the $\Delta \mathbf{g}\text{'}$$\Delta \mathbf{g}\text{'}$ with the corresponding ${\mathbf{g}}_{\alpha }\text{'}$${\mathbf{g}}_{\alpha }\text{'}$ in fcc satisfying ${\mathbf{g}}_{\alpha }\cdot {\mathbf{b}}_i=0$${\mathbf{g}}_{\alpha }\cdot {\mathbf{b}}_i=0$, where b_i is the Burgers vector (defined in fcc) of a set of dislocations in the interface [65]. Interface orientation of an O-line interface (OIF) is normal to any $\Delta \mathbf{g}\text{'}$$\Delta \mathbf{g}\text{'}$ with the corresponding ${\mathbf{g}}_{\alpha }\text{'}$${\mathbf{g}}_{\alpha }\text{'}$ satisfying ${\mathbf{g}}_{\alpha }\cdot \mathbf{b}=0$${\mathbf{g}}_{\alpha }\cdot \mathbf{b}=0$, since a single set of dislocations exists in the interface [15]. For the side interfaces that each contains two sets of dislocations, we use $\Delta {\mathbf{g}}_{(1\overline{1}1)}\text{'}$$\Delta {\mathbf{g}}_{(1\overline{1}1)}\text{'}$ and $\Delta {\mathbf{g}}_{(\overline{1}11)}\text{'}$$\Delta {\mathbf{g}}_{(\overline{1}11)}\text{'}$ as the interface normal for SIF1 and SIF2, respectively.

In the condition that the interfacial dislocations should fully accommodate the corresponding interfacial misfit, the dislocation structure can be calculated as a function of the lattice parameters, the orientation relationship (OR) between two phases and the interface orientation for semi-coherent interfaces between the simple fcc and bcc phases. With the lattice parameters from the simulation system as input, the geometry of the interfaces (including the OR and the orientation of the interfaces described in the crystal basis of either phase) was determined based on the O-line condition [15] with selected Burgers vectors and slip planes. The detailed calculation procedures are introduced as follows: (1) Dislocation line directions. According to the O-line condition [15], the dislocation line directions of all OIFs and SIFs are along the invariant line direction ${\mathbf{x}}_{\text{in}}$${\mathbf{x}}_{\text{in}}$, which can be calculated from (A-1); (2) Burgers vectors of dislocations. For OIFs, the Burgers vectors of the interfacial dislocations are just the Burgers vector we set in (A-2); For SIFs, the Burgers vectors of both sets of dislocations must be lie in ${\mathbf{g}}_{\alpha }\text{'}$${\mathbf{g}}_{\alpha }\text{'}$ that is associated with the $\Delta \mathbf{g}\text{'}$$\Delta \mathbf{g}\text{'}$ of the SIF [15]. There are six <101>_f Burgers vectors in each {111}_f plane. The two vectors that will be the real Burgers vectors of dislocations in SIF can be determined from the decomposition of the displacement field [66]. For detailed calculation procedures, the readers can refer to the original paper [66]. Here we only state the conclusion: suppose that ${\mathbf{x}}_{\perp }$${\mathbf{x}}_{\perp }$ is the unit vector lying in SIF and perpendicular to ${\mathbf{x}}_{\text{in}}$${\mathbf{x}}_{\text{in}}$; then the two Burgers vectors that have the smallest and second smallest angle with $\mathbf{T}{\mathbf{x}}_{\perp }$$\mathbf{T}{\mathbf{x}}_{\perp }$ should be the required ones, with the former as the Burgers vectors of the fine dislocations, and the latter as the Burgers vectors of the coarse dislocations. These two sets of dislocations are the most effective ones for accommodating the misfit in the SIF; (3) Dislocation spacing. For OIFs, the dislocation spacing can be calculated by [15]:

where b is the Burgers vector of the interfacial dislocation. For SIFs, the dislocation spacings of the two sets of dislocations can be given by the decomposition of the displacement field [66]:

and

where n is the unit normal of the OIF that has the same OR with the SIF, and b₁₂ and b₂₁ are effective Burgers vectors defined by:

and

where b₁ and b₂ are the Burgers vectors of the two sets of dislocations in the SIF.

These results provide the input for the interface geometry in a simulation cell, for the relaxed interfaces to contain the parallel dislocations of particular Burgers vector(s) and selected slip planes. Following the above procedures, we can calculate the crystallographic features of all IPBs constructed in the present work, as listed in Table 1. The calculated features of all IPBs are consistent with the simulated configurations after relaxation. Among the seven IPBs examined in this work, OIF1, OIF2, OIF3 and OIF5 are consistent with experimental results [2,3,5,6,9], while OIF4 is a potential preferred interface according to the O-line criterion. SIF1 or SIF2 are potential facets coexistenting with an OIF that determines the OR. Note that the same OR for the OIF1 and SIF1 and for OIF3 and SIF2.

Since the orientations of all IPBs in the present study have irrational crystal indexes with respect to both phases, the structures of these irrational interfaces are not truly periodic at the atomic scale. To meet the periodic conditions along the x- and y-directions in the simulation, an adjustment must be made on the dimension of x- and y-directions. The adjustment should be sufficiently small, otherwise the strain associated with the adjustment may change the interfacial structure [67]. To minimize the strain due to the adjustment, we have carefully selected the dimensions of the MD simulation box in the x- and y-directions as follows. Let the endpoints of the simulation box along x- and y-axis be defined by vectors ${\mathbf{l}}_{\mathbf{x}}$${\mathbf{l}}_{\mathbf{x}}$ and ${\mathbf{l}}_y$${\mathbf{l}}_y$, respectively. We require that an endpoint should be near an atom pair from the different phases in a condition that the distance between the pair of atoms reaches a local minimum (they usually meet the condition that the distance is smaller than 0.2 $\left|\mathbf{b}\right|$$\left|\mathbf{b}\right|$). Then, small strains were added on both phases such that the two atoms in a selected atom pair coincide with each other. The vectors ${\mathbf{l}}_{\mathbf{x}}$${\mathbf{l}}_{\mathbf{x}}$ and ${\mathbf{l}}_y$${\mathbf{l}}_y$ are defined by such coincidence points, and they can be described by rational crystal indexes. Because the misfit is zero at the O-line according to the O-lattice theory, an atom pair with local minimum distance is always in the closest vicinity of an O-line. After the small strain is added, the endpoint is located exactly at a shifted O-line. Therefore, | ${\mathbf{l}}_y$${\mathbf{l}}_y$ | is an integer multiple of the spacing of the modified O-line (or equivalently, dislocation spacing). Since an appropriate number of dislocations are required in a simulation cell, this sets a range for the length of | ${\mathbf{l}}_y$${\mathbf{l}}_y$ | and | ${\mathbf{l}}_y$${\mathbf{l}}_y$ | is always much larger than 0.2 $\left|\mathbf{b}\right|$$\left|\mathbf{b}\right|$. Similarly, we can also set | ${\mathbf{l}}_{\mathbf{x}}$${\mathbf{l}}_{\mathbf{x}}$ |>>0.2 $\left|\mathbf{b}\right|$$\left|\mathbf{b}\right|$. Under these conditions and the condition for the endpoints associated with local minimum distance between a pair of atoms from different phases, there are still multiple choices for ${\mathbf{l}}_{\mathbf{x}}$${\mathbf{l}}_{\mathbf{x}}$ and ${\mathbf{l}}_y$${\mathbf{l}}_y$. We selected the one with the smallest norm of the corresponding strain matrix, ensuring that the principal strains for all of the IPBs in the present study are smaller than 1%. By setting the origin in the center of the x-y plane in the simulation box, the dimension of the simulation cell along x and y directions are $2a_1$$2a_1$ and $2a_2$$2a_2$, respectively, where $a_1=\left|{\mathbf{l}}_{\mathbf{x}}\right|$$a_1=\left|{\mathbf{l}}_{\mathbf{x}}\right|$ and $a_2=\left|{\mathbf{l}}_{\mathbf{y}}\right|$$a_2=\left|{\mathbf{l}}_{\mathbf{y}}\right|$, with their values varying between different cases. The small imposed strains have a minor influence on the interfacial dislocation structure, since dislocation spacing of the simulated interfaces after relaxation is consistent with that of the calculated results in Table 1.” Since the free surface perpendicular to the z direction has little influence on interface migration prior to the interface approaching the surface, the z direction was set free and the dimension along this direction is labelled in Fig. 1. The sizes of the simulation cell and the numbers of atoms for each O-line interface are listed in Table A1.

The simulation box was filled with fcc and bcc atoms of pure iron according to the orientation relationship, interface orientation and the dimension of the simulation box, with the lattice parameters under the present potential $a_f=3.694$$a_f=3.694$ Å and $a_b=2.867$$a_b=2.867$ Å for fcc and bcc phases, respectively. Then, the atomic configurations near the interface were optimized using the method in [68]: vacancies near the interface were deleted by extending the fcc phase by 3Å to the bcc phase, which will produce many interstitial atoms. Then these interstitial atoms were removed when the intruding fcc atoms enter the Wigner-Seitz cell of the bcc atoms. After these two steps, an interface with no vacancies and interstitials is obtained.

Appendix B. Determination of Fe potential for the present simulation

Embedded atom method (EAM) potential has been widely employed in molecular dynamics (MD) simulation of solid crystalline materials [69]. However, most of the EAM potentials for Fe have a deficiency that the Gibbs free energy of bcc Fe is always lower than that of fcc Fe in the whole temperature range from 0K up to the melting point, which does not agree with the real situation. Furthermore, some of these potentials will give rise to the homogenous nucleation of bcc phase [[70], [71], [72]], disturbing the heterogeneous migration of the existing interphase boundary. To avoid these drawbacks, we selected a modified Finnis-Sinclair (F-S) potential [28] for calculating the energies and forces in the present MD simulation. The original Finnis-Sinclair potential [49] was modified by adding a cutoff function at the atomic charge density term $\varphi (r_{ij})$$\varphi (r_{ij})$, where $r_{ij}$$r_{ij}$ is the bond length between atom i and j. The cutoff function is expressed as:

where $R_c$$R_c$ is the cutoff distance and $\lambda$$\lambda$ is the half width of the cutoff region. As an adjustable parameter, $R_c$$R_c$ affects the relative stability between the bcc and fcc phases at high temperatures (as explained in the original reference [28]). The value of $R_c$$R_c$ was set 3.03Å to render a fcc/bcc transition temperature at around 1100K, close to the experimental value (1185K). Fig. B1 shows snapshots of the uniform migration of OIF1. At 900K, the interface migrates from bcc up towards fcc, while at 1300K the interface migrates in the opposite direction. The interface does not migrate when the temperature is 1100K, indicating the transition temperature is around 1100K. Fig. B1 also demonstrates that there is no homogenous nucleation event in a wide temperature range, enabling us to focus on the study of migration of the original IPBs. Validity of the selected potential was tested by calculating the lattice parameters at 0K, which are $a_f=3.694$$a_f=3.694$ Å and $a_b=2.867$$a_b=2.867$ Å for fcc and bcc phases, respectively, in reasonable agreement with results using other Fe potentials [71-73].

Although other available potentials may give a more accurate description of the dislocation core structure [71,74], we have to make a compromise for studying the migration of single interfaces, i.e., to avoid hinder of homogenous nucleation to the migration of the interface, as mentioned above. To check the influence of dislocation core structure on the interphase boundary migration, an MEAM potential [75], which can predict reasonable core structures for the fcc screw dislocations and the bcc edge dislocations and has been used to study the austenite/martensite interface migration [35], is applied to repeat the same simulations. It was found that the migration mode and the relative velocity of different interfaces are similar to what we obtained in our simulation, though the absolute value of the interface velocity may change. Therefore, we think that the influence of interface geometry on interface migration probably overwhelms the influence of dislocation core structure.

Appendix C. The calculation of macroscopic shear using the phenomenological theory of martensite crystallography (PTMC)

The phenomenological theory of martensite crystallography (PTMC) has been proved to be successful in interpreting crystallographic features as well as the macroscopic deformation associated with formation of plate martensite [42,43,76]. There are two hypotheses in this theory: 1. Habit plane of martensite is a macroscopic invariant plane, which is based on the experimental observations; 2. Interface is glissile, consistent with the high speed nature of martensitic phase transformations. Based on these two hypotheses, the overall transformation A can be decomposed into two invariant plane strain deformations:

A=RB=P₁P₂ (C-1)

where $\mathbf{B}$$\mathbf{B}$ is the Bain deformation, $\mathbf{R}$$\mathbf{R}$ is a rotation matrix, ${\mathbf{P}}_1$${\mathbf{P}}_1$ is the shear deformation and ${\mathbf{P}}_2$${\mathbf{P}}_2$ is a lattice invariant shear deformation. This implies the overall transformation deformation can be partially cancelled by the dislocations in the habit plane, and the remaining deformation gives rise to the macroscopic deformation. Mathematically, ${\mathbf{P}}_1$${\mathbf{P}}_1$ and ${\mathbf{P}}_2$${\mathbf{P}}_2$ have the same form of expression, as below

P₁=I+m₁d₁p₁' (C-2a)

P₂=I+m₂d₂p₂' (C-2b)

Here $\mathbf{I}$$\mathbf{I}$ is identity matrix, ${\mathbf{d}}_i$${\mathbf{d}}_i$ (i=1 or 2) is a unit vector to define the shear direction, ${\mathbf{p}}_i\text{'}$${\mathbf{p}}_i\text{'}$ is a unit vector to define the invariant plane normal, and $m_i$$m_i$ represents the magnitude of shear deformation.

The input parameters of the PTMC are the Bain deformation $\mathbf{B}$$\mathbf{B}$ and the slip system ( ${\mathbf{d}}_2$${\mathbf{d}}_2$, ${\mathbf{p}}_2\text{'}$${\mathbf{p}}_2\text{'}$). ${\mathbf{d}}_2$${\mathbf{d}}_2$ is parallel to the selected Burgers vector, and hence ${\mathbf{d}}_2$${\mathbf{d}}_2$ = b/|b|. Similarly ${\mathbf{p}}_2$${\mathbf{p}}_2$ is normal to the selected slip plane, so that ${\mathbf{p}}_2$${\mathbf{p}}_2$ = p/|p|. The A determined with the PTMC is the same as the one determined in Appendix A, since they are constrained by the same conditions of ${\mathbf{p}}_2\text{'}{\mathbf{x}}_{in}=0$${\mathbf{p}}_2\text{'}{\mathbf{x}}_{in}=0$ and ${\mathbf{d}}_2\text{'}{\mathbf{x}}_{in}^{*}=0$${\mathbf{d}}_2\text{'}{\mathbf{x}}_{in}^{*}=0$. Direction ${\mathbf{d}}_1$${\mathbf{d}}_1$, habit plane normal, and magnitude $m_1$$m_1$ of shear for a selected slip system is determined by [42,43,76]:

Habit plane:p₁=(p₂′A-p₂′)/|p₂′A-p₂′| (C-3)

Macroscopic shear direction: d₁=(Ad₂-d₂)/|Ad₂-d₂| (C-4)

Macroscopic shear magnitude: m₁=| (Ad₂-d₂)/p₁'d₂ | (C-5)

This is how the results of the PTMC in Table 3 have been determined. Note that the directions of the habit plane normal determined with Eq. (A-5) and (C-3) are actually parallel to each other.

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