Systematic pathway generation and sorting in martensitic transformations: Titanium $\alpha$ to $\omega$

Martensitic phase transitions appear in a diverse range of natural and engineering material systems. Examination of the energetics and kinetics of the transformation requires an understanding of the atomic mechanism for the transformation. A systematic pathway generation and sorting algorithm is presented and applied to the problem of the titanium $\alpha$ to $\omega$ transformation under pressure. The transformation pathways are separated into strain and shuffle components. All pathways are constructed within energetically motivated strain and shuffle constraints, and efficiently sorted by their energy barriers. The geometry and symmetry details of the seven lowest energy barrier pathways are given. The lack of a single simple geometric criterion for determining the lowest energy pathway shows the necessity of atomistic studies for pathway determination. The general algorithm can determine the pathway for any martensitic transformation.

Figure 1

Energy barrier at zero pressure for the seven lowest energy pathways, using ab initio NEB at 16 intermediate states. The pathways are grouped by their strains; each set has related supercell pairs. The effect of different shuffles on the energy barrier can be seen by comparing TAO-2 to TAO-5, and Silcock, TAO-3, and TAO-4 to each other. The latter three pathways have the smallest amount of strain, but the homogeneous barrier is smallest for TAO-1.

Figure 2

TAO-1 acting on a single $\alpha$ basal plane. A hexagon in the basal plane is broken into two three-atom pieces. Each three-atom piece swings out of the basal plane in opposite directions, to create half of the honeycomb lattice in parallel (0001) planes of $\omega$. The remaining atom in the center forms the $A$ sublattice lying in the (0001) planes of $\omega$. The next $\alpha$ basal layer creates the dashed atoms in the $\omega$ honeycomb. This transformation turns the (0001) plane of $\alpha$ into the $\left(01\overline{1}1\right)$ plane of $\omega$.