A molecular dynamics study on formation of the self-accommodation microstructure during phase transformation

1. Introduction

Crystallographic textures of the product phase developed from phase transformation have a strong influence on the mechanical properties of the alloys [[1], [2], [3], [4]]. A typical texture, i.e., a self-accommodation microstructure, is characterized by clustering particular variants of the product phase to accommodate the total transformation strain in a cooperative fashion. This kind of texture is frequently observed in industrial alloys, such as shape memory alloys [5,6], titanium alloys [7,8] and steels [[9], [10], [11], [12]]. An understanding of the formation and evolution of these textures is essential to a knowledge-based control of the microstructures in alloys. For many years, fruitful models have been proposed to interpret the formation of the self-accommodation microstructure [5,7,9,[13], [14], [15], [16]]. In these models, different criteria were used to evaluate the preference of a particular variant, including the strain energy determined by Eshelby inclusion method [9], the degree of incompatibility [5] and the interaction energy between variants [15,16]. In essence, these criteria are physically equivalent since they have a common basis that the clustering (or pairing) of the variants should lead to the minimization of the strain energy of the whole system. Interfacial energy has been proved to be another factor influencing the variant selection [10,17,18]. However, to authors’ knowledge, no quantitative model has been developed, which takes both the strain energy and the interfacial energy into consideration to understand the formation of the specific variants. Besides, the formation of the self-accommodation microstructures at the atomic scale remains unclear. Therefore, a clear atomistic view of how the variants evolve step by step is needed for an in-depth understanding of the self-accommodation effect.

In this work, the austenite (FCC) to ferrite/martensite (BCC) phase transformation is selected as an example to reveal the atomistic formation of the self-accommodation microstructure, since it was widely present in steels as bainitic or martensitic transformations. In an FCC/BCC system, K-S ((11$\bar{1}$)_f$\lvert$|(01$\bar{1}$)_b,[101]_f$\lvert$|[111]_b) [19] and N-W ((11$\bar{1}$)_f$\lvert$|(01$\bar{1}$)_b,[101]_f$\lvert$|[111]_b) [20,21] orientation relationship (OR) (or close to them) are the most frequently observed ORs, both of which are near the Bain ORs [22]. It is widely known that the Bain, N-W and K-S ORs have three, twelve and twenty-four variants, respectively. Near each Bain variant, there are four N-W variants and eight K-S variants. The combination of these ORs is called a “Bain zone” or “Bain circle” [10] (abbreviated as “BC” in the following text), since the four N-W variants and eight K-S variants are distributed in a circle in the pole figure, as shown in Fig. 1. The specific descriptions of these ORs are listed in Table 1. Among all these ORs, KS1-KS8 and NW1-NW4 are in BC 1, KS9-KS16 and NW5-NW8 in BC 2, while KS17-KS24 and NW9-NW12 in BC 3. Due to measurement uncertainty, the ORs are mainly classified as KS and NW, by allowing small angular deviations. Since variants of both K-S and N-W are taken into consideration, the terminology ‘OR selection’ is used to describe the selective formation of particular K-S or N-W variants, rather than ‘variant selection’ used in previous works [10,15,23,24]. For simplicity and clarity, the ferrite grain having a particular OR with the matrix is denoted by the name of the OR in the following text; e.g., KS1 represents the ferrite grain which has a KS1 OR with the matrix.

Molecular dynamics (MD) simulation is employed in this work to reveal the atomistic picture of the evolution process during the austenite to ferrite phase transformation in the pure iron system. The simulation starts from a single ferrite grain enclosed by the austenite matrix. Before a new ferrite grain forms, the growth of the initial ferrite is tracked and the evolution of the OR will be analyzed. After new grains appear, the formation sequence is carefully investigated and the growing and shrinking grains with various ORs are shown. The influence of both elastic interaction energy and interfacial energy on the formation of the self-accommodation microstructure is investigated.

2. Simulation details

The initial configuration for the MD simulation consists of a three-dimensional (3D) ferrite grain surrounded by the austenite matrix. To mimic a realistic growth of the second phase, the shape and orientation of the original ferrite grain should be close to those observed in experiments. However, previous experimental data mainly focused on the crystallography of the faceted interfaces [[25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]], while a 3D geometry of the second phase is unavailable, especially when the new phase has a small size. Therefore, in the present simulation we adopt an initial configuration from a previous simulation work, which applies the discrete atom method [38,39] to study the evolution of precipitate crystallography in a Cu-Cr system [40], as shown in Fig. 2(a). This configuration is composed of a near-ellipsoid ferrite grain enclosed by the austenite matrix and was evolved from a small-sized embryo [40], whose major crystallographic features are consistent with the experimental observation [41]. In this configuration, there are ∼9500 precipitate atoms and ∼850,000 matrix atoms, with the simulation size labeled in Fig. 2(a). The initial orientation relationship (OR) between these two phases is between K-S (KS17) and N-W (NW9), with [110]_fdeviated from [111]_f and [100]_b 1.6° and 3.7°, respectively. There are originally two sets of dislocations in the interface, with Burgers vectors 1/2[110]_f$\lvert$1/2[111]_b and 1/2[101]_f$\lvert$1/2[100]_b respectively, as identified by an automatic and simple method for specifying Burgers vectors and line directions of dislocations using the singular value decomposition of Nye tensor method [42]. The Burgers vectors of the interfacial dislocations can be double-confirmed by drawing Burgers circuits from two different views, as shown in Fig. 2(b) and (c), respectively. Atoms in the simulation are visualized using the visualization tool OVITO [43].

The potential used in the present work is a modified Finnis-Sinclair (F-S) potential for pure iron [44], with an adjustable cutoff parameter added on the charge density term. This potential not only inherits the physics of original F-S potential, but can also give a transition temperature of the FCC and BCC phase at 1100 K, which is close to the experimental value (1185 K). Besides, the homogeneous nucleation, which will interfere with the natural formation of the self-accommodation microstructure, does not occur with the present potential. The lattice parameters for FCC and BCC phases are a_f = 3.694 Å and a_b = 2.867 Å, respectively. The driving force of phase transformation is the free energy difference between the two phases, which can be determined by the Gibbs-Helmholtz equation [45], given by

$\frac{ΔG(T)}{T}=∫_T^{Tc}(H_{fcc}(T)-H_{bcc}(T))/T^2dT$ (1)

where H_fcc(T) and H_bcc(T) are enthalpy of fcc and bcc phases respectively as a function of the temperature T, and T_c is the transition temperature of the two phases. H_fcc(T) and H_bcc(T) can be obtained by the same method as used in Ref. [45]. The variation of driving force with the temperatures is plotted in Fig. 3, together with the driving force corresponding to the phase diagram and the Ackland’s Fe-Cu potential. The driving force with the present potential is close to that calculated with Ackland’s potential, while it is lower than the driving force corresponding to the Fe-C phase diagram at most temperatures [46]. Fixed boundary conditions were enforced in x, y, and z directions. The simulation was performed in an NVT ensemble at various temperatures (500 K-900 K with 100 K as the interval) via the open-sourced LAMMPS software [47], with an overall simulation time of 1 ns. The temperature was controlled by a Nosé-Hoover thermostat. To eliminate the noise raised from temperature effect, energy minimization was conducted every 10 ps. Note that all the target temperatures are below the transition temperature, thus ferrite is expected to grow at the expense of the austenite matrix at all these temperatures.

The ORs of the ferrite grains formed during the MD simulation are identified via the following three steps: (1) Search for the neighboring atoms of all ferrite atoms. A specific cutoff distance between the second- and third-nearest neighbors of the BCC atoms, 1.2a_b, is selected to define the neighboring atoms. (2) Classify all ferrite atoms into three groups according to the BC they belong to. This can be accomplished by comparing the neighboring lists of the ferrite atoms with the standard neighboring list of the Bain OR. A ferrite atom is said to be in a certain BC when at least ten of its neighboring atoms are within 0.6 Å from the corresponding atoms in the neighboring list of the standard Bain OR. The atoms that do not belong to any BC are recognized as interface atoms. (3) Classify all ferrite atoms within BCs into standard K-S or N-W ORs. In this step, the neighboring lists of all atoms within a certain BC are compared with the standard neighboring list of all 8 K-S and 4 N-W ORs in that BC. The ferrite atoms are said to near a particular standard K-S or N-W OR when the rotation between its neighboring atoms and that of the standard OR are the smallest.

3. Results

3.1. Growth of the initial ferrite before new grains form

The growth of the initial ferrite grain at 500 K prior to the formation of new ferrite grains is shown in Fig. 4. As the ferrite grows, the initial two sets of dislocation loops continuously glide along their slip planes. The gliding of interfacial dislocations leads to an anisotropic growth of the ferrite. As a result, the morphology of the ferrite at 300 ps is roughly a small plate (Fig. 4(d)). Two small facets with orientations near (1$\bar{1}$0)f and near ($\bar{5}$34)f have been developed after growth for 300 ps, as seen in Fig. 4(h). Each facet contains one set of dislocations to compensate the misfit on the facet. The major facet ($\bar{5}$34)f is in good agreement with the habit plane observed in a lath-martensite microstructure [48]. The tendency to evolve to a specific morphology that corresponds to faceted interfaces are also observed in a large number of experimental results [[29], [30], [31],33,[35], [36], [37],49] and well-interpreted in various theoretical studies [25,26,[50], [51], [52]]. However, the interfaces in these studies are presumably free of a long-range strain. In contrast, the interface in the present study possibly carries a certain degree of long-range strain, since no new misfit-compensating interfacial dislocations were observed throughout the whole growth process of the ferrite. This is probably due to the large energy barrier for generating a misfit dislocation during MD simulation. In fact, to authors’ knowledge, no newly-formed interfacial (full) dislocation has been observed in MD simulations without an external force. Consequently, the existing interfacial dislocations cannot fully accommodate the misfit between the austenite and the ferrite particle of the increased size, and the misfit strain must accumulate as ferrite grows.

As the initial ferrite grows, the OR between the initial ferrite grain and the matrix tends to evolve from a near K-S to a near N-W OR, as shown in Fig. 5. This result seems contradictory with the previous study of a Cu-Cr system [40], in which the OR evolves from N-W to near K-S with the growth of ferrite from a coherent particle to a semicoherent one. The OR evolution stems from the generation of interfacial dislocations that can approximately accommodate the interfacial misfit, as revealed by Dai et al. [40]. Inversely, the present OR evolution is accompanied by the increase of the misfit strain field accumulated during ferrite growth. This strain field may lead to a minor adjustment in the OR in favor of better elastic accommodation of the residual misfit strain. An additional reason for the discrepancy in the OR is due to the difference in the lattice parameters. The lattice parameters used in the previous study for the initial morphology are not equilibrium ones according to the selected potential in the present study. It is not surprising that the initial configuration and OR are not stable and apt to evolve to a more favorable OR associated with good matching interfaces.

3.2. Formation sequence of the new ferrite grains

New ferrite grains begin to form near the interface between the initial ferrite and the austenite when the initial ferrite grain grows to a certain size. The formation sequence of the new ferrite grains with different ORs are shown in the MD snapshots in Fig. 6 and schematically illustrated in Fig. 7. All the new ferrite grains can be classified into two groups according to their interrelations. KS3, NW3, KS16, KS1, and KS7 are in the first group since their formation is closely related to each other (with a reason to be discussed in the next section). For the same reason, KS12, NW5 and NW8 belong to the second group. In the first group, ferrite grain with the crystal orientation of KS3 firstly nucleated after 420 ps (Fig. 6(a)). Then a number of NW3 form after 450 ps (Fig. 6(b)), followed immediately by the formation of KS16. The synchronized growth of NW3 and KS16 gives rise to a local lamellar morphology, as shown in Fig. 6(c). NW3 and KS16 were identified as near twin-related (Appendix A), which are often observed experimentally [9,10]. After 550 ps, the small-sized KS1 and KS7 appear adjacent to KS3 and NW3, respectively (Fig. 6(d)). Almost simultaneously, new ferrite grain in the second group, i.e., KS12, starts to form (Fig. 6(d)), followed by the formation of NW5 next to it. At last, NW8 emerges as a connection between the early-formed KS16 and KS12 after 900 ps (Fig. 6(e)). From Fig. 6(d and e), it is found that KS7 grows larger at the expense of NW3, while KS1 and NW5 shrink slightly to smaller sizes. Consequently, there are nine types of ORs present at the end of the simulation, with KS1, KS3, KS7, and NW3 belonging to BC 1, KS12, KS16, NW5, and NW8 belonging to BC 2 and the initial grain belonging to BC 3, as also classified in Fig. 7. The interpretation of the formation of these ferrite grains and their formation sequence will be discussed in detail in the next section.

4. Discussion

4.1. Formation of the self-accommodation microstructure

It is evident that any newly-formed ferrite grain always nucleates at a position adjacent to the existing grains. The existing grains can assist the nucleation by reducing the nucleation energy barrier of both strain energy and interfacial energy. In principle, the stress field around the existing ferrite grains can cancel the transformation strain associated with the new ferrite of some particular ORs. In the meantime, interfaces of relatively small interfacial energy can be obtained by selecting special misorientations between the existing and new ferrite grains. The detailed influences of these two factors in the formation of specific self-accommodation microstructure in the present simulation will be described quantitatively below.

To evaluate the influence of the stress field on the OR selection, we use the elastic interaction energy E_inter(r) defined by [53]:

E_inter(r)=-σ_ij(r)$ε_{ij}^*$ (2)

where σ_ij(r) represents the stress field near the initial ferrite grain and can be directly obtained from the MD simulation data. $ε_{ij}^*$ is the eigenstrain of formation of a new ferrite grain of particular OR. The nucleation of new ferrite grains is favored for negative E_inter(r), and is suppressed for positive Einter(r). Here εij* is in the form of Green’s strain tensor, i.e., $ε_{ij}^*$=1/2A′A-I, where A is the deformation matrix of the ferrite formation (or phase transformation) and I is the identity matrix. Due to the fact that two phases always obey the Bain lattice correspondence in an FCC/BCC transformation system, the deformation matrix A can be factorized into the Bain deformation B and a rotationR, i.e., A=RB. Thus,

$ε_{ij}^*=1/2(B′B-I)$ (3)

This expression is different from the one used by other researchers [5,16]. In their studies, the macroscopic shape deformation in the phenomenological theory of martensite crystallography (PTMC) P₁ [54,55] was used as the deformation matrix, since martensite is not coherent with the austenite matrix. In martensitic phase transformation, there is a lattice invariant deformation (LID) P₂ that can partially cancel the transformation deformation A. The relationship between P₁, P₂ and A is A= P₁P₂ [54]. However, in the present work the new ferrite grains are coherent with the matrix, so A is directly used as the deformation matrix. Only the stress field generated by the initial large ferrite grain was taken into account. This is reasonable because all the new grains are considerably smaller than the initial one, and the stress field around them has been largely cancelled by that of the initial ferrite grain.

From Eq. (3) it is clear that the eigenstrain of formation of ferrite grains only depend on the BCs they belong to, as listed in Table 1. Therefore, ferrite grains within the same BC have exactly the same E_inter(r) values at all positions. As a result, the same amount of energy barrier for nucleation is reduced by the formation of new ferrite grains in the same BC. The elastic interaction energy field E_inter(r) of ferrite grains belonging to the three BCs near the initial ferrite grain is shown in Fig. 8(a-c). The sequence of the interaction energy of ferrite grains in different BCs is E_BC3>0>E_BC2>E_BC1. The preference order of ORs due to the influence of strain energy is summarized in the upper part of Fig. 9. The ferrite grains in BC 1 and BC 2 can probably form while the ferrite grains in BC 3 are energetically unfavorable. Therefore, new ferrite grains in BC 3 were not observed in the simulation, as seen in Fig. 6. Compared with ferrite grains in BC 2, those in BC 1 have more negative interaction energy E_inter(r) in most areas, and are thus more preferred. It explains why generally the ORs in BC 1 form earlier than that in BC 2. Moreover, from Fig. 8(a) one can also see that the elastic interaction energy E_inter(r) is not evenly distributed near the interface, with the positions near the dislocation cores associated with the most negative interaction energy. These positions are the preferred sites for the first nucleation event to occur.

While the elastic interaction energy can decide the BC in which the ORs are preferred, the selection of specific ORs from the twelve ORs within a BC was found to be mainly affected by the interfacial energy. The interfacial energy of symmetric tilt grain boundaries was calculated from the excess energy per unit area with respect to the cohesive energy of the BCC phase. For each grain boundary, the interfacial energy was calculated from the average of 100 samples, which are taken from different areas of the interface. Fig. 10 shows the variation of interfacial energy with the tilt angle with respect to the rotation axis [1$\bar{1}$0]_b, with the error bar which indicates the standard deviation from the average values of each interfacial energy. It is seen that the interfacial energy is relatively low when the tilt angle is small or when the misorientation allows the existence of low Σ coincidence site lattice (CSL) (due to a high density of CSL points at the corresponding symmetric tilt grain boundaries). The deep cusp corresponding to the Σ31$\bar{1}$2b coherent twin boundary indicates that the coherent twin boundary has much lower energy than other high-angle grain boundaries (with a misorientation larger than 15° [56]). Note that at low-angle grain boundary regime, the interfacial energy varies monotonically with the misorientation angle, consistent with the Read-Shockley formula [57]. This trend is not limited to tilt grain boundaries, but is also valid for twist boundaries, as already shown in previous studies [56,[58], [59], [60]]. Such a relationship is often used for general grain boundaries. Based on our simulation results, all boundaries between ferrite grains are low-angle grain boundaries and coherent twin boundaries, indicating that reducing interfacial energy must have played a role in the variant selection. The influence of interfacial energy on the OR selection is illustrated in the lower part of the flow chart in Fig. 9, where circled ‘L’ denotes a low-angle misorientation. The two branches in the figure correspond to the two groups defined in the previous section (Section 3.2). Note that the near-twin-related (NT) ORs are also labeled in the figure, since according to the Brandon criterion [61], if the deviation from an exact twin relationship is less than 8.6°, a coherent twin boundary can form locally between the two grains. Since the ferrite grain in the present case is small, the coherent twin structure can be maintained in the whole boundary area with a possible small elastic strain. To make a quantitative analysis of the influence of interfacial energy on OR selection, ORs with the twin or near twin relationship are listed in Table A1, while all the standard K-S or N-W ORs that permit low-angle grain boundaries can be read in the pole figure in Fig. 1 (at most the third-nearest ORs).

In our simulation, KS3 forms prior to the grains with other ORs, and shares the coherent twin boundary with the initial ferrite grain, as (1\bar{1}2) _b shown in Fig. 6(a). This can be understood based on the combining analysis of elastic interaction energy and interfacial energy discussed above. On one hand, KS3, which belongs to BC 1, has the most negative elastic interaction energy. On the other hand, among all the ferrite grains in BC 1, KS3 is the only ferrite grain that has a near-twin relationship (theoretical deviation angle of 5.26°) with the initial ferrite grain (NW9) (Table A1), with a coherent twin boundary (1$\bar{1}$2)_b between them. Therefore, KS3 not only has the most negative Einter(r) but can also share a low-energy coherent twin boundary with the initial precipitate, making it the first OR to nucleate.

The subsequent ferrite grain formation can also be interpreted similarly based on the consideration of the elastic interaction energy and interfacial energy. In the first group, several NW3 grains form subsequent to KS3 (Fig. 6(b)), followed by the formation and rapid growth of KS 16. This is because NW3 is the closest to KS3 (Fig. 1) and thus shares a low angle grain boundary (5.26°) with KS3, while KS16 and NW3 are near twin-related (Table A1) and thus form (1$\bar{1}$2)_b coherent twin boundaries at both sides of the grains, as identified in the MD simulation. NW3 is then gradually replaced by KS7 and finally forms a lamellar morphology consisting of KS7 and KS16 (Fig. 6(d) and (e)), indicating that the near twin-related NW3/KS16 OR pair tends to evolve to an exact twin-related KS7/KS16 pair by a small local rotation. This exact twin relationship with the (1$\bar{1}$2)_b coherent twinning plane can be seen in Fig. 6(f), which shows the [1$\bar{1}$0]_bview of the lamellar microstructure. In the second group, KS12 forms prior to other ORs in the same group, since it again shares a coherent twin plane (1$\bar{1}$0)_b with the initial ferrite grain. However, the nucleation of KS12 is much later than KS3 due to the smaller absolute value of the elastic interaction energy than that of ORs in BC 1. Other ORs present in the simulation, i.e., KS1, NW5 and NW8, all form low angle grain boundary with the adjacent grains, as illustrated in Fig. 9. Note that they are not necessarily the closest to the adjacent grain, probably due to the effect of complicated local stress fields from several nearby small grains. It is evident that the grains at either side of the coherent twin boundaries (KS3, KS7, KS12, KS16) can grow relatively larger than other grains, indicating the important role of the twin boundaries on the stabilizing ferrite grains in the self-accommodation microstructure.

4.2. Temperature dependence of the OR selection

The self-accommodation microstructure is affected by temperature. Fig. 11 shows the variation of atom numbers in BC 1 and 2 with the temperature range from 500 to 900 K after a simulation time of 1 ns. As the temperature elevates, the proportion of atoms in BC 2 shows an overall descending trend, while the proportion of atoms in BC 1 continuously increases. Finally, ORs in BC 2 disappear at 900 K. This trend indicates that as the driving force of phase transformation decreases with the increase of the temperature, the nucleation of ferrite has much less chance to overcome the relatively larger strain energy associated with ORs in BC2. Thus, only ferrite grains in BC 1 can survive at all these temperatures. At a temperature higher than 900 K, the driving force may not be sufficiently large to overcome the strain energy associated with the formation of any ferrite grain in BC 2.

While the driving force increases with the decrease of the temperature, the atomic mobility changes in an inverse manner. This may lead to a maximum transformation rate, as one expects from a C type of transformation curve for a typical diffusional phase transformation. This trend is reflected in Fig. 11 that the total number of atoms in the newly-formed ferrite grains appears to be the largest at 600 K. A possible reason is that both the nucleation and growth rate may be relatively high at 600 K, corresponding to the best combination of driving force and atomic mobility needed for nucleation and growth of ferrite.

4.3. Comparison with previous works

Based on the discussion above, it is found that the strain energy and interfacial energy play a role in the OR selection in two steps: In the first step, the strain energy decides in which Bain circle an OR will possibly form, with a frequency depending on the temperature. In the second step, the preference of low interfacial energy leads to specific ORs from the candidates screened by the strain energy. From this perspective, the strain energy plays a more dominant role in OR selection than the interfacial energy. That is the reason why it is possible to explain the gross trend of the formation of variant grains solely by strain energy or strain energy-related geometric factors as done in the previous studies [5,9]. Although the role of interfacial energy was not testified in these studies, the EBSD images provided in these studies clearly show the existence of a large number of low-angle grain boundaries, indicating the role of the interfacial energy in OR selection. The effect of the temperature on the ratio of BC in present simulation result is in good agreement with the experimental observations of the OR selection of bainitic ferrite [10]. According to the present results, most ORs formed are within one BC when the temperature is high. As a result, twin-related ORs, which links variant in different BC, do not exist, and all ferrite grains are separated by low-angle grain boundary, which agrees with the experimental observations [10].

5. Summary and remarks

MD simulations were conducted to provide an atomistic view of the self-accommodation microstructure development during the austenite to ferrite transformation in a pure iron system. It reveals fine details of how the ORs are affected by the strain energy and the interfacial energy. Despite a non-diffusional nature of the ferrite formation in the present simulation, the results of this study provide insights for understanding the formation of the self-accommodation microstructure in both diffusional and diffusionless phase transformations. The main results are summarized below:

1 Before the formation of new ferrite grains, the initial semicoherent ferrite evolves into a faceted particle. It consists of two facets, each containing one set of the original dislocations. The major facet ($\bar{5}$34)_f is consistent with the habit plane of lath-martensite in the experiment. No new misfit dislocation is generated. The stress field around the grown-up ferrite caused by residual misfit was evaluated from the simulation data.

2 New ferrite grains with various ORs formed upon the initial ferrite grain reaching a certain size. ORs in BC 1 and 2 were observed, but only ORs in BC 1 were observed at relatively high temperature. This result proves that the strain energy decides whether or not an OR can possibly form at a certain temperature, indicating the dominant role of strain energy in OR selection during the formation of a self-accommodation microstructure. The preference of the ORs agrees with the orders of the values (large to small) of elastic interaction energy determined for three BCs.

3 The specific ORs of the newly formed ferrite grains all related to the existing adjacent grains with either twin or near twin relationship or small angle misorientations between them. The formation sequence of ferrite grains suggests a strong influence of interfacial energy on the selection of specific ORs. However, since the OR selection according to the interfacial energy is a second step, after the selection based on strain energy, the interfacial energy plays a less dominant role in OR selection than strain energy.

Acknowledgements

The authors are grateful to Jian Wang for fruitful discussion on the related knowledge about micromechanics. 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).

Appendix A.

Twin/near-twin relationship between different variants

Given the orientation relationship matrices of any two ORs of the total 36 ORs (24 K-S variants and 12 N-W variants), M_i and M_j(i,j=1,2,…,36,andi≠j), the rotation matrix R_ij between these two ORs can be determined by R_ij=M_j/M_i. R_dev=R_ij/R_twin is the rotation matrix representing the deviation between R_ij and the standard rotation matrix corresponding to the standard twin relationship R_twin. By transforming R_dev into the axis/angle pair, we can obtain whether these two ORs are twin-related or how far they deviated from a standard twin relationship. Table A1 lists the ORs which have a twin or near-twin (with a deviation angle of 5.26°) relationship with a certain OR. The ‘twin ORs’ column (the third column) and the ‘near-twin ORs’ column (the fourth column) list the variants which have an exact twin and near-twin relationship with the corresponding OR in the first column, respectively. It is evident that all ORs have two twin/near-twin related ORs. Any KS OR has an exact twin-related KS OR and a near twin-related NW OR, while an NW OR has two near twin-related KS ORs and no exact twin-related OR.

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