Classical potential describes martensitic phase transformations between the $\alpha$, $\beta$, and $\omega$ titanium phases

A description of the martensitic transformations between the $\alpha$, $\beta$, and $\omega$ phases of titanium that includes nucleation and growth requires an accurate classical potential. Optimization of the parameters of a modified embedded atom potential to a database of density-functional calculations yields an accurate and transferable potential as verified by comparison to experimental and density-functional data for phonons, surface and stacking fault energies, and energy barriers for homogeneous martensitic transformations. Molecular-dynamics simulations map out the pressure-temperature phase diagram of titanium. For this potential the martensitic phase transformation between $\alpha$ and $\beta$ appears at ambient pressure and 1200 K, between $\alpha$ and $\omega$ at ambient conditions, between $\beta$ and $\omega$ at 1200 K and pressures above 8 GPa, and the triple point occurs at 8 GPa and 1200 K. Molecular-dynamics explorations of the kinetics of the martensitic $\alpha \text{−}\omega$ transformation show a fast moving interface with a low interfacial energy of $30\text{meV}/{\text{Å}}^2$. The potential is applicable to the study of defects and phase transformations of Ti.

Figure 1

The five cubic splines of the MEAM potential. The lines denote the spline interpolation between the spline knots represented by the solid circles. The splines are linearly extrapolated beyond the last spline knot.

Figure 2

(Color online) Fitted energies of $\alpha$, $\beta$, and $\omega \text{Ti}$ as a function of volume. (a) The MEAM potential is fit to the energy of $\alpha$, $\beta$, and $\omega$ at several volumes. The DFT results appear as symbols and the fitted MEAM curves as lines. (b) The enthalpy difference between $\alpha$ and $\omega$ as a function of pressure for MEAM. The MEAM potential predicts a transition pressure $p_{\alpha \text{−}\omega }$ of about $-5\text{GPa}$ at zero temperature similar to DFT, which predicts the $\omega$ phase to be the ground state (Ref. 7).

Figure 3

(Color online) The phonon spectra of $\alpha$, $\beta$, and $\omega \text{Ti}$ for the classical MEAM potential (black solid lines) compared to DFT results (red dashed lines) and to experimental data for the $\alpha$ and $\beta$ phases (blue circles) (Refs. 45, 46). For the $\alpha$ phase the MEAM potential accurately reproduces the low-energy acoustic branches reflecting the elastic constants. For optical and high-energy acoustic branches the potential reproduces the experimental phonon frequencies within 25%, generally underestimating the experimental values. Similarly for the $\omega$ phase the acoustic branches of the MEAM potential closely match the DFT results with larger discrepancies of up to 20% for the optical modes. The $\beta$ phase is mechanically unstable (Refs. 14, 45) at low temperatures and transforms to $\alpha$ or $\omega$. This instability is reflected in unstable (imaginary) phonon branches in the DFT and MEAM calculations. The imaginary phonon for $T$-[110] at the $N$ point corresponds to the $\beta$ to $\alpha$ transformation mechanism. The imaginary $L\text{−}\frac{2}{3}\left[111\right]$ phonon is responsible for the $\beta$ to $\omega$ transformation.

Figure 4

(Color online) Comparison of energy barriers of different pathways for the $\alpha$ to $\omega$ transformation (Refs. 15, 16) with MEAM and TB (Ref. 34). The inset shows the distribution of the deviations between the TB and the MEAM potential. The rms deviation is 5.5% and the maximum deviation 27% over the set of 977 pathways.

Figure 5

(Color online) Equilibrium phase diagram of Ti as a function of pressure and temperature. The MEAM potential accurately describes all three solid phases, $\alpha$, $\beta$, and $\omega$, of Ti and captures the martensitic phase transformations between them. Near the triple point, the $\alpha$, $\beta$, and $\omega$ phases are nearly degenerate. As a result, whether the $\beta$ phase transforms to either $\alpha$ or $\omega$ is controlled by the initial conditions of the molecular-dynamics simulations.

Figure 6

(Color online) Molecular-dynamics simulations of the $\alpha \text{−}\omega$ martensitic transformations. (a) The relaxed $\alpha \text{−}\omega$ interface is created using the TAO-1 pathway and relaxed to produce the initial interfaces. The energy for the interface is $30\text{meV}/{\text{Å}}^2$. Unidentified atoms are colored gray, while $\alpha$ and $\omega$ atoms are colored blue and green, respectively. (b) Transformed geometry at zero compression. Molecular-dynamics simulations of the relaxed interface structure at 1200 K for 200 ps transform the entire $\omega$ phase region. When the structure is fully relaxed to zero temperature, it is identified as entirely $\alpha$, except for the small region where the two interfaces merge and leave behind a number of defects. (c) Transformed geometry at 10% volume compression. The relaxed interface structure is first compressed by 10% (corresponding to $\sim 10\text{GPa}$) and then simulated at 1200 K for 200 ps. The entire $\alpha$ region transforms to $\omega$, except for a number of point defects (interstitials and vacancies). For illustration, the slab is rotated around the horizontal axis so that the $c$ axis of $\omega$ is perpendicular to the page.