New Mechanism for the $\alpha$ to $\omega$ Martensitic Transformation in Pure Titanium

We propose a new direct mechanism for the pressure driven $\alpha \to \omega$ martensitic transformation in pure titanium. A systematic algorithm enumerates all possible pathways whose energy barriers are evaluated. A new, homogeneous pathway emerges with a barrier at least 4 times lower than other pathways. The pathway is shown to be favorable in any nucleation model.

Figure 1

Our proposed $\alpha \to \omega$ transformation pathway (TAO-1). The direct six-atom transformation is visualized (i) as a two-step process and (ii) with 21 additional atoms outside the heavy gray parallelepiped supercell. In $\alpha$, atoms $1,3,5$ and $2,4,6$ are in the $A$ and $B$ stacking planes, respectively; whereas in $\omega$, atoms 1–4 are in the Wyckoff $d$ position of space group $P6/mmm$, and atoms 5–6 are in the Wyckoff $a$ position [7]. (a) The gray atoms in the $\alpha$ cell shuffle to new positions (white atoms), with atoms 1–4 shuffling $0.63\AA$, and atoms 5–6 shuffling $0.42\AA$. (b) The $\alpha$ cell is redrawn with the “bonds” of the $\omega$ structure. Step 1: shuffling the gray atoms to the white positions, the $\alpha$ cell (b) produces a strained $\omega$ cell (c), contained in the same supercell. Step 2: straining the supercell (c) $e_{xx}=-0.09$, $e_{yy}=0.12$, and $e_{zz}=-0.02$ produces the final $\omega$ supercell (d). The orientation relations connecting the $\alpha$ and $\omega$ supercells are $(0001{)}_{\alpha }\parallel (0\overline{1}11{)}_{\omega }$ and $[11\overline{2}0{]}_{\alpha }\parallel [01\overline{1}1{]}_{\omega }$.

Figure 2

Approximate elastic energy barrier ($\mathrm{m}\mathrm{e}\mathrm{V}/\mathrm{\text{atom}}$) versus landscape energy barrier ($\mathrm{m}\mathrm{e}\mathrm{V}/\mathrm{\text{atom}}$) for 359 pathways whose landscape barrier is below $200\mathrm{m}\mathrm{e}\mathrm{V}/\mathrm{\text{atom}}$. Of the 134 supercells generated, 57 had low (less than $\mathrm{100}\mathrm{m}\mathrm{e}\mathrm{V}/\mathrm{\text{atom}}$) elastic barriers. From those 57 supercells, 977 pathways were generated; for those 977 pathways, we calculate a shear/shuffle energy landscape using our tight-binding parameters. Of those 977, 359 pathways had an elastic barrier below $200\mathrm{m}\mathrm{e}\mathrm{V}/\mathrm{\text{atom}}$. We select the three pathways in the bottom left corner—TAO-1, TAO-2, and Silcock—and calculate their true barrier using the ab initio NEB method.

Figure 3

Silcock pathway for $\alpha$ to $\omega$ transformation (single $\alpha$ stacking plane shown). (a) In each stacking plane, three out of every six atoms shuffle by $0.74\AA$ along $[11\overline{2}0{]}_{\alpha }$, while the other three shuffle in the opposite direction $[\overline{1}\overline{1}20{]}_{\alpha }$. (b) This creates the $(11\overline{2}0{)}_{\omega }$ plane from $(0001{)}_{\alpha }$. The $a$ and $d$ Wyckoff positions of $P6/mmm$ are also shown. This shuffle is accompanied by a strain $e_{xx}=0.05$ along $[1\overline{1}00{]}_{\alpha }$ and $e_{yy}=-0.05$ along $[11\overline{2}0{]}_{\alpha }$ to produce a hexagonal $\omega$ cell with the correct $c/a$ ratio. The orientation relations connecting the $\alpha$ and $\omega$ supercells are $(0001{)}_{\alpha }\parallel (11\overline{2}0{)}_{\omega }$ and $[11\overline{2}0{]}_{\alpha }\parallel [0001{]}_{\omega }$.

Figure 4

Enthalpy barrier vs pressure for the three lowest energy pathways, using ab initio NEB with 16 intermediate states. The ab initio NEB gives the $T=0$ energy along the homogeneous $\alpha \to \omega$ pathway for each pathway. The enthalpy barriers decrease with increasing pressure. (a) The TAO-1 pathway gives the smallest barrier by a factor of 4 at 0 GPa. The TAO-2 pathway (b) and Silcock pathway (c) have larger barriers at 0 GPa, an ordering which continues even at 40 GPa.