vacancy-induced omega phase fluctuations in
body-centered cubic Ti and Zr

Sally Issa, Guy Tréglia, Hakim Amara, Céline Varvenne

► To cite this version:

Sally Issa, Guy Tréglia, Hakim Amara, Céline Varvenne. vacancy-induced omega phase fluctua-
tions in body-centered cubic Ti and Zr. Computational Materials Science, 2025, 260, pp.114251.
10.1016/j.commatsci.2025.114251. hal-04827146v2

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Vacancy-induced omega phase fluctuations in body-centered cubic Ti and Zr

Sally Issaª, Guy Trégliaª, Hakim Amaraᵇ,ᶜ, Céline Varvenneᵈ,\*

ªAix-Marseille Univ., CNRS, CINaM, Marseille, France

ᵇUniversité Paris-Saclay, ONERA, CNRS, LEM, Châtillon, France

ᶜUniversité Paris Cité, MPQ, CNRS-UMR7162, Paris, France

ᵈCNRS, INSA Lyon, Université Claude Bernard Lyon 1, MATEIS, UMR5510, 69621 Villeurbanne, France

**Abstract**

The precipitation of $\omega$ phase in body-centered cubic (bcc) metastable Ti and Zr alloys can occur athermally, with a typical high density of homogeneously distributed $\omega$ particles. We propose in this work that vacancies can act as an important source of $\omega$-phase fluctuations in bcc Ti and Zr matrices, which are mechanically unstable at zero temperature but dynamically stabilized at finite T. Using ab initio calculations we highlight the similarities between the relaxation field of the vacancy in a bcc mechanically unstable matrix and the LA 2/3[111] phonon mode, through a detailed analysis of the local atomic and electronic structure around the vacancy. Then, empirical potential simulations suggest a complex vacancy structure at finite temperature, alternating over time between standard cubic and linear $\omega$-like vacancy configurations. The linear, symmetry-broken, vacancy configuration involves the relaxation of many atoms, and is very close to the best structural model explaining the characteristic of the diffuse scattering distribution observed experimentally in many bcc Ti and Zr alloys. We therefore suggest that the nucleation of athermal $\omega$ precipitates could occur heterogeneously on thermal vacancies. This scenario is consistent with the spatial homogeneity of athermal $\omega$ particles seen in TEM images, and with the estimate of particle density; it also suggests a way to reduce this $\omega$ precipitation.

**Keywords:** titanium, zirconium, mechanical instability, vacancy, omega transformation, atomistic calculations

**1. Introduction**

The Mendeleev periodic table reveals, for the three series of transition metals, a crystallographic transition of the hcp to the bcc structure between group IV (Ti, Zr, Hf) and group V (V, Nb, Ta) elements, indicating a competition between these two structures for these electronic $d$ band fillings [1–3]. Thus, pure Ti and Zr elements, stable in the hcp structure at zero pressure and room temperature, are also found in a bcc metastable phase at elevated temperature. Another hexagonal phase, the so-called $\omega$ phase (see Fig. 1), is stabilized under pressure [4–6]. Competition among crystallographic structures is thus important in group IV metals [7, 8]. Alloying is a way to further stabilize the bcc phase, like in several Ti-X and Zr-X binary alloys [9, 10] or in the rather recent class of refractory high entropy alloys [2, 11], but can also promote the appearance of $\omega$ phase under thermal treatments [5, 6, 10]. These $\omega$ phase precipitates may contribute to an increased alloy brittleness [10]. Precipitation modes for the bcc $\rightarrow \omega$ transformation are usually categorized into athermal $\omega$ and isothermal $\omega$, if the $\omega$ precipitates appear upon rapid quenching or during thermal ageing, respectively. In the

former case, the transformation is diffusionless, in the displacive sense, and the resulting microstructure is made of a very high density of ellipsoidal particles with major axis lying along one of the $\langle 111 \rangle_{\text{bcc}}$ directions, and of very small dimensions [6, 12, 13]. As no clear correlation has been seen between the location of athermal $\omega$ precipitates and the presence of lattice heterogeneities such as grain boundaries or twins, this led the community to believe that athermal $\omega$ phase generation occurs rather homogeneously.

Precise understanding of these high density and homogeneous behaviors must be found in the early stages of the phase transformation. Pre-transition effects have been observed experimentally, in particular a phonon softening of the longitudinal axial (LA) 2/3 $\langle 111 \rangle$ mode [6, 14–16] that is structurally related to the collapse of $\{111\}_{\text{bcc}}$ planes, a global mechanism leading to the $\omega$ phase. More recently, the phonon softening at finite temperature has also been evidenced theoretically [17] in bcc Ti and Zr matrices. In close connection with this collapse mechanism, the presence of embryonic $\omega$ structures [6] has been suggested to explain the local starting of $\omega$ phase formation. Two structural models are discussed in the literature for the $\omega$-nuclei: (i) a four-atoms plane trigonal configuration corresponding usually to the saddle-point configuration of the vacancy migrating along the bcc first nearest neigh-

<footnote>\*Corresponding author
Email address: celine.varvenne@cnrs.fr (Céline Varvenne)</footnote>
bor path [6, 18, 19], and (ii) linear $\omega$-like configurations
in the $\langle 111 \rangle_{\text{bcc}}$ directions, described either as an $\omega$ mod-
ulation [20] or as a rigid one-dimensional shift of a single
atomic row [21]. However, both geometries present some
incompatibilities with the detailed morphology of the ob-
served shifts in the diffuse intensity maxima in Ti and
Zr bcc alloys [22]. Consequently, the precise geometrical
configurations of embryonic $\omega$ structures and mechanisms
leading to the athermal $\omega$ precipitation still remain elusive,
and require additional knowledge.

The present work aims at gaining more insights on the
possibility to start athermal $\omega$ phase transformation from
vacancies, a point defect naturally present in materials.
Using a combination of ab initio and empirical potential
atomistic calculations, we show that a vacancy placed on
a bcc atomic position in mechanically unstable bcc Ti and
Zr matrices creates very large displacement fields, leading
to a full destabilization of the bcc lattice towards $\omega$ phase.
Finite temperature simulations, on the other hand, sug-
gest the presence of a complex linear $\omega$-like vacancy struc-
ture, which better meets the requirements for diffuse scat-
tering measurements than the previous structural models,
and could act as an heterogeneous nucleation site for the
athermal $\omega$ precipitation. The remainder of the paper is
organized as follows: section 2 presents the details of our
atomistic simulations and the simulated bulk Ti and Zr
properties, section 3 is dedicated to a zero temperature
analysis, and section 4 is devoted to the study of the va-
cancy in bcc Ti and Zr at a temperature typical of that
used for the quenching of $\beta$-metastable Ti and Zr alloys,
and to a discussion of implications for $\omega$ precipitation.

## 2. Atomistic calculation details

Ab initio calculations are based on the density func-
tional theory (DFT), and are performed using the PWSCF
code of the QUANTUM ESPRESSO suite [41, 42]. The gener-
alized gradient approximation as implemented by Perdew-
Burke-Ernzerhof (GGA-PBE) is chosen for the exchange-
correlation functional. A plane wave basis set is used to
develop wave functions, with a 50 Ry (680 eV) energy cut-
off. Core electrons are treated with the projector aug-
mented waves (PAW) method. For both elements, four
valence electrons and eight semi-core electrons are consid-
ered, i.e. [Ar] 4s$^2$4p$^0$3d$^2$ with (3s, 3p) as semi-core for Ti,
and [Kr] 5s$^2$5p$^0$4d$^2$ with (4s, 4p) as semi-core for Zr. The
electronic density of state is sampled with the Methfessel-
Paxton function and a smearing of 0.15 eV. The integra-
tion is performed with a regular grid of 20 × 20 × 20 k-
points for the bcc conventional cell, and of 20 × 20 × 16
and 20 × 20 × 14 k-points for the hcp and $\omega$ conventional
cells, respectively. An equivalent density of k-points is
applied for supercell defect calculations. The number of
k-points for electronic density of states will be specified
later when needed.

Table 1 provides Ti and Zr lattice and elastic constants
at T=0 K for the bcc, hcp and $\omega$ bulk phases, all appear-

ing in the pressure-temperature phase diagrams of the pure
metals. Cohesive energy differences $\Delta E_{\text{hcp}}^{\theta} = E_{\text{coh}}^{\theta} - E_{\text{coh}}^{\text{hcp}}$,
for $\theta = \text{bcc}$ and $\omega$ are also displayed. Schemes of the $\omega$
and hcp structures are provided in Fig. 1. Two inequiva-
lent layers are present in the $\omega$ structure, and the corre-
sponding representative atoms are labelled \#A and \#B in
the following. The first nearest neighbor distance equals
$a_{\text{bcc}}\sqrt{3}/2$ for the bcc phase, $a_{\text{hcp}}$ for the hcp phase, and
$a_{\omega}/\sqrt{3}$ for the $\omega$ phase. Our ab initio calculations, as
compared to both low-temperature experiments and other
ab initio calculations, provide an overall good description
of lattice parameters and elastic constants $C_{ij}$. The hcp
phase is predicted in our calculations to be more stable
than the bcc phase for both Ti and Zr, with a larger $\Delta E_{\text{hcp}}^{\text{bcc}}$
value for Ti. The $\omega$ and hcp phases, within the precision
of our calculations, are nearly degenerated for both Ti and
Zr ($\omega$ slightly more stable for Ti). The optimized $c/a$ ra-
tio of the $\omega$ phase deviates somewhat from the ideal value
$c/a = \sqrt{3}/8 \simeq 0.612$ for both elements, in agreement with
previous literature [9, 43–45]. We finally note that, as ex-
pected, the bcc structure displays some mechanical insta-
bilities at zero temperature for both Ti and Zr [23, 30, 46],
i.e. $2C' = C_{11} - C_{12} < 0$ associated to the bcc→hcp phase
transformation, and $G_{111} = 3C_{44}C'/(2C_{44} + C') < 0$, cor-
related to the bcc→ $\omega$ phase transformation.

As ab initio calculations are typically limited to a
few hundred atoms at zero temperature, and to a few
tens of atoms and of picoseconds for finite-temperature
simulations, we also select two interatomic potentials
of embedded-atom-method (EAM) type, developed by
Mendelev and Ackland for Ti and Zr. These potentials
are labeled as Ti1 in Ref. [32], and Zr\#2 in Ref. [31], re-
spectively. Both potentials were designed to describe the
single element phase transitions, with a focus on the hcp-
bcc transition for EAM Zr\#2 and on both hcp-bcc and
hcp-$\omega$ transitions for EAM Ti1. Typically, transition tem-
peratures, latent heats and volume changes were target
data for those potentials. As seen in Table 1, the EAM
Zr\#2 potential gives accurate lattice and elastic constants
for the hcp, bcc and $\omega$ phases, and predicts the correct

<figure><img src="image_2.png" /><figcaption>Figure 1: Projection in the basal plane of the hcp (left) and $\omega$ (right) structures. Colored spheres represent atoms located at $z = 0$, and gray spheres atoms at $z = c/2$. The hcp structure has two equivalent atomic layers, and 2 atoms in its conventional cell. The $\omega$ structure has two nonequivalent atomic layers \#A and \#B, and 3 atoms in its conventional cell.</figcaption></figure>
<table><caption>Table 1: Ti and Zr bulk properties: lattice parameters *a* (in Å), *c/a* ratios, elastic constants *C*<sub>ij</sub> (in GPa) and cohesive energy differences Δ*E*<sup>θ</sup><sub>hcp</sub> = *E*<sup>θ</sup><sub>coh</sub> − *E*<sup>hcp</sup><sub>coh</sub>, for θ = bcc and ω (in meV/at). For bcc structures, 2*C'* = *C*<sub>11</sub> − *C*<sub>12</sub> and *G*<sub>111</sub> = 3*C*<sub>44</sub>*C'*/(2*C*<sub>44</sub> + *C'*') [23]. Ab initio values correspond to this work and Refs. [24–30]. Empirical potential calculations use the EAM Zr#2 and Ti1 potentials developed by Mendelev and Ackland [31, 32]. Low temperature experimental values are from Refs. [33–36], and high temperature values in bcc matrices are from Refs. [16, 37, 38].</caption><thead><tr><th>Metal</th><th></th><th>a</th><th>c/a</th><th>C<sub>11</sub></th><th>C<sub>12</sub></th><th>C<sub>13</sub></th><th>C<sub>33</sub></th><th>C<sub>44</sub></th><th>B</th><th>2C'</th><th>G<sub>111</sub></th><th>ΔE<sup>θ</sup><sub>hcp</sub></th></tr></thead><tbody><tr><td rowspan="5">Ti</td><td rowspan="4">bcc</td><td>PWSCF, this work</td><td>3.258</td><td>92</td><td>114</td><td></td><td></td><td>40</td><td>107</td><td>-22</td><td>-19</td><td>109</td></tr><tr><td>CASTEP [30]</td><td>3.258</td><td>90</td><td>113</td><td></td><td></td><td>41</td><td>105</td><td>-23</td><td>-20</td><td></td></tr><tr><td>EAM Ti1</td><td>3.251</td><td>309</td><td>307</td><td></td><td></td><td>29</td><td>308</td><td>1</td><td>1.5</td><td>29</td></tr><tr><td>Exp. [36, 37]</td><td>3.264</td><td>99</td><td>85</td><td></td><td></td><td>34</td><td>90</td><td>14</td><td>17</td><td></td></tr><tr><td rowspan="4">hcp</td><td>PWSCF, this work</td><td>2.942</td><td>1.583</td><td>169</td><td>87</td><td>76</td><td>187</td><td>42</td><td>111</td><td></td><td></td><td></td></tr><tr><td>PWSCF US 12e [27]</td><td>2.936</td><td>1.583</td><td>169</td><td>89</td><td>77</td><td>192</td><td>43</td><td>113</td><td></td><td></td><td></td></tr><tr><td>EAM Ti1</td><td>2.947</td><td>1.597</td><td>161</td><td>80</td><td>86</td><td>169</td><td>53</td><td>107</td><td></td><td></td><td></td></tr><tr><td>Exp. [34]</td><td>2.951</td><td>1.585</td><td>176</td><td>97</td><td>68</td><td>191</td><td>51</td><td>112</td><td></td><td></td><td></td></tr><tr><td rowspan="3">ω</td><td>PWSCF, this work</td><td>4.584</td><td>0.619</td><td>195</td><td>82</td><td>51</td><td>241</td><td>53</td><td>111</td><td></td><td></td><td>-5</td></tr><tr><td>EAM Ti1</td><td>4.727</td><td>0.618</td><td>194</td><td>43</td><td>40</td><td>114</td><td>86</td><td>93</td><td></td><td></td><td>5</td></tr><tr><td>Exp. [4, 39]</td><td>4.625</td><td>0.608</td><td>201</td><td>61</td><td>77</td><td>178</td><td>56</td><td>112</td><td></td><td></td><td></td></tr><tr><td rowspan="5">Zr</td><td rowspan="4">bcc</td><td>PWSCF, this work</td><td>3.574</td><td>81</td><td>91</td><td></td><td></td><td></td><td>31</td><td>88</td><td>-10</td><td>-8</td><td>84</td></tr><tr><td>CASTEP [30]</td><td>3.572</td><td>83</td><td>91</td><td></td><td></td><td></td><td>32</td><td>88</td><td>-8</td><td>-6.5</td><td></td></tr><tr><td>EAM #2</td><td>3.562</td><td>96</td><td>109</td><td></td><td></td><td></td><td>42</td><td>105</td><td>-13</td><td>-11</td><td>52</td></tr><tr><td>Exp. [16, 38]</td><td>3.615</td><td>104</td><td>93</td><td></td><td></td><td></td><td>38</td><td>97</td><td>12</td><td>15</td><td></td></tr><tr><td rowspan="5">hcp</td><td>PWSCF, this work</td><td>3.233</td><td>1.600</td><td>141</td><td>68</td><td>66</td><td>166</td><td>26</td><td>94</td><td></td><td></td><td></td></tr><tr><td>PWSCF US 12e [26]</td><td>3.230</td><td>1.601</td><td>140</td><td>70</td><td>65</td><td>168</td><td>26</td><td>93</td><td></td><td></td><td></td></tr><tr><td>VASP PAW[29]</td><td>3.235</td><td>1.598</td><td>147</td><td>70</td><td>71</td><td>164</td><td>25</td><td>96</td><td></td><td></td><td></td></tr><tr><td>EAM #2</td><td>3.220</td><td>1.619</td><td>149</td><td>82</td><td>65</td><td>180</td><td>48</td><td>104</td><td></td><td></td><td></td></tr><tr><td>Exp. [33, 35]</td><td>3.232</td><td>1.603</td><td>155</td><td>67</td><td>65</td><td>172</td><td>36</td><td>96</td><td></td><td></td><td></td></tr><tr><td rowspan="4">ω</td><td>PWSCF this work</td><td>5.048</td><td>0.623</td><td>165</td><td>74</td><td>46</td><td>199</td><td>33</td><td>96</td><td></td><td></td><td>0</td></tr><tr><td>VASP [25]</td><td>5.036</td><td>0.626</td><td>162</td><td>73</td><td>54</td><td>196</td><td>34</td><td>97</td><td></td><td></td><td></td></tr><tr><td>EAM #2</td><td>5.160</td><td>0.600</td><td>157</td><td>59</td><td>51</td><td>191</td><td>47</td><td>92</td><td></td><td></td><td>35</td></tr><tr><td>Exp. [4, 40]</td><td>5.039</td><td>0.658</td><td></td><td></td><td></td><td></td><td></td><td>104</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>5.036</td><td>0.617</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

hcp ground state. However, the hcp stabilization with respect to the bcc phase is slightly underestimated, and the ω phase falls between the hcp and bcc phases, while ab initio calculations indicate it should be very close to the hcp phase. The EAM Ti1 potential gives good lattice constants for all phases, good elastic constants for the hcp and ω phases, but the T=0 K elastic constants of the bcc phase are not well reproduced. EAM Ti1 describes well the similar stabilities of hcp and ω phases for Ti, but underestimates Δ*E*<sup>bcc</sup><sub>hcp</sub> for the bcc phase as compared to ab initio results. We also note that the mechanical instabilities of the bcc phases are well reproduced for EAM Zr#2, and acceptable for EAM Ti1 (i.e. *C'* and *G*<sub>111</sub> ≃ 0). While not perfect, these two interaction models provide a reasonable description of the various allotropic phases for both elements, and will be useful in the paper to discuss finite-size and temperature effects on the vacancy structure and behavior in the bcc phases.

Finally, finite-temperature molecular dynamics (MD) calculations are performed using the open-source package LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [47], and using empirical potential. Other

MD simulation details will be specified later, when necessary.

### 3. Analysis at zero temperature

In this section, we examine the modifications induced by a vacancy on the potential energy landscape (PES) and structure of bulk Ti and Zr at T=0 K. At zero temperature, bcc matrices for both chemical species are mechanically unstable. This means that the underlying bcc structure will be preserved when applying certain phonon modes or displacement fields followed by atomic relaxations, and not preserved for others. We perform in the following ab initio calculations for supercell sizes ranging from 2 × 2 × 2 to 5 × 5 × 5 replications of the bcc conventional cell (i.e. having 16-250 atoms in bulk). We analyze the local atomic and electronic structures around the vacancy in these small supercells, and then study the finite-size effects using empirical potentials. This allows us to establish that the static displacement field of the vacancy triggers the instability of the bcc structure towards the ω phase.
### 3.1. Local atomic structure around a vacancy

We first analyze the local atomic relaxations around the vacancy defect in its standard cubic configuration, after ab initio static relaxation of the atoms at constant supercell volume and shape. In Fig. 2(a) the relaxation profiles around the vacancy are shown by the radial displacements $\Delta d_s$ of the s successive vacancy neighbor shells, defined as $\Delta d_s = d_s - d_s^{bulk}$, where $d_s$ the distance between the vacancy and a s-shell atom after atomic relaxation, and $d_s^{bulk}$ the same quantity but in the unrelaxed system. These profiles are calculated for the vacancy in the bcc Ti and Zr matrices and with supercells of 249 atoms.

<figure><img src="image_3.png" /><figcaption>Figure 2: Relaxation profiles around the vacancy in bcc matrices: (a) radial displacements $\Delta d_s$ (in Å) of the s-neighbor shells surrounding the vacancy for both Ti and Zr, (b) scheme showing how the displacements of the nearest and next-nearest neighbors of the vacancy (empty cube) can lead to the formation of an $\omega$-embryo.</figcaption></figure>

Both elements exhibit very similar relaxation profiles, with a strong contraction of the first shell ($s=1$), followed by a dilation of the second shell ($s=2$), and then contractions of the next shells ($s=3,4,5$). The contraction of the first shell is much stronger than typically reported [48–50] for stable bcc metals, i.e. ~13–16% here, to be compared with ~5% for a vacancy in bcc Nb [51], for example. This is in line with the results of Ref. [19], where a large relaxation of ~11.5% is measured for the vacancy in bcc Hf – also a group IV element – though within a smaller supercell. A careful examination of the radial displacements of the 1st and 2nd nearest neighbors (nn) atoms having

positive coordinates indicates that these 4 atoms tend to collapse into a single plane (see red arrows on atoms labeled (1) and (2) in Fig. 2(b)). Notably, this planar trigonal arrangement of atoms could correspond to a $\omega$ phase nucleus, where the bcc-(1) atom would become a #B type atom of the $\omega$ phase (honeycomb plane of the $\omega$ structure, see notations in Figs 1 and 2).

As mentioned in the Introduction, such a nucleus with a planar trigonal configuration has already been mentioned in the literature [14, 18], but as a saddle-point configuration of the vacancy moving along its first nearest neighbor migration path. Static relaxations starting from such a configuration – where cubic symmetry is broken – gives structures in both Ti and Zr that are highly disordered, and deviate from the bcc one at every atomic site, which confirms their distinct nature. When the vacancy is placed in a bcc atomic position like here, the planar trigonal configuration arises directly from the atomic relaxation around the vacancy, and large atomic displacements of atoms not belonging to the small nucleus are involved. Moreover, due to the cubic symmetries that are preserved during the atomic relaxation in our cubic supercells, the vacancy in the relaxed system is surrounded by 8 of these trigonal ‘nuclei’, corresponding to 4 different $\langle 111 \rangle_{\text{bcc}}$ orientation variants of the $\omega$ phase. In addition, for each $\langle 111 \rangle_{\text{bcc}}$ direction, the vacancy connects two translational sub-variants of the $\omega$-phase nuclei, i.e. an initial stacking sequence in the $[111]_{\text{bcc}}$ direction ... ABCABCABC... (with the bold symbol representing the plane where the vacancy is introduced, see Fig. 2(b)) is locally transformed after atomic relaxation into the sequence ... $A\#B'A\square C\#B''C...$, thus involving an ordered sequence of two $\omega$-#B nuclei, shifted in the $[111]_{\text{bcc}}$ direction. This particular behavior is specific to a bcc vacant atomic site, and will be further discussed later on.

A more quantitative characterization of the vacancy-induced structural changes is provided in Fig. 3, where the radial distribution of atomic distances around one of the 1st nn of the vacancy – labeled (1) in Fig. 2(b) and in the following – is represented in the upper panels, before (dark) and after (green) atomic relaxation, for both Zr and Ti. As expected for a bcc matrix, before relaxation the (1)-type atom has seven 1st nn (one 1st nn bond is broken due to the vacancy), and six 2nd nn. For a vacancy in a stable bcc matrix, such as e.g. Nb, the atomic relaxation typically leads to a splitting of these peaks, obtaining two groups of neighboring atoms: the group of 1st nn with multiplicities of (3, 3, 1), and the group of 2nd nn with multiplicities of (3, 3). Here the bcc Ti and Zr matrices are mechanically unstable, and atomic relaxations induce a splitting of the (1)-type atom neighbors (see green shaded areas for the two groups of neighbors in Figs 3). Indeed, the group of 1st nn increases from 7 to 10 atoms, and is split into 4 peaks with (3, 1, 3, 3) multiplicities. The multiplicity order is changed, and there is an additional peak of 3 atoms, corresponding to half of the initial unrelaxed 2nd nearest neighbors that undergo a strong contraction upon relax-
<figure><img src="image_4.png" /><figcaption>Figure 3: Radial distributions of interatomic distances. Top panels: distributions for the (1)-atom in the bcc structure (notations in Fig. 2(b)), before (dark) and after (green) atomic relaxations around the vacancy. Bottom panels: neighbor distributions of the #B-atom in the bulk ω phase (purple) - see Fig.1 for notations, - and of atoms in the bulk hcp phase (red). Shaded areas symbolize first and second groups of neighbors as discussed in the main text.</figcaption></figure>

ation. Note that for Ti, the last two peaks are numerically
very similar, giving a multiplicity close to (3, 1, 6). An
overall contraction around the (1)-type atoms is observed,
creating a locally more densely packed structure. Finally,
the number of 2^nd^ nn of the (1)-type atom seems to be pre-
served, being split into 2 peaks with (3, 3) multiplicities,
but in fact it is formed by 3 atoms coming from the un-
relaxed 2^nd^ nn, that are shifted towards larger distances,
and by 3 additional atoms coming from the 3^rd^ nn of the
initial unrelaxed bcc structure.

Interestingly, for both Ti and Zr bcc matrices the radial
distribution of the (1)-type atom in the vacancy relaxed
state is very close to that of the #B-type atom of the ω
bulk structure shown in Fig. 3: they have fairly similar
distances and neighbor multiplicities. This is consistent
with the plane collapse mechanism displayed in Fig. 2(b).
More specifically, the group of first neighbors (see in Fig. 3
the purple shaded zone at small distances) in the #B-type
atom of the ω phase has multiplicities of (3, 2, 6), which
compares well to that of the (1)-type atom in the relaxed
states, i.e. (3, 1, 3, 3) for Zr and (3,1.6) for Ti. Note that
the multiplicity of the second peak is reduced from 2 for

the #B-type atom in the bulk ω phase to 1 for the (1)-
type atom in the bcc vacancy relaxed configuration, due to
the presence of the vacancy, and as [111]$_{bcc}$ || [0001]$_{ω}$ (see
Fig. 2). Looking now at larger distances for the #B-type
atom in the ω phase, a peak of 6 neighbors is observed
at ≃ 4.3 Å and at ≃ 3.9 Å for Zr and Ti, respectively.
Such peaks are close to the group of 2nd nn of the (1)-
type atoms at larger distance (green shaded areas), and
the total number of atoms is the same. Thus, there is
a general agreement with the ω structure. As a further
check, we provide the radial distribution of the hcp phase
in Fig. 3. The agreement is much less convincing, both in
terms of distances and neighbor multiplicities.

In summary, the atomic arrangements surrounding the
vacancy in relaxed bcc finite-size supercells for Ti and Zr
matrices are compatible with an incipient ω ordering, but
do not correspond to perfect bulk ω structures: (i) the
collapse of atomic planes in ⟨111⟩<sub>bcc</sub> is incomplete so the
trigonal configuration sketched in Fig 2(b) is not fully pla-
nar in our atomistic simulations, and (ii) the bulk ω phase
has other inequivalent atoms and is ordered at large dis-
tance. Thus, the observed local atomic structures around
the vacancy in the supercells are only ω-#B nuclei.

3.2. Local electronic structure analysis

The fingerprint of such ω-#B nuclei should be also visi-
ble in the local electronic structure of the relaxed vacancy
configurations. We thus compute the local densities of
states (LDOS) of *d*-electrons for the successive neighbor
shells surrounding the vacancy in bcc Ti and Zr, before
and after atomic relaxations, and for the same supercells
having 249 atoms (see Fig. 4). *d*-electron DOS of the bulk
bcc phase and of the bulk ω #B-type layers are superim-
posed as needed (see Appendix A for the bulk bcc, hcp
and ω DOS). A peak width of 0.015 eV is applied for the
broadening of the Methfessel-Paxton function, and the *k*-grid
is increased for the Ti matrices : 5 × 5 × 5 for the
vacancy supercell, and 20 × 20 × 20 for the reference bulk
ω structure. In all cases, the Fermi level is set to zero.

Before relaxation and for both Zr and Ti bcc matri-
ces, the LDOS of all vacancy neighbors display the typical
shape of the bulk bcc structures [43, 48, 52]. The vacancy
induces very little perturbations of the electronic struc-
ture, and the LDOS of the vacancy 3rd nearest neighbors
is exactly superimposed to the one of the bulk bcc matri-
ces. The situation strongly differs after atomic relaxation:
the electronic structure is significantly affected by the va-
cancy, and up to rather large distance. The bcc shape is
retreived only from the 3rd nn of the vacancy for Zr, and
from the 5th nn of the vacancy for Ti, and differences still
remain as compared to the bcc bulk LDOS. Very near the
vacancy and for both Zr and Ti matrices, the local den-
sity of *d* electrons is similar to the density in the ω #B-
type atoms. The bandwidth, the large peak at the top of
the band, and the details at lower eigenenergies (occupied
states) are indeed very close, as seen in the superposition
for the 1st nn and 2nd nn of the vacancy in Fig. 4, and
<figure><img src="image_5.png" /><figcaption>Figure 4: LDOS $n(E)$ of d-electrons for the successive neighbor shells of the vacancy in bcc Zr (left panels) and Ti (right panels). Labels (1) to (5) for the first to fifth nearest neighbors are explicated in Fig. 2(b). Energy is centered around the Fermi level. Upper panels correspond to unrelaxed calculations and lower panels to relaxed calculations. DOS of bulk bcc and $\omega$ \#B-type layers are indicated as needed.</figcaption></figure>

recalling that these two atom types belong to an $\omega$-\#B nucleus. Superposition with the DOS of the hcp phase or of the $\omega$ \#A-type atoms was much less satisfactory. The agreement with $\omega$ \#B-type atoms seems better for Zr than for Ti, but this is related to a finite size effect, as shown in Appendix B. An electronic signature of the $\omega$-\#B nuclei is thus obtained after relaxations in both bcc Zr and Ti.

### 3.3. Modification of bulk PES and finite-size effects

The important structural changes seen after adding the vacancy are also reflected in its energetics. We compute the following vacancy energies:

$$E^f(V) = E_{\text{tot}}(N-1) - \frac{N-1}{N} E_{\text{tot}}(N), \quad (1)$$

with $E_{\text{tot}}(N)$ the total energy of the perfect bcc bulk supercell, $E_{\text{tot}}(N-1)$ the total energy of the same system with a vacancy (and relaxing the atomic positions at fixed supercell volume and shape), and $N$ the number of atoms in the $n \times n \times n$ pristine supercell. This definition matches the one of vacancy formation energy for stable bcc crystals; we use it in the mechanically unstable matrices for convenience.

<figure><img src="image_6.png" /><figcaption>Figure 5: Ab initio vacancy energy $E^f(V)$ in bcc Ti (empty circles) and Zr (black squares) versus the number of atoms in the supercells, as defined in Eq. 1. Straight lines correspond to a fitting of ab initio data using the cohesive energy model of Eq. 2.</figcaption></figure>

#### 3.3.1. Ab initio results

Figure 5 shows the resulting ab initio vacancy energies versus supercell size for both bcc Ti and Zr, and for $N = 16, 54, 128$ and $250$ atoms. For both bcc matrices, $E^f$ drastically decreases when increasing the number of atoms in the supercell; the decay is almost linear and $E^f$ values become negative for $N > 54$. This behavior is in contrast with the usual picture seen for a vacancy in a stable crystallographic structure, for which $E^f$ is positive and converges with the supercell size. Such negative values arise through the relaxation of atomic positions: unrelaxed energies $E_0^f$ converge very quickly with the supercell size towards positive values, i.e. 2.10 eV for Ti and 2.16 eV for Zr, in good agreement with the already published ab initio literature [53, 54]. This strong relaxation effect has already been reported in the literature, either in tight-binding [48] or ab initio [19, 53] works for bcc Ti, Zr and Hf, and even described as an unphysical result [53]. Our analysis of the atomic and electronic structure around the vacancy suggests that it is connected to a transitioning towards an $\omega$-like structure. The resulting negative energies should of course not be used in any thermodynamic modeling, unlike what is usually done for defects in stable matrices.

To further support this interpretation for the negative sign and the supercell size-dependence of $E^f(V)$, we build a cohesive energy model. The $\omega$ phase is more stable than the bcc phase for both species, and the cohesive energy differences with the bcc phase are rather significant – i.e. acting as an important driving force. We thus assume that, in a vacancy supercell having $N-1$ atoms, $N_\omega$ atoms transit from the bcc phase to an $\omega$ phase. The energy defined in Eq. 1 is then rewritten as

$$ \begin{aligned} E^f(V) &= (N-1-N_\omega) E_{\text{coh}}^{\text{bcc}} + N_\omega E_{\text{coh}}^\omega - (N-1) E_{\text{coh}}^{\text{bcc}} \\ &= N_\omega \Delta E_{\text{bcc}}^\omega. \end{aligned} $$

With this simplified approach, the case $N_\omega=0$ leads to
$E^f(V) = 0$. As the unrelaxed energies are not null and nearly constant with $N$ in our calculations, we add a constant term as

$$E^f(V) = E_0^f(V) + \alpha_\omega \Delta E_{\text{bcc}}^\omega N, \qquad (2)$$

with $\alpha_\omega = N_\omega/N$ the ratio of atoms having transited from the bcc to the $\omega$ phase. If $\alpha_\omega$ is assumed to be constant with the supercell size, it can be determined from a linear fitting of the $E^f$ values using Eq. 2 and with ab initio values for $\Delta E_{\text{bcc}}^\omega$ and $E_0^f(V)$. The fitting lines are shown in Figure 5: they reproduce well the variations of $E^f$ with $N$. This supports again the transition towards the $\omega$ structure to be responsible for the divergence of $E^f$ with the supercell size. We obtain $\alpha_\omega \simeq 0.40$ for Ti and $\alpha_\omega \simeq 0.25$ for Zr. It is noteworthy that the higher $\alpha_\omega$ value obtained for Ti when fitting Eq. 2 to the ab initio $E^f$ values is consistent with the longer range perturbation of the LDOS due to the vacancy seen for Ti as compared to Zr (see Fig. 4).

### 3.3.2. Empirical potentials to study finite-size effects

Now, whether the $\omega$ phase ‘nuclei’ will remain finite in size or will induce a complete phase transition cannot be clearly established within the constraints of ab initio calculations. Indeed, due to their usual size limitations, the applied periodic boundary conditions (PBC) in simulations induce spurious image forces that hinder the real behavior of the vacancy, that would be observed otherwise in an infinite system. In the following, the empirical potentials presented in section 2, that have a low computational cost, are used to address this aspect.

<figure><img src="image_7.png" /><figcaption>Figure 6: Relaxed vacancy energies $E^f$ in bcc Ti and Zr versus the number $N$ of atoms in the supercell, computed using Zr\#2 and Ti1 EAM potentials [31, 32].</figcaption></figure>

In Fig. 6, vacancy energies in bcc Ti and Zr, computed with the EAM potentials Ti1 and Zr\#2, are displayed for supercell sizes ranging from 16 to 16 000 atoms. Supercell sizes up to 250 000 atoms were computed but do not provide additional information. The typical system size range of our ab initio calculations is indicated for comparison.

As can be seen, and similar to the ab initio results, relaxed vacancy energies become negative when increasing the system size $N$, and do not converge. The decrease is faster for Ti as compared to Zr. Note that both curves show a non-linear behavior if the system size is increased enough. The exact $E^f$ values differ from those of ab initio calculations, due to the cohesive energy differences between hcp, bcc and $\omega$ phases of Zr and Ti elements that are not perfectly reproduced by the two EAM potentials (see Table 1).

Examination of the simulation boxes after atomic relaxation using the common neighbor analysis (CNA) with OVITO [55] indicates that for sufficiently large supercells, almost no bcc atoms remain. Most atoms have transited to another phase, characterized by CNA as hcp for Zr\#2 and as other than hcp, fcc or bcc for Ti1. For intermediate supercell sizes, i.e. having negative $E^f$ and limited phase ‘nuclei’, the atomic structure around the vacancy possesses the same behaviors as in ab initio calculations, for both Zr\#2 and Ti1 potentials: a contraction is observed around the (1)-type atoms (see notation in Fig. 2(b)), and the multiplicity of its first (resp. second) group of neighbors is (3, 1, 3, 3) (resp. (3, 3)). This is close to the neighbor distributions of the \#B-type atoms in the $\omega$ phase, even if for Zr\#2, the system evolves towards an hcp structure when increasing further the box size. This set of empirical potential calculations – given their qualitative agreement with ab initio results for small supercells – allows us to say that for hypothetical $T = 0$ K ab initio calculations at large enough supercells, a complete destabilization of the bcc Ti and Zr matrices should be induced by the vacancy.

## 3.4. Summary and discussion

The finite-size $\omega$-phase embryos characterized from the local atomic and electronic structure in small ab initio supercells at $T = 0$ K are artificially stabilized by PBCs. They do not correspond to constitutive defects, as sometimes observed in low dimensional metallic systems [56, 57], where vacancies have negative $E^f$ values but do not lead to complete structural changes. Here, introducing the vacancy converts the bcc lattice (only mechanically unstable without V) into a fully unstable structure, resulting in the $\omega$ phase. There is an analogy between the vacancy static displacement field and the LA 2/3 (111) phonon mode, having imaginary frequency at zero temperature for bulk bcc Ti and Zr [17]. This mode is intrinsically anharmonic, and excitation of this mode in the right direction is generative of the $\omega$ phase [14, 58]. In other words, the vacancy creates atomic rearrangements that are locally similar to the collapse of the {111}$_{bcc}$ planes, which activates this LA unstable phonon mode and thus the phase transition in the simulations boxes. The present exploration at 0 K of bcc Ti and Zr PES in presence of a vacancy is useful to understand this connection; more physically meaningful results will be provided at finite temperature.
## 4. Finite-temperature heterophase fluctuations induced by the vacancy

In this section, we first investigate the structural details of the vacancy defect in a bcc metastable matrix at finite temperature, using the EAM Ti1 empirical potential that correctly predicts the similar stability of the $\omega$ and hcp phases, and that is very close to mechanical instability at zero temperature for the bcc structure. The insights provided by our annealing simulations – i.e. the presence of $\omega$-phase fluctuations induced by the vacancy defect – allows us to suggest possible important implications for the athermal bcc$\rightarrow\omega$ phase transformation observed in many Ti and Zr binary alloys [6, 10, 12, 13, 59].

### 4.1. Molecular dynamics setup

Each of our molecular dynamics simulation sequence starts by a 1000 ps ($N, V, E$) equilibration of the system that is then close to the target temperature. It is then followed by a 100 ns simulation in the canonical ($N, V, T$) ensemble. This choice of fixed box volume and shape is meant to suppress the occurrence of the bcc$\rightarrow$hcp phase transformation, which, in contrast to the bcc$\rightarrow\omega$ transformation, is accompanied by an important deformation. We choose cubic supercells, i.e. $3n \times 3n \times 3n$ replications of the conventional bcc cell, which ensures the commensurability of the cells with both bcc and $\omega$ structures.

Annealings of both bulk bcc cells and cells containing a single vacancy are performed at 300 K. This choice of room temperature annealing is made having in mind the general context of athermal $\omega$ precipitation in Ti and Zr alloys. This precipitation occurs when the material is rapidly quenched at room temperature from the high temperature of the thermal treatment step, i.e. well above the $\beta$ transus. So examining the behavior of the vacancy at 300 K makes sense; we will come back to that point later.

### 4.2. Annealings of small simulation boxes

We consider here EAM Ti1 MD simulations on very small $3 \times 3 \times 3$ simulation cells, containing 54 atoms. These small cell calculations provide valuable first insights. This cell size is also typically manageable for ab initio MD simulations, albeit for shorter run times (a few tens of ps in Ref. [21]), and will be useful here to further benchmark our potential.

Fig. 7 displays perfect bcc and $\omega$ structures, and snapshots of atomic configurations at different stages of ($N, V, T$) annealings, for a perfect bcc bulk configuration at 40 ns (Fig. 7(c)) and for an initially bcc structure containing a vacancy, at 0.1, 20 and 40 ns (Figs. 7(d)-(f)). The bulk remains in its bcc structure over annealing time, but when a vacancy is introduced into the supercell, the structure in the simulation box alternates over time between a bcc phase and a single $\langle 111 \rangle_{\text{bcc}}$ orientation variant of the $\omega$ phase (see perfect bcc and $\omega$ structures for reference in Figs. 7(a) and (b)). Several $\langle 111 \rangle_{\text{bcc}}$ variants can appear during a single 100 ns ($N, V, T$) run, and in the course of

other MD runs with different initial velocities. The vacancy defect is thus able to generate bcc-$\omega$ heterophase fluctuations at finite temperature, and that propagate over the whole small supercell.

In Ref. [21], ab initio ($N, V, T$) simulations were performed in bulk bcc Ti, on the same 54 atoms simulation box, at 350 K, and with a 25 ps simulation time. A defective $\omega$ structure was identified at the end of this very short run without the vacancy defect, i.e. in a small supercell of bcc bulk structure and close to room temperature. This behavior is not found in our case: this is due to the fact that $G_{111}^{\text{DFT}} < G_{111}^{\text{Ti1}}$ and $\Delta E_{\omega, \text{DFT}}^{\text{bcc}} > \Delta E_{\omega, \text{Ti1}}^{\text{bcc}}$ (see Table 1), which means that the transitioning from bcc toward $\omega$ is easier in ab initio than in Ti1 small supercells. Nevertheless, the outcome EAM calculations, i.e. increased $\omega$-phase fluctuations at finite temperature in the presence of a vacancy, is still valid, due to the easy activation of the LA phonon mode responsible for the bcc$\rightarrow\omega$ transformation

<figure><img src="image_8.png" /><figcaption>Figure 7: Various atomic configurations on a $3 \times 3 \times 3$ bcc supercell, projected in the [010] direction. (a) a perfect bcc structure, (b) a perfect $\omega$ structure corresponding to a $\langle 111 \rangle_{\text{bcc}}$ orientation variant, and several snapshots of MD ($N, V, T$) annealings at $T = 300$ K of (c) a perfect bcc bulk, and (d)–(f) an initially bcc structure containing a vacancy, using Ti1 EAM potential.</figcaption></figure>
(see discussion in section 3.4) by the vacancy displacement field.

However, such additional fluctuations could be over-
whelmed by the bulk ω-phase fluctuations in hypotheti-
cal long runs (100 ns) of finite-T ab initio simulations for
bcc Ti. As we cannot perform such runs, we repeated our
MD runs using a second EAM potential, Ti2 developed by
Mendelev [32]. At T = 0 K, it has B = 412 GPa, which
is way too high, but has ΔEωbcc = 67 meV/at, i.e. closer
to the ab initio value of 114 meV/at, and G111 = −36
GPa, i.e. ~twice lower than the ab initio value of −19 GPa.
With Ti2, both bulk and vacancy simulations in small su-
percells immediately display transitioning towards a single
⟨111⟩bcc orientation variant of the ω-phase, with no alter-
nance over time. In other words, with Ti2 the effect of
the vacancy is overwhelmed by bulk fluctuations. Con-
sequently, as the ab initio value of G111 is bracketed by
those of Ti1 and Ti2 potentials, the ratio of vacancy ω-
fluctuations over bulk ω-fluctuations in ‘real’ finite-T ab
initio Ti calculations should lie somewhere between the
ones observed with Ti1 and Ti2.

4.3. *Annealings of large simulation boxes*

The vacancy concentration in the previous small super-
cell simulations is unrealistically high. Here, we repeat
with Ti1 the same (N, V, T) sequence, but with 30×30×30
supercells (54 000 atoms). This simulation size is large
enough to reproduce the isolated vacancy behavior, and
its vacancy concentration is 1, 8.10<sup>-5</sup>, i.e. typical of the
equilibrium ones at the temperatures used for the materi-
als thermal treatment (c<sub>V</sub> ~ 10<sup>-5</sup>-10<sup>-4</sup>, see estimates in
section 4.5).

Figure 8 provides representative snapshots of the 300 K vacancy annealing. Only atoms recognized as ‘others’ using the polyhedral template matching method (PTM) [60] are represented. As can be seen, there are two types of vacancy configurations, both involving a finite number of non-bcc atoms. The first one, in Fig. 8(a), has an octahedral shape and involves 14 atoms, corresponding to the vacancy first and second nearest neighbors, and similar to the standard cubic vacancy configuration. The second type has a linear shape parallel to the [111]bcc direction, and involves a larger number of non-bcc atoms, typically 19-32 atoms. A representative example is shown in Fig 8(b): it is different from the collective atomic motion observed in Ref. [61] that do not involve a vacancy. Analysis of the stacking sequence normal to [111]bcc of the linear vacancy configuration is given in Fig. 8(c), and corresponds locally to the sequence of planes #B'A#B'A□C#BC#B, thus revealing an ordered sequence of two translation subvariants of the same orientation variant of ω-phase (shift in the [111]bcc direction), and separated by the vacancy. Note that the various nuclei of #B ω-plane are not fully planar in MD simulations: their thickness is typically ~ 0.3-0.4 Å, i.e. still way smaller than the equivalent in the perfect bcc structure (0.94 Å with Ti1). Other ⟨111⟩bcc orientation variants of this linear configuration are randomly seen

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

Figure 8: Representative snapshots of the two different vacancy con-
figurations observed in a 300 K (N, V, T) annealing using Ti1: (a) a
classical bcc-type vacancy configuration and (b) a linear ω-like con-
figuration oriented along the [111] direction. (c) sequence of (111)
planes in the linear vacancy configuration, showing locally a sequence
of two translation subvariants of ω phase. Only atoms identified as
'others' using the PTM analysis are displayed, and a 30 × 30 × 30
replication of the bcc conventional cell is used.

during the annealing. Thermal fluctuations are thus able –
temporarily – to break the symmetry of the octahedral bcc
vacancy configuration, and to generate linear ω-phase fluc-
tuations. Their typical life time is short: about 30-50 ps,
occasionally ~ 100-200 ps, and a rough estimate indicates
that the linear vacancy configuration is present less than
6% of the time. Similar MD simulations using Ti2 lead
for both bulk and vacancy cases to an immediate switch
from the bcc structure to a single ⟨111⟩<sub>bcc</sub> variant of the
ω-phase, maintained until the end of each MD run.

4.4. Discussion on the isolated vacancy structure

In hypothetical *ab initio* 100 ns MD runs at *T* = 300 K of bcc Ti with 54 000 atoms, as again *G*<sub>111</sub><sup>Ti2</sup> *<i>G*<sub>111</sub><sup>DFT</sup> *<i>G*<sub>111</sub><sup>Ti1</sup>*<br>ω-phase fluctuations should be more (resp. less) developed than seen in Ti1 (resp. Ti2) simulations. The coexistence of two vacancy configurations – *i.e.* classical bcc-type or linear *ω*-like as shown in Fig. 8 – would thus be possible in simulation conditions where the bcc→hcp transition is blocked like in the present work, but likely with a linear configuration more extended along the ⟨111⟩<sub>bcc</sub> direction.

At temperatures where the bcc phase is stable for both
Ti and Zr, experimental and theoretical results [14–17]
indicate that the LA 2/3 (111) becomes positive while re-
maining soft up to very elevated temperatures. So we
expect that the vacancy defect will act as a source of
heterophase fluctuations in bcc Ti and Zr over a large T
range, possibly displaying the structural characteristics de-
scribed above. Note that, without any structural details,
the coexistence of two different vacancy configurations was
sometimes mentioned [14] to explain part of the anomalous
diffusion behavior of β-metastable Ti and Zr metals and
alloys. We gave here their possible atomic configurations.

Besides, complex vacancy defect structures, involving
the relaxation of a large number of atoms, were already
postulated to explain both internal friction results [12] and
the morphology of diffuse scattering distributions [22] in
bcc Ti and Zr binary alloys. In particular, Kuan and
Sass [22] explained the diffuse peak shifts observed at 300 K
in these alloys by postulating the existence of a linear
ω-like vacancy defect, involving the relaxation of many
atoms. More specifically, the presence of a vacancy con-
necting two ω-subvariants along a given (111)_{bcc} direc-
tion in an ordered sequence, together with the existence of
atomic displacements normal to that direction, were both
needed to reproduce the detailed morphology of the dif-
fuse intensity. The vacancy configuration of figure 8(b)
and (c) displays such behaviors, as well as an elongated
shape along (111)_{bcc}. Other similar features include a dis-
placement magnitude of ~0.17a_{bcc} of the (1)-type atoms
along the (111)_{bcc} direction, as well as inward displace-
ments of the (3)-type atoms normal to the (111)_{bcc} direc-
tion (see notations and relaxations given in Fig. 2). The
linear vacancy configuration obtained in our MD anneal-
ings using Ti1 seems however less elongated and involving
less non-bcc atoms than the best vacancy structural model
of Ref. [22]. It should be noted that the diffuse peaks ap-
pear in experiments for rather large solute concentrations
in bcc metastable Ti and Zr alloys [22], and not in pure
bcc Ti and Zr. Alloying with group V and VI elements
has the overall effect of stabilizing the bcc phase and in-
creasing G<sub>111</sub> [62]), but should also have a heterogeneous
effect at concentrations of 10-20 at%. The local chemical
environment of a vacancy will include at least one solute
that will locally break the symmetry. If the vacancy-solute
interaction is attractive, the presence of solutes will sup-
press other (111)_{bcc} orientation variants of the ω nuclei,
pin the linear ω-like vacancy configurations and thus en-
hance their life time and statistical relevance. Note finally
that linear ω defect structures have been suggested to ex-
ist in bcc bulk Ti and Zr at finite temperature [20, 21]<sup>1</sup>,
but do not involve the required ordered sequence of two ω
subvariants [22], that is automatically implied in the linear
ω-like vacancy configuration.

4.5. Possible consequences for athermal ω precipitation

The heterophase fluctuations induced by the vacancy
defect from the lattice positions of the bcc matrices could
act as nucleation sites for the athermal ω transforma-
tion, as suggested by Doherty et al. [12]. Indeed, the
very high density of fine ω particles that is obtained
upon quenching in finite – but low – solute concentra-
tion alloys seems incompatible with a homogeneous nu-
cleation from the bcc phases [12]. In contrast, a hetero-
geneous nucleation from the linear ω-like vacancy con-
figurations would be more compatible with this obser-
vation. Evaluation of the amount of thermal vacancies
from the relation c<sub>V</sub>(T) = exp(-H<sup>f</sup>(V)-TS<sup>f</sup>)/k<sub>B</sub>T), and us-
ing the ab initio finite-temperature vacancy formation en-
thalpies H<sup>f</sup>(V) and entropies S<sup>f</sup> from Ref. [17], gives
c<sub>V</sub>(1700 K) = 2, 4.10<sup>-4</sup> for Ti and 2.5.10<sup>-4</sup> for Zr. Conver-
sion into volumic densities gives ~ 10<sup>19</sup> cm<sup>-3</sup>, i.e. an order
of magnitude close to the typical density of ω particles ob-
tained from high-temperature quenching of low solute con-
centration Ti and Zr alloys [12, 13]. The same evaluation
at a lower temperature gives c<sub>V</sub>(1300 K) = 1, 7.10<sup>-5</sup> for
Ti and 1, 3.10<sup>-5</sup> for Zr, allowing to reduce the number of
thermal vacancies by more than a factor of 10. A quench-
ing from this lower temperature of the bcc metastable al-
loys should thus diminish accordingly the concentration
of ω precipitates, and would allow to further confirm this
scenario for the athermal ω phase precipitation from the
unusual vacancy configuration characterized in this work.
Importantly, as thermal vacancies diffuse into the bcc ma-
trix, they are uniformly distributed over time in the ma-
trix. Therefore, a heterogeneous nucleation of the ω phase
on these point defects will lead to ω precipitates that are
homogeneously distributed in space, in agreement with ex-
perimental observations [6, 10, 12, 13, 59].

The role of intrinsic point defects has been discussed
previously in the case of the *martensitic hcp* → *ω* transformation in Ti. Under zero pressure and temperature, vacancy (V) and self-interstitial (SIA) defects increase the energy barrier of the transformation [45], but due to their low concentration in usual conditions, no specific influence is expected for this hcp → *ω* transformation. Under irradiation however, a significant amount of V and SIA is created and leads to the formation of *ω* precipitates in the hcp Ti matrix: is it interpreted through a mechanism involving point defects and the activation of three phonons [63, 64]. The mechanism we propose here for the *reconstructive bcc* → *ω* transformation and involving thermal vacancies thus belongs to this body of literature on the initiation of displacive phase transitions.

5. Conclusion

We performed in the present work a combination of ab
initio and empirical potential calculations to better un-
derstand the behavior of the vacancy in bcc Ti and Zr
matrices. Within the mechanically unstable matrices at

<footnote n="1">¹The linear chain defects evidenced in AIMD simulations in Ref. [21] are in ω phase, but can be similarly envisioned in bcc.</footnote>
$T = 0$ K, the static displacement field of the vacancy activates the LA $2/3\langle 111 \rangle$ phonon mode responsible for the bcc$\rightarrow \omega$ transformation, thus making the bcc lattice fully unstable in presence of the vacancy. We expect similar effect of the vacancy in the bcc Hf matrix [19]. Our classical MD simulations at finite temperature allowed us to identify two alternating vacancy configurations: the standard cubic configuration, and a linear, symmetry-broken configuration, locally corresponding to two connected translation sub-variants of a single $\langle 111 \rangle_{\text{bcc}}$ orientation variant of the $\omega$ phase. Thus, the vacancy acts as a source of $\omega$ heterophase fluctuations in the bcc matrix. The complex linear vacancy structure characterized in our simulations is in good agreement with the best structural model explaining experimental diffuse scattering observations [22]. Consequently, at temperature high enough to stabilize the bcc phases, we suggest that thermal vacancies should act as heterogeneous nucleation sites for the $\omega$ athermal transformation. The role of point defects was suggested earlier in the literature [9], but without the clear identification of the $\omega$-like fluctuations induced by the vacancy. This scenario aligns with the observed homogeneous distribution of $\omega$ particles [6, 13], and with the similar densities of vacancies at high temperature and of athermal $\omega$ particles after quenching from this high temperature. Moreover, such a mechanism involving instability/heterophase fluctuations could be compatible with an athermal process.

Insights from our work call for additional studies to verify the scenario envisioned here for the displacive athermal $\omega$ precipitation in bcc Ti and Zr alloys. In particular, trying to better correlate experimentally the density of athermal $\omega$ precipitates to the quenching temperature would be of interest. Besides, a better understanding of the role of alloying effect – and of the vacancy/solute interactions – would be useful to (i) fully characterize the possible nucleation processes of $\omega$ phase from linear defect configurations (vacancies or possibly contraction centers) and (ii) better understand the links between alloy composition and the occurrence of athermal $\omega$ precipitation. This would require to use more accurate (machine learning) potentials or to combine classical potentials with a careful sensitivity analysis to their parameters [65, 66]. Finally, it is worth mentioning that bcc-$\omega$ heterophase fluctuations can be found to exist intrinsically in high concentration materials like refractory bcc high entropy alloys [67, 68] mixing group IV and group V & VI elements, and displaying particularly large atomic displacements. We will report on these directions in future work.

**Acknowledgements** - Support for this work was provided by the French national funding agency through the grant CoCoA ANR-18-CE08-0002, and by national HPC resources of TGCC CEA under the allocation grants No.2021-0912054, 2023-A0150913812 and 2024-A0170913812 made by GENCI. We are grateful to F. Ducastelle for pointing out the importance of the $\omega$ phase in group IV transition metals, and to B. Appolaire and M. Cottura for useful discussions.

**Data availability** - The raw/processed data required to reproduce these findings, and that are not included into the article main text and appendices, are available upon reasonable request to the corresponding author.

## Appendix A. Electronic DOS for bulk phases

The total DOS and partial DOS of d-electrons from our ab initio calculations are displayed in Figure A.9 for the bcc, hcp and $\omega$ structures of pure Ti and Zr. As the $\omega$ phase involves two inequivalent atomic layers, the partial DOS of d-electrons for each atomic layer are also provided, together with the weighted average between the two.

The two bcc phases both display the usual rather narrow d-band lying in the middle of the broader s-p band, with a triple hump shape involving a pronounced pseudogap. The Fermi levels are located in the central peak of $n(E)$, i.e. well below the pseudogap, as expected for group IV metals, and in contrast to group V and VI metals [43, 48, 52]. The bulk hcp phases also display rather narrow d bands as compared to the s-p ones, and the top of the bands display a double peak behavior for both metals. The Fermi levels are located close to rather sharp minima of $n(E)$, in accordance with the hcp structure being more stable than bcc for Ti and Zr. Finally, the $\omega$ phases have d-electron DOS rather close to the hcp phase for the #A-type layers, but different for the #B-type layers. The total and partial average densities both display a broad band [43] with the Fermi level lying close to a more shallow local minimum, and a single peak at the top of the band, mainly originating from the #B-type layer atoms.

## Appendix B. LDOS supercell size effect in bcc Ti

This appendix provides in Figure B.10 the LDOS of the d-electrons on the first and second nearest neighbors of the vacancy in bcc Ti, in a supercell of 431 atoms, after performing atomic relaxations at constant supercell volume and shape. These two vacancy neighbors pertain to the $\omega$-#B nuclei (see Figure 3(b)). Two different k-grid are tested ($2 \times 2 \times 2$ and $4 \times 4 \times 4$) and give very similar LDOS, indicating convergence with this respect.

The d-electron DOS of the #B-type layers of the $\omega$ phase is superimposed, and compares very well with the LDOS on both vacancy neighbors: the bandwidth is very similar, and the shape details are also very similar, especially for the 2nd nearest neighbors. Comparison with the d-electron DOS of the bulk hcp and #A-type layers of the $\omega$ phase have also been performed, and the comparison with the LDOS on the vacancy 1st and 2nd nearest neighbors is not as satisfactory.

The agreement with the #B-type layer atoms of the bulk $\omega$ phase is better for this 431 atom vacancy calculation than for the 249 atoms vacancy calculation of Figure 4. This is because the larger supercell allows more atomic relaxations (less effect of the PBCs), and then, the distances between atoms pertaining to the $\omega$-#B nuclei in the 431 vacancy supercell are closer to the one in pure #B layers in bulk $\omega$, as compared to ones in the 249 atoms vacancy supercell.
<figure><img src="image_12.png" /></figure>

Figure A.9: Total DOS and partial DOS of *d*-electrons *n*(*E*) for Zr (top panels) and Ti (bottom panels), for the bcc, hcp and *ω* crystallographic structures. Energy is centered around the Fermi level. For the *ω* phase, the partial DOS of the #A and #B layers are also shown, and the weighted average 〈*ω*〉 between the two as well.

<figure><img src="image_13.png" /></figure>

Figure B.10: Local DOS of the *d*-electrons *n*(*E*) on the first and second nearest neighbors of the vacancy in a bcc Ti matrix, for a supercell of 431 atoms and after atomic relaxations. Energy is centered around the Fermi level. The partial DOS of the #B-type layer of the *ω* phase is superimposed in the two pictures, showing good agreement.

References

[1] F. Ducastelle, Order and Phase Stability in Alloys, North-Holland, Amsterdam, 1991.

[2] G. D. Samolyuk, Y. N. Osetsky, G. M. Stocks, J. R. Morris, Role of static displacements in stabilizing body centered cubic high entropy alloys, Phys. Rev. Lett. 126 (2021) 025501. doi: 10.1103/PhysRevLett.126.025501.

[3] A. Aguayo, G. Murrieta, R. de Coss, Elastic stability and electronic structure of fcc Ti, Zr, and Hf: A first-principles study,

Phys. Rev. B 65 (2002) 092106. doi:10.1103/PhysRevB.65.092106.

[4] S. K. Sikka, Y. K. Vohra, R. Chidambaram, Omega phase in materials, Prog. Mater. Sci. 27 (3) (1982) 245–310. doi:https://doi.org/10.1016/0079-6425(82)90002-0.

[5] B. S. Hickman, The formation of omega phase in titanium and zirconium alloys: A review, J. Mater. Sci. 4 (1969) 554–563. doi:10.1007/BF00550217.

[6] S. Banerjee, R. Tewari, G. K. Dey, Omega phase transformation - morphologies and mechanisms, Int. J. Mater. Res. 97 (7) (2006) 963–977. doi:10.3139/146.101327.

[7] H. Xia, J. Duclos, A. L. Ruoff, Y. K. Vohra, New high-pressure phase transition in zirconium metal, Phys. Rev. Lett. 64 (2) (1990) 204–207.

[8] H. Xia, G. Parthasarathy, H. Luo, Y. K. Vohra, A. L. Ruoff, Crystal structures of group iva metals at ultrahigh pressures, Phys. Rev. B 42 (1990) 6736–6738. doi:10.1103/PhysRevB.42.6736.

[9] H. Ezaki, M. Morinaga, M. Kato, N. Yukawa, Electronic and atomic-size effects on the omega phase formation in transition-metal based BCC alloys, Acta Metall. Mater. 39 (8) (1991) 1755–1761. doi:https://doi.org/10.1016/0956-7151(91)90143-0.

[10] J. Ballor, T. Li, F. Prima, C. J. Boehlert, A. Devaraj, A review of the metastable omega phase in beta titanium alloys: the phase transformation mechanisms and its effect on mechanical properties, International Materials Reviews 68 (1) (2023) 26–45. arXiv:https://doi.org/10.1080/09506608.2022.2036401, doi:10.1080/09506608.2022.2036401.

[11] O. N. Senkov, D. B. Miracle, K. J. Chaput, J.-P. Couzinie, Development and exploration of refractory high entropy alloys – a review, J. Mater. Res. 33 (19) (2018) 3092–3128. doi:10.1557/jmr.2018.153.

[12] J. Doherty, D. Gibbons, Internal friction studies in some titanium and zirconium based alloys, Acta Metall. 19 (4) (1971) 275–282. doi:https://doi.org/10.1016/
0001-6160(71)90093-9.

[13] M. J. Blackburn, J. C. Williams, Phase transformations in Ti-Mo and Ti-V alloys, Trans. Met. Soc. AIME 242 (1968) 2461-9. URL https://www.osti.gov/biblio/4806773

[14] C. Herzig, U. Köhler, Anomalous self-diffusion in bcc ivb metals and alloys, Mater. Sci. Forum 15-18 (1987) 301-322. doi:10.4028/www.scientific.net/MSF.15-18.301

[15] W. Petry, A. Heiming, J. Trampenau, M. Alba, C. Herzig, H. R. Schober, G. Vogl, Phonon dispersion of the bcc phase of group-iv metals. i. bcc titanium, Phys. Rev. B 43 (1991) 10933-10947. doi:10.1103/PhysRevB.43.10933.

[16] A. Heiming, W. Petry, J. Trampenau, M. Alba, C. Herzig, H. R. Schober, G. Vogl, Phonon dispersion of the bcc phase of group-iv metals. ii. bcc zirconium, a model case of dynamical precursors of martensitic transitions, Phys. Rev. B 43 (1991) 10948-10962. doi:10.1103/PhysRevB.43.10948.

[17] S. Kadkhodaei, A. Davariashtiyani, Phonon-assisted diffusion in bcc phase of titanium and zirconium from first principles, Phys. Rev. Materials 4 (2020) 043802. doi:10.1103/PhysRevMaterials.4.043802.

[18] J. M. Sanchez, D. de Fontaine, Model for anomalous self-diffusion in group-IVB transition metals, Phys. Rev. Lett. 35 (1975) 227-230. doi:10.1103/PhysRevLett.35.227.

[19] F. Willaime, A. Satta, S. de Gironcoli, Structure and energetics of vacancies in body centered cubic hafnium under pressure: First-principles study, in: Diffusion in Materials DIMAT2000, Vol. 194 of Defect and Diffusion Forum, Trans Tech Publications Ltd, 2001, pp. 295-302. doi:10.4028/www.scientific.net/DDF.194-199.295.

[20] D. Schryvers, L. E. Tanner, High resolution electron microscopy observations of athermal omega phase in Ti-Mo alloys, Mater. Sci. Forum 56-58 (1991) 329-334. doi:https://doi.org/10.4028/www.scientific.net/msf.56-58.329.

[21] D. Korbmacher, A. Glensk, A. I. Duff, M. W. Finnis, B. Grabowski, J. Neugebauer, Ab initio based method to study structural phase transitions in dynamically unstable crystals, with new insights on the $\beta$ to $\omega$ transformation in titanium, Phys. Rev. B 100 (2019) 104110. doi:10.1103/PhysRevB.100.104110.

[22] T. Kuan, S. Sass, The structure of a linear omega-like vacancy defect in Zr-Nb BCC solid solutions, Acta Metall. 24 (11) (1976) 1053-1059. doi:https://doi.org/10.1016/0001-6160(76)90136-X.

[23] G. Grimvall, B. Magyari-Köpe, V. Ozoliņš, K. A. Persson, Lattice instabilities in metallic elements, Rev. Mod. Phys. 84 (2012) 945-986. doi:10.1103/RevModPhys.84.945.

[24] C. Varvenne, O. Mackain, E. Clouet, Vacancy clustering in zirconium: An atomic-scale study, Acta Mater. 78 (2014) 65-77. doi:10.1016/j.actamat.2014.06.012.

[25] B.-T. Wang, P. Zhang, H.-Y. Liu, W.-D. Li, P. Zhang, First-principles calculations of phase transition, elastic modulus, and superconductivity under pressure for zirconium, J. Appl. Phys. 109 (6), 063514. arXiv:https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/1.3556753/15064910/063514\_1\_online.pdf, doi:10.1063/1.3556753.

[26] E. Clouet, Screw dislocation in zirconium: An ab initio study, Phys. Rev. B 86 (2012) 144104. doi:10.1103/PhysRevB.86.144104. URL http://link.aps.org/doi/10.1103/PhysRevB.86.144104

[27] N. Chaari, E. Clouet, D. Rodney, First-principles study of secondary slip in zirconium, Phys. Rev. Lett. 112 (2014) 075504. doi:10.1103/PhysRevLett.112.075504.

[28] W. Szewc, L. Pizzagalli, S. Brochard, E. Clouet, Onset of plasticity in zirconium in relation with hydrides precipitation, Acta Mater. 114 (2016) 126-135. doi:https://doi.org/10.1016/j.actamat.2016.05.025.

[29] B. Yin, Z. Wu, W. A. Curtin, Comprehensive first-principles study of stable stacking faults in hcp metals, Acta Mater. 123 (2017) 223-234. doi:https://doi.org/10.1016/j.actamat.2016.10.042.

URL Zr

[30] M. Liao, Y. Liu, P. Cui, N. Qu, F. Zhou, D. Yang, T. Han, Z. Lai, J. Zhu, Modeling of alloying effect on elastic properties in bcc nb-ti-v-zr solid solution: From unary to quaternary, Comput. Mater. Sci. 172 (2020) 109289. doi:https://doi.org/10.1016/j.commatsci.2019.109289.

[31] M. I. Mendelev, G. J. Ackland, Development of an interatomic potential for the simulation of phase transformations in zirconium, Philos. Mag. Lett. 87 (2007) 349-359. doi:10.1080/09500830701191393.

[32] M. I. Mendelev, T. L. Underwood, G. J. Ackland, Development of an interatomic potential for the simulation of defects, plasticity, and phase transformations in titanium, J. Chem. Phys. 145 (2016) 154102. doi:https://doi.org/10.1063/1.4964654.

[33] E. S. Fisher, C. J. Renken, Single-crystal moduli and the hcp → bcc transformation in Ti, Zr and Hf, Phys. Rev. 135 (2A) (1964) 482-494.

[34] C. S. G. Cousins, Electrostatic contributions to the second- and third-order inner elastic constants of hcp metals for a range of axial ratios, J. Phys. C Solid State 12 (1979) 989-994. doi:10.1088/0022-3719/12/6/011.

[35] P. Villars, L. D. Calvert, Pearson's handbook of crystallographic data for intermetallic phases, Am. Soc. Metals, Materials Park, OH, USA, 1985.

[36] J. Spreadborough, J. W. Christian, The measurement of the lattice expansions and debye temperatures of titanium and silver by x-ray methods, Proceedings of the Physical Society 74 (5) (1959) 609. doi:10.1088/0370-1328/74/5/314.

[37] E. Fisher, D. Dever, The single crystal elastic moduli of beta-titanium and titanium-chromium alloys, in: R. Jaffee, N. Promisel (Eds.), The Science, Technology and Application of Titanium, Pergamon, 1970, pp. 373-381. doi:https://doi.org/10.1016/B978-0-08-006564-9-50047-6.

[38] A. Heiming, W. Petry, J. Trampenau, W. Miekeley, J. Cockcroft, The temperature dependence of the lattice parameters of pure bcc zr and bcc zr-2 at. Phys.: Condens. Matter 4 (3) (1992) 727. doi:10.1088/0953-8984/4/3/012.

[39] M. Tane, Y. Okuda, Y. Todaka, H. Ogi, A. Nagakubo, Elastic properties of single-crystalline $\omega$ phase in titanium, Acta Mater. 61 (20) (2013) 7543-7554. doi:https://doi.org/10.1016/j.actamat.2013.08.036.

[40] Y. Zhao, J. Zhang, C. Pantea, J. Qian, L. L. Daemen, P. A. Rigg, R. S. Hixson, G. T. Gray, Y. Yang, L. Wang, Y. Wang, T. Uchida, Thermal equations of state of the $\alpha$, $\beta$, and $\omega$ phases of zirconium, Phys. Rev. B 71 (2005) 184119. doi:10.1103/PhysRevB.71.184119.

[41] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, R. M. Wentzcovitch, Quantum ESPRESSO: a modular and open-source software project for quantum simulations of materials, J. Phys.: Condens. Matter 21 (39) (2009) 395502 (19pp). URL http://www.quantum-espresso.org

[42] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buonigio, Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. Dal Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. Oterode-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, S. Baroni, Advanced capabilities for materials modelling with quantum ESPRESSO, J. Phys.: Condens. Matter 29 (46) (2017) 465901.
doi:10.1088/1361-648x/aa8f79.

[43] Y. K. Vohra, S. K. Sikka, R. Chidambaram, Electronic structure of ω phase of titanium and zirconium, J. Phys. F: Met. Phys. 9 (9) (1979) 1771-1782. doi:10.1088/0305-4608/9/9/008.

[44] R. G. Hennig, T. J. Lenosky, D. R. Trinkle, S. P. Rudin, J. W. Wilkins, Classical potential describes martensitic phase transformations between the α, β, and ω titanium phases, Phys. Rev. B 78 (2008) 054121. doi:10.1103/PhysRevB.78.054121.

[45] D. R. Trinkle, R. G. Hennig, S. G. Srinivasan, D. M. Hatch, M. D. Jones, H. T. Stokes, R. C. Albers, J. W. Wilkins, New mechanism for the α to ω martensitic transformation in pure titanium, Phys. Rev. Lett. 91 (2003) 025701. doi:10.1103/PhysRevLett.91.025701.
URL https://link.aps.org/doi/10.1103/PhysRevLett.91.025701

[46] F. Ducastelle, Modules élastiques des métaux de transition, J. Phys. 31 (1970) 1055-1062. doi:https://doi.org/10.1051/jphys:019700031011-120105500.

[47] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117 (1) (1995) 1-19. doi:http://dx.doi.org/10.1006/jcph.1995.1039.

[48] F. Willaime, A. Satta, M. Nastar, O. Le Bacq, Electronic structure calculations of vacancy parameters in transition metals: Impact on the bcc self-diffusion anomaly, Int. J. Quantum Chem. 77 (2000) 927-939. doi:https://doi.org/10.1002/(SICI)1097-461X(2000)77:6<927::AID-QUA1>3.0.CO;2-5.

[49] S. N. Khanna, B. K. Rao, P. Jena, D. Esterling, M. J. Puska, Atomic relaxations around vacancy clusters in molybdenum and their effects on trapped-positron lifetime, Phys. Rev. B 37 (1988) 6-11. doi:10.1103/PhysRevB.37.6.

[50] K. Masuda, R. Yamamoto, M. Doyama, Application of tight-binding type electronic theory to lattice defect problems in transition metals, J. Phys. F: Met. Phys. 13 (7) (1983) 1407-1413. doi:10.1088/0305-4608/13/7/011.

[51] P.-W. Ma, S. L. Dudarev, Effect of stress on vacancy formation and migration in body-centered-cubic metals, Phys. Rev. Materials 3 (2019) 063601. doi:10.1103/PhysRevMaterials.3.063601.

[52] X. Li, Phase stability and micromechanical properties of tizrhf-based refractory high-entropy alloys: A first-principles study, Phys. Rev. Mater. 7 (2023) 113604. doi:10.1103/PhysRevMaterials.7.113604.

[53] D. Connetable, J. Huez, E. Andrieu, C. Mijoule, First-principles study of diffusion and interactions of vacancies and hydrogen in hcp-titanium, J. Phys. Condens. Matter 23 (40). doi:10.1088/0953-8984/23/40/405401.

[54] G. Vérité, F. Willaime, C. C. Fu, Anisotropy of the vacancy migration in Ti, Zr and Hf hexagonal close-packed metals from first principles, Solid State Phenom. 129 (2007) 75-81. doi: 10.4028/www.scientific.net/SSP.129.75.

[55] A. Stukowski, Visualization and analysis of atomistic simulation data with ovito—the open visualization tool, Modelling Simul. Mater. Sci. Eng. 18 (1) (2010) 015012.
URL http://stacks.iop.org/0965-0393/18/i=1/a=015012

[56] C. Mottet, G. Tréglia, B. Legrand, New magic numbers in metallic clusters: an unexpected metal dependence, Surf. Sci. 383 (1) (1997) L719-L727. doi:https://doi.org/10.1016/S0039-6028(97)00226-4.

[57] I. Meunier, G. Tréglia, B. Legrand, R. Tétot, B. Aufray, J.-M. Gay, Molecular dynamics simulations for the ag/cu(111) system: from segregated to constitutive interfacial vacancies, Appl. Surf. Sci. 162-163 (2000) 219-226. doi:https://doi.org/10.1016/S0169-4332(00)00195-1.

[58] D. De Fontaine, Mechanical instabilities in the BCC lattice and the β to ω phase transformation, Acta Metall. 18 (2) (1970) 275-279. doi:https://doi.org/10.1016/0001-6160(70)90035-0.
URL omega,phasetransformation

[59] A. Devaraj, S. Nag, R. Srinivasan, R. Williams, S. Banerjee, H. Banerjee, R. and Fraser, Experimental evidence of concurrent compositional and structural instabilities leading to ω precipita-
tion in titanium-molybdenum alloys, Acta Mater. 60 (2) (2012)

596-609. doi:https://doi.org/10.1016/j.actamat.2011.10.008.

[60] P. M. Larsen, S. Schmidt, J. Schiøtz, Robust structural identification via polyhedral template matching, Modell. Simul. Mater. Sci. Eng. 24 (5) (2016) 055007. doi:10.1088/0965-0393/24/5/055007.

[61] D. G. Sangiovanni, J. Klarbring, D. Smirnova, N. V. Skripnyak, D. Gambino, M. Mrovec, S. I. Simak, I. A. Abrikosov, Superioniclike diffusion in an elemental crystal: bcc titanium, Phys. Rev. Lett. 123 (2019) 105501. doi:10.1103/PhysRevLett.123.105501.

[62] W. Zhou, R. Sahara, K. Tsuchiya, First-principles study of the phase stability and elastic properties of ti-x alloys (x = mo, nb, al, sn, zr, fe, co, and o), J. Alloys Compd. 727 (2017) 579-595. doi:https://doi.org/10.1016/j.jallcom.2017.08.128.

[63] H. Dammak, A. Dunlop, D. Lesueur, Phase transformation induced by swift heavy ion irradiation of pure metals, Nucl. Instrum. Methods Phys. Res., Sect. B 107 (1) (1996) 204-211. doi:https://doi.org/10.1016/0168-583X(95)01032-7.

[64] H. Dammak, A. Dunlop, D. Lesueur, Study of the irradiation-induced α → ω phase transformation in titanium: Kinetics and mechanism, Philos. Mag. A 79 (1) (1999) 147-166. arXiv: https://doi.org/10.1080/01418619908214280, doi:10.1080/01418619908214280.
URL https://doi.org/10.1080/01418619908214280

[65] G. Dhaliwal, P. B. Nair, C. V. Singh, Uncertainty and sensitivity analysis of mechanical and thermal properties computed through embedded atom method potential, Comp. Mater. Sci. 166 (2019) 30-41. doi:https://doi.org/10.1016/j.commatsci.2019.03.060.

[66] A. Del Masto, J. Baccou, G. Tréglia, F. Ribeiro, C. Varvenne, Insights on the capabilities and improvement ability of classical many-body potentials: Application to α-zirconium, Comput. Mater. Sci. 231 (2024) 112544. doi:https://doi.org/10.1016/j.commatsci.2023.112544.

[67] Y. Ikeda, K. Gubaev, J. Neugebauer, B. Grabowski, F. Körmann, Chemically induced local lattice distortions versus structural phase transformations in compositionally complex alloys, npj Comp. Mater. 7 (34). doi:https://doi.org/10.1038/s41524-021-00502-y.

[68] P. P. Borges, R. O. Ritchie, M. Asta, Local lattice distortions and the structural instabilities in bcc nb-ta-ti-hf high-entropy alloys: An ab initio computational study, Acta Mater. 262 (2024) 119415. doi:https://doi.org/10.1016/j.actamat.2023.119415.