High Accuracy Ab Initio Prediction of Phase Transitions in Ti and TiNb Alloys

Ab initio methods based on DFT are now routinely used to investigate T=0 K properties. In contrast, finite temperature DFT studies are rare and, in particular, limited to approximations such as the quasiharmonic model. In the present study, we investigate the influence of anharmonic excitations—which correspond to phonon-phonon interactions beyond the simple quasiharmonic picture—on the thermodynamics of Ti and TiNb alloys.

Motivation

Fig. 1: bcc Ti at 1000 K obtained from ab initio molecular dynamics (MD) simulations. For each atom a sequence of 1000 MD steps is shown. MD simulations as exemplified in this figure can provide access to thermodynamic properties including phonon-phonon interactions if special speed up techniques are used (see text).

Dominique Korbmacher, Max-Planck-Institut für Eisenforschung GmbH

Fig. 1: bcc Ti at 1000 K obtained from ab initio molecular dynamics (MD) simulations. For each atom a sequence of 1000 MD steps is shown. MD simulations as exemplified in this figure can provide access to thermodynamic properties including phonon-phonon interactions if special speed up techniques are used (see text).

Dominique Korbmacher, Max-Planck-Institut für Eisenforschung GmbH

Ab initio methods based on density-functional theory (DFT) are now routinely used to investigate T=0 K properties such as, e.g., the influence of alloying elements on titanium’s mechanical properties. In contrast, finite temperature DFT studies are rare and, in particular, limited to approximations such as the quasiharmonic model. In the present study, we investigate the influence of anharmonic excitations—which correspond to phonon-phonon interactions beyond the simple quasiharmonic picture—on the thermodynamics of Ti and TiNb alloys (Fig. 1) which are of high technological relevance (see related project Intrinsic electronic interactions in TiNb alloys).

Theory

At low temperatures pure titanium is stable in the hcp (α) phase and transforms to the high-temperature bcc (β) phase at a transition temperature of 1155 K. To get the full thermodynamics the free energy F (V, T) depending on the volume V and temperature T needs to be computed. This energy is composed of the following contributions

where E_0K is the total energy at T=0 K, F^el the electronic free energy, F^qh the quasiharmonic and F^ah the anharmonic free energy. The electronic free energy is calculated using finite temperature DFT, the quasiharmonic contribution is obtained from phonon frequencies based on the direct force constant method, and the anharmonic contribution is computed based on thermodynamic integration using molecular dynamics.

Methodology

Dynamically unstable systems, such as the bcc phase of Ti, cannot be studied with current state-of-the-art approaches. Using for example the quasiharmonic approach is not sufficient to investigate these unstable systems, and in fact leads to diverging thermodynamic properties. One has to consider anharmonicities which, however, typically require long CPU times. To overcome this difficulty we use here our recently developed TU-TILD method [1] which is based on our previous development, the UP-TILD method [2], a coarse graining scheme allowing to reduce the computation time to days while keeping high accuracy. The TU-TILD method employs as a reference optimized embedded atom potentials that are fitted to reproduce ab initio molecular dynamics data for a narrow volume and temperature range.

Results

Fig. 1 shows the expansion coefficient of pure Ti where we used two different exchange correlation functionals. We get a similarly good agreement with experimental data for both functionals including the anharmonic contribution in the free energy. Fig. 2 shows the free energy for all three phases appearing in pure Ti. With our methods we are able to identify the low temperature stable omega phase which is not measurable in experiments under ambient pressure. All phases are correctly predicted using our methods and the hcp to bcc transition temperature occurs at 1253K, which agrees well with the experimentally observed temperature of 1150K.

Fig. 1: Expansion coefficient depending on the temperature for hcp and bcc phases using two different exchange correlation functionals.

Dominique Korbmacher, Max-Planck-Institut für Eisenforschung GmbH

Fig. 1: Expansion coefficient depending on the temperature for hcp and bcc phases using two different exchange correlation functionals.

Dominique Korbmacher, Max-Planck-Institut für Eisenforschung GmbH

Fig. 2: The free energy for all three phases appearing in pure Ti.

Dominique Korbmacher, Max-Planck-Institut für Eisenforschung GmbH

Fig. 2: The free energy for all three phases appearing in pure Ti.

Dominique Korbmacher, Max-Planck-Institut für Eisenforschung GmbH

References

Andrew Ian Duff, Theresa Davey, Dominique Korbmacher, Albert Glensk, Blazej Grabowski, Jörg Neugebauer, and Michael W. Finnis

Improved method of calculating ab initio high-temperature thermodynamic properties with application to ZrC

Physical Review B 91, 214311 (2015)

Blazej Grabowski, Lars Ismer, Tilmann Hickel, and Jörg Neugebauer, "Ab initio up to the melting point: Anharmonicity and vacancies in aluminum," Physical Review B 79 (13), 134106 (2009).

MPG.PuRe

DOI

Other group activities

● complementary project: Intrinsic electronic interactions in TiNb alloys

● other projects: Adaptive Structural Materials