Crystal structure and free energy of Ti2Ni3 precipitates in Ti–Ni alloys from first principles
Graphical abstract
Introduction
Near-equiatomic TiNi is an important structural material for shape memory alloy applications. The concentration of Ni in typical alloys of this type is between 49 and 51 at.%. Buehler at US Naval Ordnance Laboratory first investigated TiNi shape memory alloys in 1962 [1]. These alloys have the ability to “remember” their original, cold-forged shape. This shape memory mechanism occurs due to a martensitic transformation between the B2-ordered high temperature austenitic parent phase and low-temperature martensite with monoclinic B19′ structure. During heating of deformed TiNi beyond a critical temperature, the shape reverts to the undeformed state of the parent austenite. Precipitation of intermetallic phases during controlled heat treatments allows for adjustment of the transformation behavior [2], [3], [4], [5] and contributes to strengthening of the material. The processing of TiNi alloys typically includes solution annealing and subsequent aging, where diffusional transformations involving intermetallic phases occur in the following order with increasing aging temperature and time: metastable Ti3Ni4 → metastable Ti2Ni3 → equilibrium TiNi3 [6]. That is, at lower aging temperature and shorter aging time, the Ti3Ni4 phase appears, while at higher aging temperature and longer aging time, the TiNi3 phase appears and, at intermediate temperature and time, Ti2Ni3 phase precipitates. Koskimaki et al. [7] detected a metastable phase in aged Ti–54.4 at.%Ni, which they called “X-phase”. At that time, the atomic constitution in the intermetallic phase was not clear. Saburi et al. [8] described the crystal structure, the morphology and the chemistry of the metastable “X-phase” as having a rhombohedral structure with space group R-3. The unit cell contains six titanium atoms and eight nickel atoms (Ti3Ni4) with a lattice parameters a = 0.6704 nm and α = 113.85°. Tadaki et al. [9] revealed the structure at the same time independently. The metastable phase equilibrium between the TiNi matrix and the Ti3Ni4 phase has been investigated in detail and structural and thermodynamic data are given in Refs. [10], [11], [12], [13]. Nishida et al. [14] first confirmed the composition Ti2Ni3 of a previously unascertained intermediate phase next to Ti3Ni4. Hara et al. [15] studied the structure of Ti2Ni3 and observed that the precipitate Ti2Ni3 exists in two different structures as a function of temperature, and one form transforms diffusionless into the other as a function of temperature. Ti2Ni3 is orthorhombic at low temperatures and tetragonal at high temperatures [14], [15]. Hara et al. [15] analyzed X-ray diffractograms of aged Ti–52.0 at.%Ni during heating from 298 K and 373 K and determined space group, lattice parameters and atomic coordinates.
In the present paper, we use DFT analysis to study the enthalpy of formation of the two Ti2Ni3 modifications. This study is part of a project aimed at the optimization of a multi-component thermodynamic CALPHAD-database for kinetic phase transformation simulations in shape memory alloys. In order to assure internal consistency of the enthalpies of formation from DFT, which is important as they are used as key data for the optimization of thermodynamic model parameters, we also re-evaluate enthalpies of formation at 0 K for the B2, Ti3Ni4 and Ti3Ni phases.
Section snippets
Computational details
First principles total energy calculations are carried out with the Vienna Ab-Initio Simulation Package (VASP 5.2.8) [16], [17], [18]. The calculations are performed in the density functional theory (DFT) context using the projector augmented wave (PAW) method in the generalized gradient approximation (GGA) with the Perdew, Burke and Ernzerhof (PBE) [19] exchange correlation functional. The p-electrons of Ti (default energy cut-off value is 222.338 eV for Ti) and Ni (default energy cut-off value
DFT calculation of the Ti2Ni3 phase
The DFT refined full structure and the enthalpy of formation, ΔHf, of the low-temperature Ti2Ni3-L and high-temperature Ti2Ni3-H are presented in the following Tables.
Discussion
The converged unit cell structures for both modifications of Ti2Ni3 are in good agreement with experiment, as shown in Table 2, Table 4. The enthalpy of formation of Ti2Ni3-L is 0.004 eV/atom (386 J/mol) lower than Ti2Ni3-H, lying inside the accuracy of our high precision DFT calculations. This indicates that the orthorhombic structure is energetically slightly more favourable than the tetragonal structure, and Ti2Ni3-L (orthorhombic structure) is thermodynamically stabilised at low temperatures.
Conclusion
Enthalpies of formation at 0 K are re-evaluated for the TiNi, Ti3Ni4 and TiNi3 phases using DFT calculations. The results agree well with previously published data, assuring internal consistency of our DFT calculations of the Ti2Ni3 phase. Energy calculations determine the crystal structure parameters of the orthorhombic low-temperature and the tetragonal high-temperature Ti2Ni3 phase, respectively. DFT evaluated atomic coordinates in the orthorhombic and tetragonal unit cell structures of the Ti
Acknowledgements
This research was supported in part by the Austrian FWF under the Projects No. P24681-N20 and within the SFB ViCoM (F41). The computational results presented have been achieved [in part] using the Vienna Scientific Cluster (VSC-1).
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