Origin of phase stability in Fe with long-period stacking order as an intermediate phase in cyclic $\gamma \text{−}\epsilon$ martensitic transformation

A class of Fe-Mn-Si–based alloys exhibit a reversible martensitic transformation between the $\gamma$ phase with a face-centered cubic (fcc) structure and an $\epsilon$ phase with a hexagonal close-packed (hcp) structure. During the deformation-induced $\gamma \text{−}\epsilon$ transformation, we identified a phase that is different from the $\epsilon$ phase. In this phase, the electron diffraction spots are located at the 1/3 positions that correspond to the $\left\{0002\right\}$ plane of the $\epsilon$ (hcp) phase with 2H structure, which suggests long-period stacking order (LPSO). To understand the stacking pattern and explore the possible existence of an LPSO phase as an intermediate between the $\gamma$ and $\epsilon$ phases, the phase stability of various structural polytypes of iron was examined using first-principles calculations with a spin-polarized form of the generalized gradient approximation in density functional theory. We found that an antiferromagnetic ordered $6{\mathrm{H}}_2$ structure is the most stable among the candidate LPSO structures and is energetically closest to the $\epsilon$ phase, which suggests that the observed LPSO-like phase adopts the $6{\mathrm{H}}_2$ structure. Furthermore, we determined that the phase stability can be attributed to the valley depth in the density of states, close to the Fermi level.

Figure 1

Atomic displacement and formation of partial dislocation-stacking fault units associated with (a) $\gamma \to$ (b) $\epsilon$ martensitic transformation.

Figure 2

(a) Atomic configuration of the hcp lattice. The tetrahedron marked in bold green lines is the spin frustration unit in the hcp lattice. (b) An example of spin frustration in a tetrahedron into which an hcp lattice can be decomposed. Arrows show the spin orientation on an isolated tetrahedron in the hcp lattice. (c) Spin configuration of the AFM-II phase in hcp-Fe, which is projected on the ${\left(0001\right)}_{\text{hcp}}$ plane. Up and down arrows indicate the spin orientation. The solid line connects in-plane sites, and the dashed line connects sites to those in the adjacent layer. The open circles with red arrows show the internal atomic coordinates at $z=1/4$, and the filled circles with blue arrows show those at $z=3/4$.

Figure 3

(a) Dimensions of a low-cycle fatigue specimen. (b) Photograph of the fatigue test setup.

Figure 4

(a) Bright-field image and (b) SAED pattern of the LPSO–like (${\epsilon }^{\prime }$) phase found in the fatigue-failed Fe-33Mn-4Si (mass %) alloy [encircled area in (a)]. The red arrows indicate the extra spots at the 1/6 positions that correspond to the ${\left(\overline{1}\overline{1}1\right)}_{\gamma }$ plane [the 1/3 positions corresponding to the $\left\{0002\right\}$ plane of the $\epsilon$ phase], the latter of which corresponds to the $\left(0006\right){\epsilon }^{\prime }$. (c) Dark-field image of (a).

Figure 5

(a) Bright-field image for crossing variant plates in an LPSO-like (${\epsilon }^{\prime }$) phase of the Fe-33Mn-4Si alloy, and both observed in the SN orientation. (b) The (0006) planes of the variants, where ${\epsilon }_1^{\prime }$ and ${\epsilon }_2^{\prime }$ are parallel to the $\left(1\overline{1}1\right)\gamma$ and $\left(\overline{1}\overline{1}1\right)\gamma$ planes, respectively. The red arrows indicate the extra spots at the 1/6 positions that correspond to the ${\left(1\overline{1}1\right)}_{\gamma }$ plane (the 1/3 positions correspond to the $\left\{0002\right\}$ plane of the $\epsilon$ phase), the latter of which corresponds to the $\left(0006\right){\epsilon }_1^{\prime }$. Dark-field images of the (c) $\left(01\overline{1}3\right){\epsilon }_1^{\prime }$ and (d) $\left(01\overline{1}\overline{3}\right){\epsilon }_2^{\prime }$ spots. The parallel plates of the ${\epsilon }_1^{\prime }$ and ${\epsilon }_2^{\prime }$ phases are shown in bright contrast in those figures. The streaks along the $\left(0006\right){\epsilon }^{\prime }$ spots suggest a high concentration of stacking faults in the ${\epsilon }^{\prime }$ planes.

Figure 6

(a) Projection of the hcp lattice with AFM-II order on the (0001) plane, (b) $6{\mathrm{H}}_1\text{−}\mathrm{AFM}2$, and (c) $6{\mathrm{H}}_2$-AFM structures projected on the (111) plane of the fcc lattice, which is parallel to the (0001) plane of the hcp lattice. The arrows denote the direction and magnitude of magnetic moments, and the magnetic moments on the $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{C}$ layers are shown with red, blue, and green arrows, respectively. The orthorhombic cell is shown as bold dashed lines. The spin alternation along the $\left[0\overline{1}1\right]$ and $\left[10\overline{1}\right]$ directions of the fcc lattice is realized with this unit cell. The $\mathsf{A}$ layer is on top of the $\mathsf{B}$ layer, and the $\mathsf{C}$ layer is on top of the $\mathsf{B}$ layer. The parallelogram shown in the background is the $\mathsf{B}$ layer. The planes shown in (b) and (c) correspond to the bottom three layers of the orthorhombic cells shown in Figs. 7 and 7, respectively.

Figure 7

Schematic illustrations of the spin arrangements for the (a) $6{\mathrm{H}}_1\text{−}\mathrm{AFM}2$ and (b) $6{\mathrm{H}}_2$-AFM structures with the spin direction (red and blue circles represent Fe atoms with up and down spins, respectively). Orthorhombic cells are shown as bold dashed lines. The $ab$ planes in (a) and (b) corresponds to the {111} plane of the $\gamma$ phase. The relative position of hexagonal units due to Shockley partial dislocations are displayed; In (a) and (b), the lattice vector $b$ corresponds to the direction of the Shockley partial dislocation along $\left[11\overline{2}\right]$ of the fcc structure.

Figure 8

Body-centered-tetragonal (bct) cells for the (001)-type AFM order structures of fcc-Fe; (a) AFM single-layered (AFM-S) and (b) AFM double-layered (AFM-D) structures. Black spheres with red up arrows represent Fe atoms with up spin, and those with blue down arrows represent those with down spin. The lattice constants in the bct cell were used; (a) as $a=a_0\sqrt{2}$ and $c=a_0$ [= $2a_0$ in (b)], where $a_0$ is the cubic lattice constant. Solid and dashed lines indicate the original cell with an fcc lattice and the magnetic unit cell with the bct lattice, respectively. The bct cell contains two and four atoms to describe the (001) layered structure of AFM-S and AFM-D, respectively.

Figure 9

Volume dependence of the total energy difference in structural polytypes of Fe with respect to the hcp AFM-II phase. The total energies of each phase are calculated as a function of the lattice volume per atom within the GGA-PBE functional. Open and solid squares on black lines represent the AFM-S and AFM-D states in the fcc structure. Solid orange and blue triangles on dashed lines represent the AFM-1 and AFM-2 states for the $6{\mathrm{H}}_1$ structures, respectively. Open and solid triangles on red dashed lines show NM and AFM states for the $6{\mathrm{H}}_1$ structures, respectively. Open squares show the fcc structure with the AFM-D ordered state. Open and solid circles show NM and AFM-II states of the 2H (hcp) structures, respectively. Open green squares on the green dotted line represent the ferromagnetic (FM) state of bcc Fe.

Figure 10

Difference of the total energies for different structural polytypes of pure iron as a function of hexagonality, $\beta$. $\mathrm{\Delta }E$ was calculated with respect to hcp Fe with the AFM-II phase and are listed in Table 2. Open and solid circles represent NM and AFM states, respectively. The values of $\beta$ are listed in Table 1.

Figure 11

The total DOS for the AFM state of the (a) 2H (hcp), (b) $6{\mathrm{H}}_2$, (c) $6{\mathrm{H}}_1$ with AFM2, and (d) $6{\mathrm{H}}_1$ with AFM1 structures of pure iron at ambient pressure. Dashed lines represent the Fermi level.