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Martensitic transformation between competing phases in Ti–Ta alloys: a solid-state nudged elastic band study
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 115401 (http://iopscience.iop.org/0953-8984/27/11/115401) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 115401 (8pp)
doi:10.1088/0953-8984/27/11/115401
Martensitic transformation between competing phases in Ti–Ta alloys: a solid-state nudged elastic band study Tanmoy Chakraborty, Jutta Rogal and Ralf Drautz Interdisciplinary Centre for Advanced Materials Simulation, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany E-mail: [email protected] Received 10 November 2014, revised 19 January 2015 Accepted for publication 11 February 2015 Published 4 March 2015 Abstract
A combined density functional theory and solid-state nudged elastic band study is presented to investigate the martensitic transformation between β → (α , ω) phases in the Ti–Ta system. The minimum energy paths along the transformation are calculated and the transformation mechanisms as well as relative stabilities of the different phases are discussed for various compositions. The analysis of the transformation paths is complemented by calculations of phonon spectra to determine the dynamical stability of the β, α , and ω phase. Our theoretical results confirm the experimental findings that with increasing Ta concentration there is a competition between the destabilisation of the α and ω phase and the stabilisation of the high-temperature β phase. Keywords: Ti–Ta alloys, density functional theory, solid-state nudged elastic band, martensitic transformation, phonon calculations, shape memory alloys (Some figures may appear in colour only in the online journal)
for a stable shape memory effect at high temperatures therefore depends on the competition between the transformation from the β to the α and ω phase. Only few theoretical studies have investigated the structure and stability of the involved phases in Ti–Ta from first-principles calculations. Earlier studies focussed on the structural and elastic properties of the β phase only [7, 8]. The relative stability of the α , α , β, and ω phases in binary Ti alloys was recently investigated using density functional theory (DFT) calculations [9]. The composition range in this study was restricted to low concentrations and it was shown that for a low Ta content the ω phase is indeed more stable than the α phase which is in agreement with experimental observations of the ω phase for low solute concentrations. In the study by Li et al [10] Ti-X (X = V, Nb, Ta, Mo) solid solutions were modelled within the coherent potential approximation (CPA) [11, 12]. They reported lattice parameters as well as the relative stability of the β, α, and α phases in a concentration range of 0–30 at.% in these binary Ti alloys. For both Ti–Ta and Ti–Nb the calculated trend in lattice parameters is in good agreement with experimental data [6, 13–16]. They
1. Introduction
Alloying titanium with elements that stabilise the bodycentered cubic (BCC) β phase, such as vanadium, niobium, molybdenum and tantalum, facilitates a reversible martensitic transformation between the high-temperature β phase and the orthorhombic α phase. Recently, Buenconsejo et al [1] have studied the shape memory effect in Ti–Ta over a composition range of 30–40 at.% Ta showing a stable, reversible β ↔ α transition with a martensitic transformation start temperature of Ms = 440 K for 32 at.% Ta. This rather high value of Ms makes Ti–Ta attractive as a base system for the development of new high-temperature shape memory alloys (HTSMA) [1–4]. The shape memory effect can, however, be destroyed by the formation of the ω phase (having either a hexagonal or trigonal structure). The formation of the ω phase is observed at high Ti concentrations and can be suppressed by increasing the concentration of β stabilising elements where the addition of Ta was shown to be much more effective than that of Nb [1, 5]. On the other hand it was found that Ms decreases by 30 K per 1 at.% Ta (40 K per 1 at.% Nb) [1, 6]. The optimal alloy composition 0953-8984/15/115401+08$33.00
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J. Phys.: Condens. Matter 27 (2015) 115401
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also find that the relative stability of the β phase increases with increasing Ta content, but up to 30 at.% the α phase is energetically more favourable than the β phase. As discussed by Lazar et al [17] in their study on Ti–Nb alloys the CPA might, however, fail to accurately describe interatomic forces due to mass disorder and the influence of the local environment on the local force constants. The transformation between the β, α and ω phases was not discussed in previous first-principles studies. Here, we use density functional theory calculations in combination with the solid-state nudged elastic band method (SSNEB) [18] to investigate the martensitic transformation β → (α , ω) in binary Ti–Ta alloys over a wide range of compositions. We focus in particular on changes to the minimum energy path (MEP) along the transformation as a function of Ta content and possible changes in the transformation mechanism. The endpoints of the transformation paths represent the β, α , and ω phase at each composition and we carefully investigate their relative stability and structural properties. Our results on the influence of the Ta content on the transformation paths and the difference in formation energies of the β, α and ω phases provide a first step towards a more detailed understanding of the influence that Ta has on Ms and the formation of the ω phase. The paper is organised as follows: in section 2 we describe our computational setup. Section 3 contains the results on the stability of different configurations in the Ti2 Ta β phase which is close to the experimentally identified composition of 32 at.% that shows a stable shape memory effect with high Ms . The transformation pathways between β → α and β → ω obtained using SSNEB calculations as well as the change in formation energies are discussed in section 4. Phonon calculations to assess the dynamical stability of the three phases are presented in section 5. We summarise our results in section 6.
within 4 meV/atom. The optimised lattice vectors of the β and ω phases were obtained by fitting energy versus volume curves with Murnaghan’s equation of state. For the hexagonal structures the c/a ratio was optimised for each volume. For the orthorhombic α phase lattice vectors were optimised by allowing a full relaxation of the cell shape and volume. Unless stated otherwise atomic positions were fully relaxed until all forces were below 10−3 eV Å−1 . 2.2. Solid-state nudged elastic band calculations
For the calculation of the minimum energy paths along the transformation between different phases in the Ti–Ta system, we employed the generalised solid-state nudged elastic band [18] method as implemented in the VTST package [24] for VASP. Within the SSNEB method atomic and cell degrees of freedom (DOF) are simultaneously relaxed during the minimisation. It is thus possible to determine the MEP between structures with different unit cells which is essential to describe the phase transformations in the Ti–Ta system. The initial path was created by a linear interpolation of cell vectors and atomic positions between the initial and final state and requires a one-to-one mapping between the atoms in each structure. For all calculations five images were used between the initial and final state. The structures of the end points of the SSNEB calculations, i.e. the β, α , and ω phase at the corresponding compositions, were fully relaxed with respect to atomic positions and all cell vectors were optimised unless indicated otherwise. The MEPs were relaxed until the forces on each image were below 10−3 eV Å−1 . 2.3. Phonon calculations
All phonon calculations were performed using the supercell approach with the small displacement method [25] as implemented in the PHONOPY [26] code. Forces were obtained from VASP calculations. The k-point density in the DFT calculations was increased to ensure accurate force constants. The k-point meshes used for each supercell are given in the corresponding section. The number of unique displacements for the different structures are determined by PHONOPY depending on the crystal symmetry. The magnitude of the atomic displacements was kept at the default value of 0.01 Å. The convergence of phonon frequencies with respect to supercell size was checked and supercells containing up to 128 atoms were considered.
2. Computational details 2.1. Density functional theory calculations
The density functional theory calculations were performed with the Vienna ab initio Simulation Package (VASP) [19–21]. In all calculations the projector augmented wave (PAW) method [21, 22] was used including the 3p electrons for Ti and 5p electrons for Ta in the valence configuration. The Perdew–Burke–Ernzerhof gradient corrected approximation to exchange and correlation (PBE-GGA) [23] was employed and a minimum cutoff of 225 eV for the plane waves was applied. For an accurate calculation of the phonon frequencies the cutoff was increased to 250 eV. Test calculations with a cutoff of 500 eV showed only numerically insignificant changes. Unless noted otherwise the Methfessel–Paxton scheme with a smearing of σ = 0.05 eV was used to integrate the k-space. A large k-point density is needed to obtain accurate energies and forces and the k-point meshes were adjusted for each supercell, respectively. The k-point meshes are explicitly mentioned for each supercell in the corresponding sections. Within our computational setup total energy differences are converged to
3. Stability of Ti2 Ta β phase
Experimentally Ti–Ta alloys are found to form solid solutions at low temperature. This agrees with the small ordering energies that we predict from our calculations. In the following we use the lowest energy configurations from our calculations to analyse phase stability, and also transformation paths and phonons at low temperature. We checked with a few calculations that our results do not depend strongly on the order, but did not investigate disordered configurations systematically, for example, with special quasirandom structures [27] or a cluster expansion approach [28]. 2
J. Phys.: Condens. Matter 27 (2015) 115401
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The effect of the order on the energetics of different ordered structures has been discussed in great detail for the Ti3 Nb β phase and it was shown that all symmetry inequivalent configuration within a 2×2×2 supercell lie within 12 meV/atom [17]. For Ti–Ta the most promising composition for HTSMA applications contains around 32 at.% Ta and we therefore focus here on the influence of different configurations on the energetics in the Ti2 Ta β phase. We use a 3 × 3 × 3 supercell containing 54 atoms (36 Ti and 18 Ta atoms) and a [6 × 6 × 6] k-point mesh. Within such a supercell the number of possible, symmetry inequivalent configurations is by far too large to allow for the calculation of all configurations using DFT. A more complete assessment of possible configurations would require much faster energy evaluations, e.g. within a cluster expansion approach [28] as discussed for the Ti3 Nb β phase [17], but this is beyond the scope of the work presented here. We nevertheless would like to obtain a good idea about the energetics of different configurations. We therefore set up a total number of 81 symmetry inequivalent configurations in the 3×3×3 supercell and allow for relaxation of the cell shape and volume. For this set of structures the atomic positions were not relaxed in order to keep the overall body-centered lattice intact and a smearing of σ = 0.2 eV was used for the Methfessel– Paxton integration scheme. The 81 different configurations comprise structures with different numbers of Ti and Ta atoms in each layer in [0 0 1] direction. One configuration with 9 Ti atoms in layer A, 9 Ta atoms in layer B and a layer sequence AABAAB was included. 10 configurations were created with 6 Ta and 3 Ti atoms in layer A, 9 Ti atoms in layer B and a ABABAB layer sequence. The remaining 70 configurations contained 6 Ti and 3 Ta atoms in each layer in [0 0 1] direction with a random layer stacking. Similarly as it was observed for Ti3 Nb small tetragonal and orthorhombic distortions of the cubic cell are found for most configurations but the overall change in volume/atom is small ( 2. We do want to stress that the absolute difference between our results and experimental values for the lattice vectors are small and that we also obtain the correct ordering, i.e. a < c < b. The MEPs along the transformation from β → α from our SSNEB calculations for the different compositions are shown in figure 3. On the y axis we plot the formation energy, Ef , with tot tot (Ti) − yEBCC (Ta), Ef = E tot (Tix Tay ) − xEHCP
4.2. Martensitic transformation β → ω
The low temperature ω phase occurs in two different crystallographic structures in Ti alloys [33]—hexagonal and trigonal. The hexagonal structure has the space group P6/mmm with three atoms per unit cell. The Wyckoff positions of the atoms are given by (0,0,0), (2/3,1/3,1/2) and (1/3,2/3,1/2), and c/a ≈ 0.62. The orientation relationship between the β and ω phase is schematically shown in figure 4. The lattice vector correspondence between these two phases is given by [1 1 1] β || [0 0 0 1] ω, [1 1¯ 0] β || [1 0 0 0] ω, and ¯ β || [0 1 0 0] ω. As shown in figure 4 the distance [0 1 1] between √ two layers in the [1 1 1] direction in the β phase is aBCC 3/6. In the ω√phase two layers collapse by moving the atoms by ±aBCC √ 3/12. The idealised lattice √ vectors of 3/2 with an the ω phase are aω = 2aBCC and cω = aBCC √ interlayer distance in [0 0 0 1] direction of aBCC 3/4. The trigonal ω phase only forms √ when the displacement of the atomic layers is less than aBCC 3/2. For calculations of the ω phase with 25 and 75 at.% Ta we used a 2 × 2 × 1 supercell (with 12 atoms), for 0, 33 and 100 at.% Ta a 1×1×1 supercell (with 3 atoms), and for 50 at.% Ta a 2 × 1 × 1 supercell (with 6 atoms). The respective k-point meshes for each supercell are [6 × 6 × 20], [12 × 12 × 20], and [6 × 12 × 20] and all structures were fully relaxed. Again we chose one representative configuration for each composition
(1) 4
J. Phys.: Condens. Matter 27 (2015) 115401
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Table 2. Optimised lattice constants (in Å), ratios of lattice vectors, volume/atom (in Å3 ) as well as the y-parameter of the α phase for different Ta concentration.
Lattice constants (Å) Composition
a0
b0
c0
b0 /a0
c0 /a0
0
y Parameter
Ti Ti–25Ta Ti–33Ta Ti–50Ta Ti–75Ta Ta
3.104 3.266 3.325 3.295 3.274 3.289
4.898 4.776 4.693 4.695 4.703 4.678
4.553 4.462 4.503 4.572 4.696 4.832
1.578 1.462 1.411 1.425 1.436 1.422
1.467 1.366 1.354 1.388 1.434 1.469
17.30 17.41 17.57 17.69 18.08 18.59
0.200 0.200 0.215 0.225 0.250 0.200
Note: For elemental Ti and Ta the y parameter was not relaxed. 140
Table 3. Optimised lattice constants (in Å), c/a ratio and volume/atom for the ω phase in Ti–Ta binary alloys with different compositions.
120 Pure Ti
Ef (meV/atom)
100
Composition
a0
c0
c0 /a0
0
Ti Ti–25Ta Ti–33Ta Ti–50Ta Ti–75Ta Ta
4.579 4.651 4.673 4.723 4.809 4.872
2.829 2.799 2.787 2.770 2.761 2.728
0.618 0.602 0.596 0.586 0.574 0.560
17.13 17.48 17.57 17.84 18.43 18.70
25% Ta 80 33% Ta 60 40
50% Ta 75% Ta
20 Pure Ta 0
250 -20 β
Pure Ta Reaction coordinate
α"
200
75% Ta
Figure 3. Minimum energy path between β and α phase for Ef (meV/atom)
different Ta concentrations.
150 50% Ta
100
33% Ta 50
25% Ta
0 Pure Ti -50 β
Reaction coordinate
ω
Figure 5. Minimum energy path for the transformation β → ω for different Ta concentrations.
The optimised lattice parameters for these configurations of the ω phase are compiled in table 3 as a function of Ta content. As for the β and α phase 0 increases with increasing Ta content. The c/a ratio slightly decreases with Ta concentration, as a increases and c decreases. The changes in lattice vectors as compared to the β phase are moderate and thus also 0 is very comparable for the β and ω phase at the same composition. In general we observe that a increases from β → ω and c decreases. The MEPs for the β → ω transformation obtained from our SSNEB calculations are shown in figure 5. We again plot the formation energy, Ef , as defined in equation (1) along the transformation paths. Similar as observed for the β → α transformation the stability of the β phase increases whereas the stability of the ω phase decreases with increasing Ta content. For 25 at.% Ta the formation energy of the ω
Figure 4. Schematic representation of the orientation relationship between β and ω phase.
and check that changes in volume and energy are small for different configurations. For 25 at.% the 3 Ta atoms are placed at (0,0,0), (0,1/2,0) and (1/2,0,0) in the 2 × 2 × 1 supercell, and for 75 at.% Ta the 3 Ti atoms are placed at these positions. Since for 33 at.% the configuration discussed in section 3 could not be realised in a small supercell we here placed the Ta atom at (0,0,0) in the 1×1×1 supercell. In the 2×1×1 supercell for 50 at.% the 3 Ta atoms are at (0,0,0), (0,1/2,0) and (1/6,2/3,1/3) in the β phase. 5
J. Phys.: Condens. Matter 27 (2015) 115401
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hexagonal α phase is observed which is not included in our study. The composition range from 20–45 at.% Ta appears to be promising for a stable shape memory effect as the α phase is favoured over the ω phase at low temperatures. Furthermore, we expect that Ms will be higher for compositions that show a more negative energy difference between β and α at 0 K. From the fit of our data points in figure 6 we find that α −β increases by 2 meV/at.% Ta (α is destabilised) which Ef corresponds to a temperature of T = 24 K/at.% Ta. This is in good agreement with the experimental observation that Ms decreases by 30 K/at.% Ta [1].
300 ∆Ef
∆Ef (meV/atom)
200
∆Ef
α"-β ω-β
∆T/at.% Ta = 42 K
100
∆T/at.% Ta = 24 K 0
-100
-200 0
20
40
60
80
100
5. Phonons of β , α and ω phases at T = 0 K
at.% Ta
Figure 6. Difference in formation energy between β and α (black, circles), respectively β and ω (red, squares), as a function of Ta concentration. The solid lines mark a linear fit to the data points.
Since we do not find significant barriers along the MEPs for any of the compositions during the β → (α , ω) transformations we will discuss the dynamical stability at T = 0 K of the end-points of the SSNEB calculations in more detail. The phonon band structure can be used to evaluate the dynamical stability of a crystal structure as for a stable structure all phonon frequencies must be real. For the Ti–Ta system we have calculated the phonon frequencies for the three phases in each composition. Our results were obtained within the harmonic approximation. For the β phase with 25, 50 and 75 at.% a 4 × 4 × 4 supercell with 128 atoms was used to calculate the force constants. For the 33 at.% Ta we used a 6×3×3 supercell with 108 atoms that represents the configuration shown in figure 1. A 3 × 3 × 3 supercell was used for the α phase force constant calculations with 25, 50 and 75 at.% of Ta whereas a 4 × 3 × 2 supercell was used for 33 at.% Ta. For the ω phase a 4×4×2 supercell was used for 25 and 75 at.% Ta and a 3×3×3 and 4 × 2 × 2 supercell for 33 and 50 at.% Ta, respectively. To maintain the same configurational ordering as for the SSNEB calculations the supercells used in the phonon calculations are multiples of the smaller cells discussed in section 4. The phonon band structures and density of states (DOS) are shown in figure 7. As discussed before the β phase is stable only for high Ta concentrations and α and ω for low Ta concentrations. For the β phase imaginary phonon frequencies are observed up to 50 at.% Ta. For 50 at.% Ta (green lines in figure 7) the negative part is, however, very small, indicating the increasing stability. We also calculated the phonon spectra for different configurations in the 50 at.% Ta β phase but no significant changes were observed. For all tested configurations a small number of imaginary phonon frequencies was found. The phonon band structures of the α phase for 25 and 33 at.% Ta look very similar and clearly show the dynamical stability of the α in this composition range. For larger concentrations imaginary frequencies are observed in [ξ ξ ξ ] and [ξ ξ 0] direction, respectively. Also the phonon band structures for α in Ti–Ta are similar to the ones reported for Ti3 Nb [17]. For the α phase a clear shift towards higher phonon frequencies is observed with a decrease of the heavier Ta. The ω phase is dynamically stable only for 25 at.% Ta. At 33 at.% Ta an imaginary frequency is observed in [ξ ξ 0].
phase is 26 meV/atom lower than of the β phase and also lower than of α (see figure 3) which is consistent with the experimental observation of ω phase formation for low solute concentrations. With increasing Ta concentration the ω phase is, however, much stronger destabilised (52 meV/atom less stable than the β phase at 50 at.% Ta) which is in agreement with the experimental finding that the addition of Ta suppresses the formation of the ω phase. For 33 at.% both the ω and α phase are more stable than the β phase, but also the α is more stable than the ω phase. This supports the experimental observation that for Ta concentrations around 30 at.% a reversible martensitic transformation between β → α is possible without the formation of the athermal ω phase [1]. For the β → ω transformation we also do not observe any significant energy barriers along the MEPs, only for 33 at.% a small barrier of ∼7 meV/atom is found. The transformation shows a shuffle of atomic planes without notable changes in the lattice vectors. 4.3. Trends in formation energies
As the β phase is the high temperature phase a stable martensitic transformation will not be observed for any composition where the β phase is more stable than the α phase at T = 0 K. To discuss the change in relative stability of the different phases we show in figure 6 the difference in formation energy between β and α as well as β and ω as a function of Ta content, where α −β
Ef
β
= Efα − Ef ,
(2)
for each composition. For low Ta concentrations the ω phase is most stable but is then much stronger destabilised than the α phase with increasing Ta concentration. We find that up to ∼20 at.% Ta the ω phase is the most stable phase, from 20–45 at.% Ta the α phase and for larger Ta concentrations the β phase. Our estimate of the stability range of the β and α phase is in reasonable agreement with experimental findings [14, 15, 34]. For low Ta concentrations usually the 6
J. Phys.: Condens. Matter 27 (2015) 115401
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Ti-25Ta [ξ −ξ ξ] [0 0 ξ]
8
Ti-33Ta [ξ ξ ξ]
[ξ −ξ ξ] [0 0 ξ]
Ti-50Ta [ξ ξ ξ]
[ξ −ξ ξ] [0 0 ξ]
Ti-75Ta [ξ ξ ξ]
[ξ −ξ ξ] [0 0 ξ]
DOS [ξ ξ ξ]
β
Frequency (THz)
6 4 2 0 -2 -4
Γ
Frequency (THz)
N
Γ
PΓ
[ξ ξ 0] [ξ 0 0]
[ξ ξ ξ]
8
α"
H
H [ξ ξ ξ]
N
Γ
PΓ
[ξ ξ 0] [ξ 0 0]
H
N
[ξ ξ ξ]
Γ
PΓ
[ξ ξ 0] [ξ 0 0]
H
N
[ξ ξ ξ]
Γ
P0
0.5 1 1.5 DOS/atom (1/THz)
[ξ ξ 0] [ξ 0 0]
6 4 2 0 Γ
R [ξ ξ ξ]
S
XR
[ξ ξ 0][ξ 0 0]
Γ
Γ
S XR
[ξ ξ ξ] [ξ ξ 0][ξ 0 0]
[ξ ξ ξ]
S
XR
[ξ ξ 0][ξ 0 0]
Γ [ξ ξ ξ]
S
X 0
0.5 1 1.5 DOS/atom (1/THz)
[ξ ξ 0][ξ 0 0]
ω
Frequency (THz)
8 6 4 2 0 -2 -4
H
Γ
K MH
Γ
Γ
K MH
K
MH
Γ
K M0
0.5 1 1.5 DOS/atom (1/THz)
Figure 7. Phonon band structure and DOS of the β (top), α (middle) and ω (bottom) phase for 25 at.% (black), 33 at.% (red), 50 at.% (green) and 75 at.% (blue) Ta content (from left to right, respectively) in the binary Ti–Ta alloy.
As observed from the formation energies the ω phase is more strongly destabilised with increasing Ta content than the α phase. For 50 and 75 at.% Ta a considerable contribution to the imaginary phonon frequencies can be seen in the DOS. As for the α phase a shift towards higher phonon frequencies is observed with decreasing Ta content. The phonon dispersions shown in figure 7 were calculated for T = 0 K, while the stability of the different structures can be influenced significantly by finite temperatures. For Ti3 Nb it was shown that the negative phonon frequencies of the β phase disappear already at a temperature of T = 300 K [17]. In Ti–Ta a stabilisation of the different, unstable structures is also expected at finite temperatures. A direct evaluation of the phonon frequencies at finite temperature would require e.g. ab initio molecular dynamics simulations, which are computationally very demanding, or the determination of the phonon band structure in a self-consistent manner [35]. These finite temperature effects constitute a future step in our investigation of the Ti–Ta system. The phonon calculations confirm the trend in stability for the β, α and ω phase as a function of composition inferred from our SSNEB calculations. At T = 0 K the β and α /ω phases are only stable within limited composition ranges. The dynamical instability observed for the β phase
up to 50 at.% Ta is at variance with previous reports of elastic constants [7, 8] where the stability criteria of the cubic phase were already fulfilled for only 18.75 at.% Ta. We also calculated the elastic constants of our structures and found mechanical stability of the β phase for compositions with less than 50 at.% Ta. Similarly, our elastic constant calculations indicate stable α and ω phases even for high Ta concentrations. The discrepancy between the results of the phonon and elastic constant calculations concerning the stability of the different phases is an artefact of the computational setup, i.e. to observe the dynamical instability correctly in the elastic constants larger supercells would be required. 6. Summary
We have investigated the martensitic transformation paths between β → (α , ω) in the Ti–Ta system over a wide range of compositions based on DFT calculations and the SSNEB approach. The application of the SSNEB method is essential as it allows us to change cell and atomic degrees of freedom in a consistent framework along the MEP. We find that the stability of the different phases strongly depends on Ta concentration. With increasing Ta content the ω phase is more rapidly destabilised than the α phase which is in 7
J. Phys.: Condens. Matter 27 (2015) 115401
T Chakraborty et al
agreement with the experimental findings that the addition of Ta suppresses the formation of the ω phase in Ti–Ta alloys. The stabilisation of the β phase with increasing Ta content leads, however, to a decrease in Ms . The change in Ms with increasing Ta concentration is reflected in the difference in formation energies between the β and α phase. We found that the energy difference decreases by about 2 meV/at.% Ta which corresponds to a temperature change of T = 24 K/at.% Ta. The trends in stability of the different phases observed in our SSNEB calculations were further supported by calculations of phonon spectra for all compositions and we find clear differences in the dynamical stability at T = 0 K. Our results confirm the hypothesis that a suitable range of Ta concentrations to enable a stable shape memory effect may be found as a trade-off between the destabilisation of the ω phase, which destroys the shape memory effect, and a stabilisation of the β phase, which lowers Ms . From our theoretical prediction, a composition between 25 and 33 at.% Ta appears most suitable.
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Acknowledgments
Financial support by Deutsche Forschungsgemeinschaft within TP3 of the Research Unit FOR1766 is gratefully acknowledged. References [1] Buenconsejo P J S, Kim H Y, Hosoda H and Miyazaki S 2009 Acta Mater. 57 1068 [2] Ma J, Karaman I and Noebe R D 2010 Int. Mater. Rev. 55 257 [3] Buenconsejo P J S, Kim H Y and Miyazaki S 2009 Acta Mater. 57 2509 [4] Buenconsejo P J S, Kim H Y and Miyazaki S 2011 Scr. Mater. 64 1114 [5] Kim H Y, Hashimoto S, Kim J I, Inamura T, Hosoda H and Miyazaki S 2006 Mater. Sci. Eng. A 417 120 [6] Kim H Y, Ikehara Y, Kim J I, Hosoda H and Miyazaki S 2006 Acta Mater. 54 2419 [7] Ikehata H, Nagasako N, Furuta T, Fukumoto A, Miwa K and Saito T 2004 Phys. Rev. B 70 174113
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Martensitic transformation between competing phases in Ti–Ta alloys: a solid-state nudged elastic band study
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 115401 (8pp)
doi:10.1088/0953-8984/27/11/115401
Martensitic transformation between competing phases in Ti–Ta alloys: a solid-state nudged elastic band study Tanmoy Chakraborty, Jutta Rogal and Ralf Drautz Interdisciplinary Centre for Advanced Materials Simulation, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany E-mail: [email protected] Received 10 November 2014, revised 19 January 2015 Accepted for publication 11 February 2015 Published 4 March 2015 Abstract
A combined density functional theory and solid-state nudged elastic band study is presented to investigate the martensitic transformation between β → (α , ω) phases in the Ti–Ta system. The minimum energy paths along the transformation are calculated and the transformation mechanisms as well as relative stabilities of the different phases are discussed for various compositions. The analysis of the transformation paths is complemented by calculations of phonon spectra to determine the dynamical stability of the β, α , and ω phase. Our theoretical results confirm the experimental findings that with increasing Ta concentration there is a competition between the destabilisation of the α and ω phase and the stabilisation of the high-temperature β phase. Keywords: Ti–Ta alloys, density functional theory, solid-state nudged elastic band, martensitic transformation, phonon calculations, shape memory alloys (Some figures may appear in colour only in the online journal)
for a stable shape memory effect at high temperatures therefore depends on the competition between the transformation from the β to the α and ω phase. Only few theoretical studies have investigated the structure and stability of the involved phases in Ti–Ta from first-principles calculations. Earlier studies focussed on the structural and elastic properties of the β phase only [7, 8]. The relative stability of the α , α , β, and ω phases in binary Ti alloys was recently investigated using density functional theory (DFT) calculations [9]. The composition range in this study was restricted to low concentrations and it was shown that for a low Ta content the ω phase is indeed more stable than the α phase which is in agreement with experimental observations of the ω phase for low solute concentrations. In the study by Li et al [10] Ti-X (X = V, Nb, Ta, Mo) solid solutions were modelled within the coherent potential approximation (CPA) [11, 12]. They reported lattice parameters as well as the relative stability of the β, α, and α phases in a concentration range of 0–30 at.% in these binary Ti alloys. For both Ti–Ta and Ti–Nb the calculated trend in lattice parameters is in good agreement with experimental data [6, 13–16]. They
1. Introduction
Alloying titanium with elements that stabilise the bodycentered cubic (BCC) β phase, such as vanadium, niobium, molybdenum and tantalum, facilitates a reversible martensitic transformation between the high-temperature β phase and the orthorhombic α phase. Recently, Buenconsejo et al [1] have studied the shape memory effect in Ti–Ta over a composition range of 30–40 at.% Ta showing a stable, reversible β ↔ α transition with a martensitic transformation start temperature of Ms = 440 K for 32 at.% Ta. This rather high value of Ms makes Ti–Ta attractive as a base system for the development of new high-temperature shape memory alloys (HTSMA) [1–4]. The shape memory effect can, however, be destroyed by the formation of the ω phase (having either a hexagonal or trigonal structure). The formation of the ω phase is observed at high Ti concentrations and can be suppressed by increasing the concentration of β stabilising elements where the addition of Ta was shown to be much more effective than that of Nb [1, 5]. On the other hand it was found that Ms decreases by 30 K per 1 at.% Ta (40 K per 1 at.% Nb) [1, 6]. The optimal alloy composition 0953-8984/15/115401+08$33.00
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also find that the relative stability of the β phase increases with increasing Ta content, but up to 30 at.% the α phase is energetically more favourable than the β phase. As discussed by Lazar et al [17] in their study on Ti–Nb alloys the CPA might, however, fail to accurately describe interatomic forces due to mass disorder and the influence of the local environment on the local force constants. The transformation between the β, α and ω phases was not discussed in previous first-principles studies. Here, we use density functional theory calculations in combination with the solid-state nudged elastic band method (SSNEB) [18] to investigate the martensitic transformation β → (α , ω) in binary Ti–Ta alloys over a wide range of compositions. We focus in particular on changes to the minimum energy path (MEP) along the transformation as a function of Ta content and possible changes in the transformation mechanism. The endpoints of the transformation paths represent the β, α , and ω phase at each composition and we carefully investigate their relative stability and structural properties. Our results on the influence of the Ta content on the transformation paths and the difference in formation energies of the β, α and ω phases provide a first step towards a more detailed understanding of the influence that Ta has on Ms and the formation of the ω phase. The paper is organised as follows: in section 2 we describe our computational setup. Section 3 contains the results on the stability of different configurations in the Ti2 Ta β phase which is close to the experimentally identified composition of 32 at.% that shows a stable shape memory effect with high Ms . The transformation pathways between β → α and β → ω obtained using SSNEB calculations as well as the change in formation energies are discussed in section 4. Phonon calculations to assess the dynamical stability of the three phases are presented in section 5. We summarise our results in section 6.
within 4 meV/atom. The optimised lattice vectors of the β and ω phases were obtained by fitting energy versus volume curves with Murnaghan’s equation of state. For the hexagonal structures the c/a ratio was optimised for each volume. For the orthorhombic α phase lattice vectors were optimised by allowing a full relaxation of the cell shape and volume. Unless stated otherwise atomic positions were fully relaxed until all forces were below 10−3 eV Å−1 . 2.2. Solid-state nudged elastic band calculations
For the calculation of the minimum energy paths along the transformation between different phases in the Ti–Ta system, we employed the generalised solid-state nudged elastic band [18] method as implemented in the VTST package [24] for VASP. Within the SSNEB method atomic and cell degrees of freedom (DOF) are simultaneously relaxed during the minimisation. It is thus possible to determine the MEP between structures with different unit cells which is essential to describe the phase transformations in the Ti–Ta system. The initial path was created by a linear interpolation of cell vectors and atomic positions between the initial and final state and requires a one-to-one mapping between the atoms in each structure. For all calculations five images were used between the initial and final state. The structures of the end points of the SSNEB calculations, i.e. the β, α , and ω phase at the corresponding compositions, were fully relaxed with respect to atomic positions and all cell vectors were optimised unless indicated otherwise. The MEPs were relaxed until the forces on each image were below 10−3 eV Å−1 . 2.3. Phonon calculations
All phonon calculations were performed using the supercell approach with the small displacement method [25] as implemented in the PHONOPY [26] code. Forces were obtained from VASP calculations. The k-point density in the DFT calculations was increased to ensure accurate force constants. The k-point meshes used for each supercell are given in the corresponding section. The number of unique displacements for the different structures are determined by PHONOPY depending on the crystal symmetry. The magnitude of the atomic displacements was kept at the default value of 0.01 Å. The convergence of phonon frequencies with respect to supercell size was checked and supercells containing up to 128 atoms were considered.
2. Computational details 2.1. Density functional theory calculations
The density functional theory calculations were performed with the Vienna ab initio Simulation Package (VASP) [19–21]. In all calculations the projector augmented wave (PAW) method [21, 22] was used including the 3p electrons for Ti and 5p electrons for Ta in the valence configuration. The Perdew–Burke–Ernzerhof gradient corrected approximation to exchange and correlation (PBE-GGA) [23] was employed and a minimum cutoff of 225 eV for the plane waves was applied. For an accurate calculation of the phonon frequencies the cutoff was increased to 250 eV. Test calculations with a cutoff of 500 eV showed only numerically insignificant changes. Unless noted otherwise the Methfessel–Paxton scheme with a smearing of σ = 0.05 eV was used to integrate the k-space. A large k-point density is needed to obtain accurate energies and forces and the k-point meshes were adjusted for each supercell, respectively. The k-point meshes are explicitly mentioned for each supercell in the corresponding sections. Within our computational setup total energy differences are converged to
3. Stability of Ti2 Ta β phase
Experimentally Ti–Ta alloys are found to form solid solutions at low temperature. This agrees with the small ordering energies that we predict from our calculations. In the following we use the lowest energy configurations from our calculations to analyse phase stability, and also transformation paths and phonons at low temperature. We checked with a few calculations that our results do not depend strongly on the order, but did not investigate disordered configurations systematically, for example, with special quasirandom structures [27] or a cluster expansion approach [28]. 2
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The effect of the order on the energetics of different ordered structures has been discussed in great detail for the Ti3 Nb β phase and it was shown that all symmetry inequivalent configuration within a 2×2×2 supercell lie within 12 meV/atom [17]. For Ti–Ta the most promising composition for HTSMA applications contains around 32 at.% Ta and we therefore focus here on the influence of different configurations on the energetics in the Ti2 Ta β phase. We use a 3 × 3 × 3 supercell containing 54 atoms (36 Ti and 18 Ta atoms) and a [6 × 6 × 6] k-point mesh. Within such a supercell the number of possible, symmetry inequivalent configurations is by far too large to allow for the calculation of all configurations using DFT. A more complete assessment of possible configurations would require much faster energy evaluations, e.g. within a cluster expansion approach [28] as discussed for the Ti3 Nb β phase [17], but this is beyond the scope of the work presented here. We nevertheless would like to obtain a good idea about the energetics of different configurations. We therefore set up a total number of 81 symmetry inequivalent configurations in the 3×3×3 supercell and allow for relaxation of the cell shape and volume. For this set of structures the atomic positions were not relaxed in order to keep the overall body-centered lattice intact and a smearing of σ = 0.2 eV was used for the Methfessel– Paxton integration scheme. The 81 different configurations comprise structures with different numbers of Ti and Ta atoms in each layer in [0 0 1] direction. One configuration with 9 Ti atoms in layer A, 9 Ta atoms in layer B and a layer sequence AABAAB was included. 10 configurations were created with 6 Ta and 3 Ti atoms in layer A, 9 Ti atoms in layer B and a ABABAB layer sequence. The remaining 70 configurations contained 6 Ti and 3 Ta atoms in each layer in [0 0 1] direction with a random layer stacking. Similarly as it was observed for Ti3 Nb small tetragonal and orthorhombic distortions of the cubic cell are found for most configurations but the overall change in volume/atom is small ( 2. We do want to stress that the absolute difference between our results and experimental values for the lattice vectors are small and that we also obtain the correct ordering, i.e. a < c < b. The MEPs along the transformation from β → α from our SSNEB calculations for the different compositions are shown in figure 3. On the y axis we plot the formation energy, Ef , with tot tot (Ti) − yEBCC (Ta), Ef = E tot (Tix Tay ) − xEHCP
4.2. Martensitic transformation β → ω
The low temperature ω phase occurs in two different crystallographic structures in Ti alloys [33]—hexagonal and trigonal. The hexagonal structure has the space group P6/mmm with three atoms per unit cell. The Wyckoff positions of the atoms are given by (0,0,0), (2/3,1/3,1/2) and (1/3,2/3,1/2), and c/a ≈ 0.62. The orientation relationship between the β and ω phase is schematically shown in figure 4. The lattice vector correspondence between these two phases is given by [1 1 1] β || [0 0 0 1] ω, [1 1¯ 0] β || [1 0 0 0] ω, and ¯ β || [0 1 0 0] ω. As shown in figure 4 the distance [0 1 1] between √ two layers in the [1 1 1] direction in the β phase is aBCC 3/6. In the ω√phase two layers collapse by moving the atoms by ±aBCC √ 3/12. The idealised lattice √ vectors of 3/2 with an the ω phase are aω = 2aBCC and cω = aBCC √ interlayer distance in [0 0 0 1] direction of aBCC 3/4. The trigonal ω phase only forms √ when the displacement of the atomic layers is less than aBCC 3/2. For calculations of the ω phase with 25 and 75 at.% Ta we used a 2 × 2 × 1 supercell (with 12 atoms), for 0, 33 and 100 at.% Ta a 1×1×1 supercell (with 3 atoms), and for 50 at.% Ta a 2 × 1 × 1 supercell (with 6 atoms). The respective k-point meshes for each supercell are [6 × 6 × 20], [12 × 12 × 20], and [6 × 12 × 20] and all structures were fully relaxed. Again we chose one representative configuration for each composition
(1) 4
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Table 2. Optimised lattice constants (in Å), ratios of lattice vectors, volume/atom (in Å3 ) as well as the y-parameter of the α phase for different Ta concentration.
Lattice constants (Å) Composition
a0
b0
c0
b0 /a0
c0 /a0
0
y Parameter
Ti Ti–25Ta Ti–33Ta Ti–50Ta Ti–75Ta Ta
3.104 3.266 3.325 3.295 3.274 3.289
4.898 4.776 4.693 4.695 4.703 4.678
4.553 4.462 4.503 4.572 4.696 4.832
1.578 1.462 1.411 1.425 1.436 1.422
1.467 1.366 1.354 1.388 1.434 1.469
17.30 17.41 17.57 17.69 18.08 18.59
0.200 0.200 0.215 0.225 0.250 0.200
Note: For elemental Ti and Ta the y parameter was not relaxed. 140
Table 3. Optimised lattice constants (in Å), c/a ratio and volume/atom for the ω phase in Ti–Ta binary alloys with different compositions.
120 Pure Ti
Ef (meV/atom)
100
Composition
a0
c0
c0 /a0
0
Ti Ti–25Ta Ti–33Ta Ti–50Ta Ti–75Ta Ta
4.579 4.651 4.673 4.723 4.809 4.872
2.829 2.799 2.787 2.770 2.761 2.728
0.618 0.602 0.596 0.586 0.574 0.560
17.13 17.48 17.57 17.84 18.43 18.70
25% Ta 80 33% Ta 60 40
50% Ta 75% Ta
20 Pure Ta 0
250 -20 β
Pure Ta Reaction coordinate
α"
200
75% Ta
Figure 3. Minimum energy path between β and α phase for Ef (meV/atom)
different Ta concentrations.
150 50% Ta
100
33% Ta 50
25% Ta
0 Pure Ti -50 β
Reaction coordinate
ω
Figure 5. Minimum energy path for the transformation β → ω for different Ta concentrations.
The optimised lattice parameters for these configurations of the ω phase are compiled in table 3 as a function of Ta content. As for the β and α phase 0 increases with increasing Ta content. The c/a ratio slightly decreases with Ta concentration, as a increases and c decreases. The changes in lattice vectors as compared to the β phase are moderate and thus also 0 is very comparable for the β and ω phase at the same composition. In general we observe that a increases from β → ω and c decreases. The MEPs for the β → ω transformation obtained from our SSNEB calculations are shown in figure 5. We again plot the formation energy, Ef , as defined in equation (1) along the transformation paths. Similar as observed for the β → α transformation the stability of the β phase increases whereas the stability of the ω phase decreases with increasing Ta content. For 25 at.% Ta the formation energy of the ω
Figure 4. Schematic representation of the orientation relationship between β and ω phase.
and check that changes in volume and energy are small for different configurations. For 25 at.% the 3 Ta atoms are placed at (0,0,0), (0,1/2,0) and (1/2,0,0) in the 2 × 2 × 1 supercell, and for 75 at.% Ta the 3 Ti atoms are placed at these positions. Since for 33 at.% the configuration discussed in section 3 could not be realised in a small supercell we here placed the Ta atom at (0,0,0) in the 1×1×1 supercell. In the 2×1×1 supercell for 50 at.% the 3 Ta atoms are at (0,0,0), (0,1/2,0) and (1/6,2/3,1/3) in the β phase. 5
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hexagonal α phase is observed which is not included in our study. The composition range from 20–45 at.% Ta appears to be promising for a stable shape memory effect as the α phase is favoured over the ω phase at low temperatures. Furthermore, we expect that Ms will be higher for compositions that show a more negative energy difference between β and α at 0 K. From the fit of our data points in figure 6 we find that α −β increases by 2 meV/at.% Ta (α is destabilised) which Ef corresponds to a temperature of T = 24 K/at.% Ta. This is in good agreement with the experimental observation that Ms decreases by 30 K/at.% Ta [1].
300 ∆Ef
∆Ef (meV/atom)
200
∆Ef
α"-β ω-β
∆T/at.% Ta = 42 K
100
∆T/at.% Ta = 24 K 0
-100
-200 0
20
40
60
80
100
5. Phonons of β , α and ω phases at T = 0 K
at.% Ta
Figure 6. Difference in formation energy between β and α (black, circles), respectively β and ω (red, squares), as a function of Ta concentration. The solid lines mark a linear fit to the data points.
Since we do not find significant barriers along the MEPs for any of the compositions during the β → (α , ω) transformations we will discuss the dynamical stability at T = 0 K of the end-points of the SSNEB calculations in more detail. The phonon band structure can be used to evaluate the dynamical stability of a crystal structure as for a stable structure all phonon frequencies must be real. For the Ti–Ta system we have calculated the phonon frequencies for the three phases in each composition. Our results were obtained within the harmonic approximation. For the β phase with 25, 50 and 75 at.% a 4 × 4 × 4 supercell with 128 atoms was used to calculate the force constants. For the 33 at.% Ta we used a 6×3×3 supercell with 108 atoms that represents the configuration shown in figure 1. A 3 × 3 × 3 supercell was used for the α phase force constant calculations with 25, 50 and 75 at.% of Ta whereas a 4 × 3 × 2 supercell was used for 33 at.% Ta. For the ω phase a 4×4×2 supercell was used for 25 and 75 at.% Ta and a 3×3×3 and 4 × 2 × 2 supercell for 33 and 50 at.% Ta, respectively. To maintain the same configurational ordering as for the SSNEB calculations the supercells used in the phonon calculations are multiples of the smaller cells discussed in section 4. The phonon band structures and density of states (DOS) are shown in figure 7. As discussed before the β phase is stable only for high Ta concentrations and α and ω for low Ta concentrations. For the β phase imaginary phonon frequencies are observed up to 50 at.% Ta. For 50 at.% Ta (green lines in figure 7) the negative part is, however, very small, indicating the increasing stability. We also calculated the phonon spectra for different configurations in the 50 at.% Ta β phase but no significant changes were observed. For all tested configurations a small number of imaginary phonon frequencies was found. The phonon band structures of the α phase for 25 and 33 at.% Ta look very similar and clearly show the dynamical stability of the α in this composition range. For larger concentrations imaginary frequencies are observed in [ξ ξ ξ ] and [ξ ξ 0] direction, respectively. Also the phonon band structures for α in Ti–Ta are similar to the ones reported for Ti3 Nb [17]. For the α phase a clear shift towards higher phonon frequencies is observed with a decrease of the heavier Ta. The ω phase is dynamically stable only for 25 at.% Ta. At 33 at.% Ta an imaginary frequency is observed in [ξ ξ 0].
phase is 26 meV/atom lower than of the β phase and also lower than of α (see figure 3) which is consistent with the experimental observation of ω phase formation for low solute concentrations. With increasing Ta concentration the ω phase is, however, much stronger destabilised (52 meV/atom less stable than the β phase at 50 at.% Ta) which is in agreement with the experimental finding that the addition of Ta suppresses the formation of the ω phase. For 33 at.% both the ω and α phase are more stable than the β phase, but also the α is more stable than the ω phase. This supports the experimental observation that for Ta concentrations around 30 at.% a reversible martensitic transformation between β → α is possible without the formation of the athermal ω phase [1]. For the β → ω transformation we also do not observe any significant energy barriers along the MEPs, only for 33 at.% a small barrier of ∼7 meV/atom is found. The transformation shows a shuffle of atomic planes without notable changes in the lattice vectors. 4.3. Trends in formation energies
As the β phase is the high temperature phase a stable martensitic transformation will not be observed for any composition where the β phase is more stable than the α phase at T = 0 K. To discuss the change in relative stability of the different phases we show in figure 6 the difference in formation energy between β and α as well as β and ω as a function of Ta content, where α −β
Ef
β
= Efα − Ef ,
(2)
for each composition. For low Ta concentrations the ω phase is most stable but is then much stronger destabilised than the α phase with increasing Ta concentration. We find that up to ∼20 at.% Ta the ω phase is the most stable phase, from 20–45 at.% Ta the α phase and for larger Ta concentrations the β phase. Our estimate of the stability range of the β and α phase is in reasonable agreement with experimental findings [14, 15, 34]. For low Ta concentrations usually the 6
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Ti-25Ta [ξ −ξ ξ] [0 0 ξ]
8
Ti-33Ta [ξ ξ ξ]
[ξ −ξ ξ] [0 0 ξ]
Ti-50Ta [ξ ξ ξ]
[ξ −ξ ξ] [0 0 ξ]
Ti-75Ta [ξ ξ ξ]
[ξ −ξ ξ] [0 0 ξ]
DOS [ξ ξ ξ]
β
Frequency (THz)
6 4 2 0 -2 -4
Γ
Frequency (THz)
N
Γ
PΓ
[ξ ξ 0] [ξ 0 0]
[ξ ξ ξ]
8
α"
H
H [ξ ξ ξ]
N
Γ
PΓ
[ξ ξ 0] [ξ 0 0]
H
N
[ξ ξ ξ]
Γ
PΓ
[ξ ξ 0] [ξ 0 0]
H
N
[ξ ξ ξ]
Γ
P0
0.5 1 1.5 DOS/atom (1/THz)
[ξ ξ 0] [ξ 0 0]
6 4 2 0 Γ
R [ξ ξ ξ]
S
XR
[ξ ξ 0][ξ 0 0]
Γ
Γ
S XR
[ξ ξ ξ] [ξ ξ 0][ξ 0 0]
[ξ ξ ξ]
S
XR
[ξ ξ 0][ξ 0 0]
Γ [ξ ξ ξ]
S
X 0
0.5 1 1.5 DOS/atom (1/THz)
[ξ ξ 0][ξ 0 0]
ω
Frequency (THz)
8 6 4 2 0 -2 -4
H
Γ
K MH
Γ
Γ
K MH
K
MH
Γ
K M0
0.5 1 1.5 DOS/atom (1/THz)
Figure 7. Phonon band structure and DOS of the β (top), α (middle) and ω (bottom) phase for 25 at.% (black), 33 at.% (red), 50 at.% (green) and 75 at.% (blue) Ta content (from left to right, respectively) in the binary Ti–Ta alloy.
As observed from the formation energies the ω phase is more strongly destabilised with increasing Ta content than the α phase. For 50 and 75 at.% Ta a considerable contribution to the imaginary phonon frequencies can be seen in the DOS. As for the α phase a shift towards higher phonon frequencies is observed with decreasing Ta content. The phonon dispersions shown in figure 7 were calculated for T = 0 K, while the stability of the different structures can be influenced significantly by finite temperatures. For Ti3 Nb it was shown that the negative phonon frequencies of the β phase disappear already at a temperature of T = 300 K [17]. In Ti–Ta a stabilisation of the different, unstable structures is also expected at finite temperatures. A direct evaluation of the phonon frequencies at finite temperature would require e.g. ab initio molecular dynamics simulations, which are computationally very demanding, or the determination of the phonon band structure in a self-consistent manner [35]. These finite temperature effects constitute a future step in our investigation of the Ti–Ta system. The phonon calculations confirm the trend in stability for the β, α and ω phase as a function of composition inferred from our SSNEB calculations. At T = 0 K the β and α /ω phases are only stable within limited composition ranges. The dynamical instability observed for the β phase
up to 50 at.% Ta is at variance with previous reports of elastic constants [7, 8] where the stability criteria of the cubic phase were already fulfilled for only 18.75 at.% Ta. We also calculated the elastic constants of our structures and found mechanical stability of the β phase for compositions with less than 50 at.% Ta. Similarly, our elastic constant calculations indicate stable α and ω phases even for high Ta concentrations. The discrepancy between the results of the phonon and elastic constant calculations concerning the stability of the different phases is an artefact of the computational setup, i.e. to observe the dynamical instability correctly in the elastic constants larger supercells would be required. 6. Summary
We have investigated the martensitic transformation paths between β → (α , ω) in the Ti–Ta system over a wide range of compositions based on DFT calculations and the SSNEB approach. The application of the SSNEB method is essential as it allows us to change cell and atomic degrees of freedom in a consistent framework along the MEP. We find that the stability of the different phases strongly depends on Ta concentration. With increasing Ta content the ω phase is more rapidly destabilised than the α phase which is in 7
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agreement with the experimental findings that the addition of Ta suppresses the formation of the ω phase in Ti–Ta alloys. The stabilisation of the β phase with increasing Ta content leads, however, to a decrease in Ms . The change in Ms with increasing Ta concentration is reflected in the difference in formation energies between the β and α phase. We found that the energy difference decreases by about 2 meV/at.% Ta which corresponds to a temperature change of T = 24 K/at.% Ta. The trends in stability of the different phases observed in our SSNEB calculations were further supported by calculations of phonon spectra for all compositions and we find clear differences in the dynamical stability at T = 0 K. Our results confirm the hypothesis that a suitable range of Ta concentrations to enable a stable shape memory effect may be found as a trade-off between the destabilisation of the ω phase, which destroys the shape memory effect, and a stabilisation of the β phase, which lowers Ms . From our theoretical prediction, a composition between 25 and 33 at.% Ta appears most suitable.
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Acknowledgments
Financial support by Deutsche Forschungsgemeinschaft within TP3 of the Research Unit FOR1766 is gratefully acknowledged. References [1] Buenconsejo P J S, Kim H Y, Hosoda H and Miyazaki S 2009 Acta Mater. 57 1068 [2] Ma J, Karaman I and Noebe R D 2010 Int. Mater. Rev. 55 257 [3] Buenconsejo P J S, Kim H Y and Miyazaki S 2009 Acta Mater. 57 2509 [4] Buenconsejo P J S, Kim H Y and Miyazaki S 2011 Scr. Mater. 64 1114 [5] Kim H Y, Hashimoto S, Kim J I, Inamura T, Hosoda H and Miyazaki S 2006 Mater. Sci. Eng. A 417 120 [6] Kim H Y, Ikehara Y, Kim J I, Hosoda H and Miyazaki S 2006 Acta Mater. 54 2419 [7] Ikehata H, Nagasako N, Furuta T, Fukumoto A, Miwa K and Saito T 2004 Phys. Rev. B 70 174113
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