††thanks: Corresponding author

# Impact of oxygen ordering on titanium lattice parameters

Martin S. Talla Noutack  Univ. Paris-Saclay, CEA, Service de recherche en
Corrosion et Comportement des Matériaux, SRMP, F-91191 Gif-sur-Yvette, France
Fabienne Amann  Univ. PSL, Chimie ParisTech, CNRS, IRCP (UMR 8247), F-75005,
Paris, France  Univ. Paris Est Creteil, CNRS, ICMPE (UMR 7182), 2 rue Henri
Dunant, F-94320, Thiais, France  Sophie Nowak  Univ. Paris Cité, CNRS, ITODYS
(UMR 7086), F-75013, Paris, France  Régis Poulain  Univ. Paris Est Creteil,
CNRS, ICMPE (UMR 7182), 2 rue Henri Dunant, F-94320, Thiais, France  Raphaëlle
Guillou  Univ. Paris-Saclay, CEA, Service de Recherche sur les Matériaux et
procédés Avancés, F-91191 Gif-sur-Yvette, France  Stéphanie Delannoy  Univ.
PSL, Chimie ParisTech, CNRS, IRCP (UMR 8247), F-75005, Paris, France  Biotech
Dental SAS, 305 allées de Craponne, F-13300, Salon-de-Provence, France  Ivan
Guillot  Univ. Paris Est Creteil, CNRS, ICMPE (UMR 7182), 2 rue Henri Dunant,
F-94320, Thiais, France  Frédéric Prima  Univ. PSL, Chimie ParisTech, CNRS,
IRCP (UMR 8247), F-75005, Paris, France  Emmanuel Clouet
<EMAIL_ADDRESS>Univ. Paris-Saclay, CEA, Service de recherche en
Corrosion et Comportement des Matériaux, SRMP, F-91191 Gif-sur-Yvette, France

###### Abstract

Variations with oxygen concentration of titanium lattice parameters are
obtained by means of ab initio calculations, considering the impact of oxygen
ordering. The quasi harmonic approximation is used to take into account the
thermal expansion at finite temperature. Results show that lattice parameters
depend mainly on oxygen concentration and to a lesser extent to the ordering
state. Knowing these theoretical variations, one can get insights on the
composition of ordered compounds existing in Ti-O binary alloys from their
lattice mismatch measured experimentally by X-ray diffraction. The approach is
used in a binary alloy containing 6000 ppm in weight of oxygen. It is
concluded that the ordered compounds, which are observed after a
recrystallization heat treatment, do not have the expected Ti6O stoichiometry
but have a composition close to the nominal concentration. Oxygen ordering
proceeds therefore before oxygen partitioning in titanium.

##  I Introduction

Oxygen is one of the major hardening elements of titanium, used to strengthen
the hexagonal close-packed (hcp) $\alpha$ phase [1]. With a high solubility
limit, more than 30 at.% at room temperature [2], titanium can incorporate a
large amount of oxygen in its $\alpha$ matrix. But suboxide compounds with a
Ti6O and a Ti3O stoichiometry are known to exist below the solubility limit
[3, 4, 5]. These suboxides correspond to an ordering of the oxygen atoms which
occupy the octahedral interstitial sites of the hcp lattice [6, 7]. Ab initio
calculations predict that these ordered compounds are the ground states in the
titanium rich part of the Ti-O phase diagram [8, 9, 10]. Although known for a
long time now, one hardly considers the existence of these ordered compounds
when discussing the effect of oxygen on titanium properties, in particular
mechanical properties [11, 12, 13, 14, 15, 16]. One generally assumes that all
oxygen is dissolved in solid solution for the low concentrations encountered
in titanium alloys which are usually lower than 1 at.% (0.5 and 1 at.%
respectively in grade-2 and -4 commercially pure Ti). But recent experiments
have shown that ordered compounds can exist at room temperature in titanium
for an oxygen concentration as low as 0.5 at.% [17], with potentially
important consequences on titanium mechanical properties [18, 6, 7, 19]. These
ordered compounds could be imaged by transmission electron microscopy through
the superlattice reflections arising from oxygen ordering. Lattice mismatches
between the ordered precipitates and the $\alpha$ matrix could also be
measured at room temperature with X-ray diffraction (XRD). But no information
on their composition could be obtained. In particular, it remains unclear if
these ordered precipitates have a Ti6O composition, as expected from
thermodynamics [10], or if their oxygen content is lower, with oxygen ordering
proceeding before oxygen partitioning i.e. with the formation of compounds
having the same long-range order as Ti6O ordered compounds but without
reaching the Ti6O stoichiometry. As oxygen addition is known to increase the
lattice parameters of the Ti $\alpha$-phase, one can attempt to deduce the
composition of ordered compounds from the lattice mismatches measured by XRD.
But one needs first to better know the impact of oxygen ordering on titanium
lattice parameters, in particular to be able to separate variations arising
from oxygen ordering and from oxygen enrichment.

The aim of this article is to use ab initio calculations to fully relate the
variations of titanium lattice parameters to oxygen concentration and order
state. We first perform ab initio calculations at 0 K and then extend them at
finite temperature, considering thermal expansion through the quasi-harmonic
approximation. The theoretical variations obtained for lattice parameters are
finally compared to the ones measured by XRD in a Ti-O binary alloy at
different temperatures, allowing to conclude on the composition of the ordered
compounds observed in this alloy.

##  II Phase stability and lattice parameters at 0 K

###  II.1 DFT parameters

Ab initio calculations are performed within non-spin polarized density
functional theory (DFT) using the Vasp code [20]. The exchange and correlation
functional is described with the generalized gradient approximation (GGA) as
formulated by Perdew, Burke and Ernzerhof (PBE) [21]. Considering the local
density approximation (LDA) instead of GGA leads to the same impact of oxygen
on titanium lattice parameters at 0 K (appendix B). Interactions between core
and valence electrons are accounted for by the projector augmented wave (PAW)
method [22], with $3s$, $3p$, $3d$, and $4s$ orbitals treated as valence
states for Ti and $2s$ and $2p$ for O. A plane wave cutoff energy equal to 500
eV is used for wave functions. A 24$\times$24$\times$18 $k$-point grid
generated with the Monkhorst-Pack [23] method is used for primitive hcp Ti
unit cell and grids with similar $k$-point density for supercells. Integration
is performed with the Methfessel-Paxton broadening scheme, using a 0.1 eV
width. According to our convergence tests, this leads to a precision better
than 1 meV per atom on total energy differences. Each structure is fully
relaxed, i.e. both periodicity vectors and atomic coordinates, to achieve a
force convergence lower than 0.02 eV/Å and an energy convergence of 10-6 eV.
Although the supercells can theoretically become monoclinic during the
relaxation, the underlying lattices remain always close to the initial HCP
crystal, with only a small orthorhombicity, which may appear when the symmetry
of the initial structure allows for different relaxations in different
directions of the basal plane.

###  II.2 Phase stability

Figure 1: Structures of ordered TinO compounds. (a) Projection in the basal
plane showing the periodic occupation by O atoms of octahedral interstitial
sites for one basal layer. (b) $AVBV$ and (c) $AVAV$ stacking along the ¡c¿
axis of partially occupied basal layers. O atoms and empty octahedral sites
are shown respectively with balls and cubes, using a different color for $A$,
$B$ and $C$ octahedral sites. $B$ and $C$ empty octahedral sites are shown
only for Ti6O compound in (a).

Oxygen occupies the octahedral interstitial sites of the hcp lattice formed by
Ti atoms. The ground state with the lowest oxygen content is known to have
Ti6O stoichiometry [6, 7, 8, 9, 10]. It corresponds to a stacking along the
¡c¿ axis of basal layers with one third of their octahedral sites occupied by
oxygen atoms alternating with oxygen free layers. Enriched basal layers are
ordered with occupied octahedral sites separated by ¡1-100¿ periodicity
vectors (see Ti6O structure in Fig. 1a). Denoting $A$, $B$, and $C$ the three
different octahedral sites in a basal layer, the Ti6O ground state can be
described by a periodic $AVBV$ stacking along the ¡c¿ axis, with $A$
corresponding to a basal layer where $A$ octahedral sites are occupied by O
atoms and $V$ meaning a layer where all octahedral sites are empty (Fig. 1b).

We check that our ab initio approach predicts the right ground state in the
concentration range $0\leq x_{\rm O}\leq 1/6$, where $x_{\rm O}=N_{\rm
O}/N_{\rm Ti}$ is the occupation of the octahedral interstitial sites 111One
can also define the oxygen concentration, $c_{\rm O}=N_{\rm O}/(N_{\rm
Ti}+N_{\rm O})$, i.e. the ratio of the number of oxygen atoms to the total
number of atoms.. To this aim we calculate the formation energy of different
ordered and disordered compounds. Disordered structures are built from a
supercell corresponding to the $4\times 4\times 3$ repetition of the
conventional hcp unit cell, with 96 Ti atoms. Oxygen atoms are placed on the
octahedral sites with the special quasirandom structures (SQS) method [24] to
obtain a random Ti-O solid solution. Ordered TinO structures with
$n=6,12,24,48$ are built in the same supercell, starting from the Ti6O ordered
compound and depleting oxygen atoms regularly in the basal layer, with either
a $AVBV$ or $AVAV$ stacking along the ¡c¿ axis (Fig. 1). Finally, some
partially ordered structures are also considered, using a larger supercell
with 216 Ti atoms and incorporating some oxygen vacancies in the ordered Ti6O
compound.

Figure 2: Formation energies of different TinOm compounds as a function of
oxygen occupation $x_{\rm O}=m/n$ of interstitial sites. Symbols correspond to
ab initio results for disordered, partly ordered or completely ordered
compounds. The solid black line indicates the convex hull linking most stable
structures.

Formation energies are calculated with reference to pure Ti and to TiO
compound, i.e. structures with all octahedral sites either empty ($x_{\rm
O}=0$) or occupied by oxygen atoms ($x_{\rm O}=1$). Considering that the alloy
is actually an oxygen-vacancy binary system on the octahedral interstitial
sites of the hcp lattice formed by the Ti atoms, all formation energies are
defined per interstitial site, leading to

$E^{\rm f}[{\rm Ti}_{n}{\rm O}_{m}]=\frac{1}{n}\left(E[{\rm Ti}_{n}{\rm
O}_{m}]-\frac{n-m}{n}E[{\rm Ti}_{n}]-\frac{m}{n}E[{\rm Ti}_{n}{\rm
O}_{n}]\right),$

where $E[{\rm Ti}_{n}]$ and $E[{\rm Ti}_{n}{\rm O}_{n}]$ are the energy of
pure Ti and TiO in the same supercell as the TinOm compound.

Fig. 2 shows that ordered structures are always more stable than disordered
solid solutions, confirming the ordering tendency of oxygen in titanium as
seen experimentally [4, 5, 25, 6, 17]. No intermediate compound is found below
the convex hull linking Ti and Ti6O, indicating that a supersaturated Ti(O)
solid solution should decompose in an oxygen-depleted solution and an ordered
Ti6O compound at low temperature.

Table 1: Energy variation $\Delta E$ of the Ti6O structures for different stackings of the oxygen enriched basal layers along the ¡c¿ direction.  Stacking |  $\Delta E$ (meV/site)
---|---
$AVBV$ | $0.$
$AVAV$ | $14.72$
$AVBVCV$ | $-0.93$
$AVBVCVBV$ | $-0.66$

Considering now the different possible stackings along the ¡c¿ axis of oxygen
enriched basal layers, ordered compounds with an $AVBV$ stacking are always
more stable than $AVAV$ stacking, in agreement with previous ab initio
calculations [8, 9, 10] and with the stable structure known experimentally for
Ti6O. Nevertheless, the stacking of these basal layers offers more
possibilities. Ab initio calculations for Ti6O show that stackings in which
two successive occupations of oxygen layers are different, i.e. $AVBV$,
$AVBVCV$, $AVBVCVBV$, lead almost to the same formation energy, with an energy
difference lower than 1 meV/site, thus below the expected DFT accuracy (Tab.
1). These ordered structures appear therefore degenerate, with $AVBV$ stacking
having the smallest wave-length along ¡c¿ direction, leading to Devil’s
staircases [9] disordering at finite temperature.

###  II.3 Variations of lattice parameters

Ab initio calculations show that both $a$ and $c$ lattice parameters increase
with the oxygen content (Fig. 3), with the parameter $c$ increasing more
rapidly than $a$, in agreement with experiments [2, 26]. The comparison
between experimental and theoretical values (Fig. 3) shows that ab initio
calculations correctly predict the increase rates of both lattice parameters
with the oxygen concentration, despite a slight underestimation of the
absolute value. As ab initio calculations are performed at 0 K while
experimental data have been obtained at room temperature, such an
underestimation is indeed expected. Besides, DFT approximation is known to be
not fully quantitative when predicting absolute values for lattice parameters,
with a dependence on the functional used for the electronic exchange and
correlation (see appendix B).

Figure 3: Lattice parameters determined at 0 K by ab initio calculations for
different compounds of the Ti-O binary alloy (square and circle symbols)
compared to the experimental values determined at room temperature by
Wiedemann [26]. Similar experimental variations are shown in the review of
Murray and Wriedt [2].

According to these ab initio calculations, the lattice parameters mainly
depend on the oxygen concentration, with a marginal impact of the ordering
state: for a same oxygen composition, a random solid solution and an ordered
compound have almost the same lattice parameters at 0 K (Fig. 3). The only
noticeable impact is for the highest oxygen concentrations, in particular for
Ti6O, where larger $c$ and smaller $a$ parameters are obtained for ordered
compounds than for disordered solid solutions.

##  III Modeling at finite temperatures

###  III.1 Methods

To go beyond 0 K calculations, we now take into account thermal expansion by
including the contribution of atomic vibrations in the formation free energy
of the different compounds. Calculations are performed in the quasi-harmonic
approximation, as described below, considering different structures in the
same supercell containing 72 Ti atoms and defined by the periodicity vectors
[-2200], [0-220], and [-1103].

This is done by coupling Phonopy [27] with Vasp DFT calculations, so as to
obtain phonons for different volumes $V$ and ratios $\gamma=c/a$ of the
supercell. A finite displacement of 0.01 Å is used in Phonopy to calculate the
supercell force constants. Dynamical matrices $D(\vec{q})$ are then calculated
by Fourier transformation on a regular $\vec{q}$ grid with $N_{q}$ points and
diagonalized to obtain the pulsations $\omega_{\vec{q},s}$ of the eigenmodes.
The vibration energy, defined per interstitial octahedral site, of a supercell
containing $n$ Ti atoms and $m$ O atoms is finally given by

$F^{\rm
vib}(V,\gamma,T)=\frac{1}{n}\frac{1}{N_{\vec{q}}}\sum_{\vec{q},s}{\left\\{\frac{\hbar\omega_{\vec{q},s}(V,\gamma)}{2}+kT\ln{\left[1-\exp\left(-\frac{\hbar\omega_{\vec{q},s}(V,\gamma)}{kT}\right)\right]}\right\\}}.$
(1)

A $15\times 15\times 15$ $\vec{q}$ grid is used for Fourier transform.
Comparison with results obtained with $10\times 10\times 10$ and $20\times
20\times 20$ grids shows that this is enough for the supercell considered in
this work.

Table 2: Contributions to the free energy of electronic excitations $F^{\rm el}$ and of atomic vibrations $F^{\rm vib}$ calculated at different temperatures in pure Ti and in ordered Ti6O compound. Calculations are performed at the equilibrium volume and $c/a$ ratio of the cohesive energy $E^{\rm coh}$ at 0 K. | Free energies (meV/site)
---|---
| Pure Ti | Ti6O
$T$(K) | $F^{\rm el}$ | $F^{\rm vib}$ | $F^{\rm el}$ | $F^{\rm vib}$
0 | $0.0$ | $0.7$ | $0.0$ | $46.8$
300 | $-0.9$ | $-45.7$ | $-1.5$ | $5.9$
600 | $-4.1$ | $-169.2$ | $-3.8$ | $-115.8$
900 | $-10.0$ | $-332.1$ | $-8.7$ | $-283.0$

The total free energy $F(V,\gamma,T)=E^{\rm coh}(V,\gamma)+F^{\rm
vib}(V,\gamma,T)$ is obtained by adding the cohesive energy $E^{\rm
coh}(V,\gamma)$ directly given by DFT calculations. One could also consider
the contribution $F^{\rm el}$ of electronic excitations to this free energy
[28, 29, 30], but calculations performed for pure Ti and for the ordered Ti6O
compound show that this contribution can be neglected compared to the cohesive
and vibrational contributions (Tab. 2). The same conclusion was reached by
Argaman et al. in pure Ti [31].

a)

b)

Figure 4: Variation of the free energy with a) the volume and b) the
$\gamma=c/a$ ratio for pure Ti (left), the ordered Ti6O compound (center) and
a disordered solid solution of composition Ti6O (right). Free energies are
shown every 100 K. Symbols are the direct results of ab initio calculations
with quasi harmonic approximation and lines their interpolation. The $c/a$
ratio and volume have been fixed to their 0 K equilibrium values respectively
in a) and b). The dashed line in a) shows the position of the equilibrium
volume for each temperature.

For each structure, the phonon spectrum is calculated on a $4\times 4$ grid
corresponding to 4 different volumes $V$ and 4 different $\gamma=c/a$ ratios
choosing a variation $\sim\pm 10\%$ around the equilibrium values of the
cohesive energy. The free energy can then be defined on this $(V,\gamma)$ grid
for each temperature and a continuous representation obtained thanks to a
smooth interpolation with second rank polynomials (Fig. 4). Minimization of
this free energy finally leads to the equilibrium lattice parameters $a(T)$
and $c(T)$. As shown in appendix A, this approach predicts lattice expansion
in pure Ti in agreement with experimental data (Fig. 10).

###  III.2 Variations of lattice parameters

Figure 5: Lattice expansion in Ti-O alloys as a function of oxygen content
$x_{\rm O}$. Red disks correspond to ordered compounds and black squares to
disordered solid solutions, both calculated for the compositions Ti12O and
Ti6O. Results in absence of atomic vibrations, without zero-point energy, are
indicated at 0 K with open symbols and dashed lines. Solid and dashed lines
are linear fits of the corresponding data, imposing the lattice parameter of
pure titanium in $x_{\rm O}=0$.

Variations of lattice parameters with temperature are calculated for two
different oxygen compositions, Ti6O and Ti12O, in addition to pure Ti. For
each composition, we considered a random solid solution, as given by SQS
method, and a fully ordered structure with the most stable $AVBV$ stacking of
basal planes (Fig. 1). At 0 K, in absence of atomic vibrations, the same
variations of the lattice parameters with O concentration and ordering state
(open symbols in Fig. 5) are observed as in previous section (Fig. 3), thus
showing that the obtained lattice parameters do not depend on the supercell
used to model the different structures. Accounting for atomic vibrations does
not modify how these lattice parameters vary qualitatively. Results confirm
that lattice parameters increase with the oxygen content at all temperatures
(Fig. 5), with variations which can be considered linear both in the
disordered and the ordered states. The impact of oxygen ordering is the same
as the tendency observed at 0 K for the highest concentrations: the ordered
state has larger $c$ and lower $a$ lattice parameter than the disordered solid
solution at the same oxygen concentration. This impact of oxygen ordering
slightly increases with the temperature, as can be seen from the values of the
slopes $\alpha^{\rm O}_{a}=1/a\ \partial a/\partial x_{\rm O}$ and
$\alpha^{\rm O}_{c}=1/c\ \partial c/\partial x_{\rm O}$ given in table 3. The
difference between the values for the ordered and the disordered states
increases with the temperature both for $\alpha^{\rm O}_{a}$ and $\alpha^{\rm
O}_{c}$. But this variation remains small and can be considered as second
order compared to the variation with the oxygen concentration.

Table 3: Slopes characterizing linear variations of lattice parameters $a$ and $c$ with O concentration, $\alpha^{\rm O}_{a}=1/a_{\rm Ti}\ \partial a/\partial x_{\rm O}$ and $\alpha^{\rm O}_{c}=1/c_{\rm Ti}\ \partial c/\partial x_{\rm O}$, shown in Fig. 5 for different temperatures. |  | 0 K | 300 K | 600 K | 900 K
---|---|---|---|---|---
$\alpha^{\rm O}_{a}$ | Ordered TiOx | 0.040 | 0.035 | 0.028 | 0.022
| Disordered TiOx | 0.055 | 0.055 | 0.049 | 0.048
$\alpha^{\rm O}_{c}$ | Ordered TiOx | 0.139 | 0.147 | 0.155 | 0.159
| Disordered TiOx | 0.106 | 0.111 | 0.115 | 0.116

To a good approximation, one can neglect the variations with the temperature
of the increase rates $\alpha^{\rm O}_{a}$ and $\alpha^{\rm O}_{c}$. The
variations of the lattice parameters with the temperature and with the oxygen
concentration are then reasonably well described by the equations

$\displaystyle a(T,x_{\rm O})$ $\displaystyle=a^{0}_{\rm
Ti}\left(1+\alpha_{a}^{T}\,T\right)\left(1+\alpha_{a}^{\rm O}\,x_{\rm
O}\right),$ (2) $\displaystyle c(T,x_{\rm O})$ $\displaystyle=c^{0}_{\rm
Ti}\left(1+\alpha_{c}^{T}\,T\right)\left(1+\alpha_{c}^{\rm O}\,x_{\rm
O}\right),$ (3)

showing that the increases with temperature $T$ and with oxygen concentration
$x_{\rm O}$ can be factorized. Parameters given in Table 4 obtained through a
fit of ab initio calculations confirm that $a$ and $c$ lattice parameters
increase with the oxygen content, with a slightly higher (respectively lower)
increase rate of the $c$ (respectively $a$) parameter for the ordered than the
disordered states.

Table 4: Parameters entering Eq. 3 and describing variations of Ti lattice parameters with temperature and oxygen content for the ordered and disordered states. Pure Ti |  $a^{0}_{\rm Ti}=2.940$ Å |  $c^{0}_{\rm Ti}=4.660$ Å
---|---|---
|  $\alpha_{a}^{T}=9.4\times 10^{-6}$ K-1 |  $\alpha_{c}^{T}=8.3\times 10^{-6}$ K-1
Ordered TiOx | $\alpha_{a}^{\rm O}=0.031$ | $\alpha_{c}^{\rm O}=0.150$
Disordered TiOx | $\alpha_{a}^{\rm O}=0.052$ | $\alpha_{c}^{\rm O}=0.112$

##  IV Experiments and discussion

###  IV.1 Material and methods

X-ray diffraction (XRD) measurements are performed on titanium and a Ti-O
binary alloy. The titanium, hereafter denoted pure titanium, contains only 600
wppm oxygen, corresponding to an atomic composition $c_{\rm O}=N_{\rm
O}/(N_{\rm Ti}+N_{\rm O})=0.18$ at.% and to an occupation of interstitial
octahedral sites $x_{\rm O}=N_{\rm O}/N_{\rm Ti}=0.18$%, while the oxygen
nominal concentration in the Ti-O alloy is 6000 wppm ($c_{\rm O}=1.77$ at.%
and $x_{\rm O}=1.81$%). These concentrations are confirmed by inert gas fusion
analysis. Both pure Ti and Ti-O alloy are first hot- and cold-rolled with a
thickness reduction of 75% and 40% respectively to finally obtain 1 mm thick
plates. The samples are then recrystallized in molten salt baths (Li2CO3,
Na2CO3, K2CO3) at 1023 K during 600 s. After water quenching, both pure
titanium and Ti-O samples present fully recrystallized $\alpha$ grains with a
90 and 30 $\mu$m size respectively. The samples are then polished with SiC
grinding papers, before a final polishing with colloidal silica suspension.

XRD patterns are obtained with an XPert Pro Panalytical instrument operating
at 40 kV and 40 mA. A cobalt source ($K_{\alpha 1}=1.789$ Å and $K_{\alpha
2}=1.793$ Å) is used. A first diffractogram is recorded at room temperature
without the heating chamber, so that the sample height can be adjusted after
installing the chamber by comparing the peak positions. After the measurement
at room temperature, an Anton Paar HTK 1200N furnace with a resistive system
is used to heat the sample holder up to 473 K and then to different
temperatures between 473 and 773 K at 10 K/min. The temperature measured on
the sample holder could slightly differ from the sample temperature. The
heating chamber is kept under argon flow to prevent oxidation. After one hour
of heating stabilization for each temperature, diffractograms are recorded in
the 40-90∘ $2\theta$-range with a 0.016∘ step size during 80 s. 5 scans are
recorded at each temperature step and then summed up for the analysis.

Lattice parameters of both $\alpha$ matrix and ordered precipitates are
determined thanks to a Rietveld refinement using the Maud software. The
refinement takes account of the non-monochromatic feature of the X-ray beam
and allows to assign the shoulder on the right of the peaks to the $K_{\alpha
2}$ ray of the cobalt source. The Caglioti parameters of the diffractometer
are first calibrated with a LaB6 powder. The COD (Crystallographic Open
Database) ID #9008517 is chosen for the crystal structure of the $\alpha$
matrix. Ordered precipitates have a Ti6O structure, which corresponds to an
ordering of the oxygen atoms in the octahedral interstitial sites of the hcp
Ti-lattice [17]. XRD pattern of the precipitate phase should theoretically
display the same principal peaks as the $\alpha$ matrix, with some additional
peaks arising from oxygen ordering. However, a previous study on the same Ti-O
samples [17] has shown that these additional peaks cannot be detected by XRD
because of their too low intensity and that Ti6O-type precipitates can be
characterized only by shoulders on the left of the matrix peaks, corresponding
to slightly larger $a$ and $c$ lattice parameters of the underlying hcp
lattice. Therefore, the same COD ID is used in the Rietveld refinement for the
ordered precipitates as for the $\alpha$ matrix. Finally, the sample holder in
alumina leads to some additional peaks which are taken into account in the
Rietveld refinement by considering an Al2O3 phase (COD ID #1000059).

Transmission electron microscopy (TEM) observations of the Ti-O alloys are
performed with a JEOL TEM after the recrystallization heat treatment, using
the same sample preparation as in Refs. [17, 19].

###  IV.2 Results

TEM observations confirm the presence of precipitates with a nanometric size
(Fig. 6). These precipitates, which have been previously imaged in dark-field
conditions using the superlattice diffraction of the Ti6O ordered structure
[32, 17], can also be imaged in bright field mode, thanks to their elastic
strain field. This strain field arises from the lattice misfit between the
ordered precipitates and the surrounding titanium matrix.

Figure 6: TEM observation in bright field mode of the Ti-O alloy. Figure 7:
X-ray diffraction patterns obtained at three different temperatures and their
Rietveld refinement showing the different contributions of the $\alpha$-Ti
matrix, the ordered compounds, and the alumina sample holder. These
refinements lead to goodness of fit (GOF) values [33] between 1.1 and 1.5.

This lattice misfit leads to the presence of shoulders on the diffraction
peaks in the XRD diffractograms (Fig. 7 and Fig. 12 in appendix C). This
shoulder can be clearly seen on the left of the diffraction peak corresponding
to the basal 0002 planes ($44^{\circ}<2\theta<45^{\circ}$). It is the
signature of the larger $c$ lattice parameter of the ordered compounds
compared to the one of the matrix. Note that this shoulder appearing for the
basal 0002 planes cannot be explained by a martensitic transformation of the
hexagonal $\alpha$ phase to the orthorhombic $\alpha^{\prime\prime}$ phase as
such a transformation will lead to a splitting of all diffraction peaks except
the ones corresponding to basal planes. This shoulder on 0002 diffraction
peaks can be explained only by a mixture of two phases with different $c$
lattice parameters. Such a left shoulder is also visible on the peak
corresponding to pyramidal 10-11 planes ($46.5^{\circ}<2\theta<47^{\circ}$).
On the other hand, because of the sample texture, the intensity of the peaks
corresponding to 10-10 prismatic planes is not high enough to detect any
shoulder. It is also observed that the signature of the ordered compounds is
not very intense and almost disappears at 723 K and above (Fig. 12 in appendix
C), making analysis at these temperatures difficult. This is an indication
that the ordered compounds have started to dissolve and that O solubility
limit at these temperatures is higher than the sample nominal concentration,
i.e. 6000 wppm, which looks reasonable according to the theoretical phase
diagram [10]. Such a dissolution of ordered precipitates have also been seen
recently in a Ti-Zr-O alloy using TEM and resistivity with in situ heating
experiments [34].

Figure 8: Increase of lattice parameters $a$ and $c$ induced by oxygen as a function of the temperature. Lines are the predictions of the ab initio model for the ordered Ti6O ordered compound (purple line) and for a random solid solution of composition $x_{\rm O}=1.81$% (green line). Symbols are results of experiments for the ordered compounds and for the matrix. Table 5: Experimental values of lattice parameters deduced from Maud Rietveld refinement and oxygen content $x_{\rm O}=N_{\rm O}/N_{\rm Ti}$ deduced from these measurements using Eq. 4. $T$ (K) | sample | phase |  $a$ (Å) |  $c$ (Å) | $x_{\rm O}$
---|---|---|---|---|---
298 | pure Ti | matrix |  2.9536 | $\pm$ | $1\times 10^{-4}$ |  4.6850 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9501 | $\pm$ | $1\times 10^{-4}$ |  4.6929 | $\pm$ | $1\times 10^{-4}$ |  0.012
| Ti-O | compounds |  2.9575 | $\pm$ | $2\times 10^{-4}$ |  4.7061 | $\pm$ | $2\times 10^{-4}$ |  0.030
473 | pure Ti | matrix |  2.9586 | $\pm$ | $1\times 10^{-4}$ |  4.6913 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9551 | $\pm$ | $1\times 10^{-4}$ |  4.6999 | $\pm$ | $1\times 10^{-4}$ |  0.013
| Ti-O | compounds |  2.9566 | $\pm$ | $3\times 10^{-4}$ |  4.7125 | $\pm$ | $2\times 10^{-4}$ |  0.029
523 | pure Ti | matrix |  2.9596 | $\pm$ | $1\times 10^{-4}$ |  4.6937 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9567 | $\pm$ | $1\times 10^{-4}$ |  4.7022 | $\pm$ | $1\times 10^{-4}$ |  0.013
| Ti-O | compounds |  2.9642 | $\pm$ | $6\times 10^{-4}$ |  4.7146 | $\pm$ | $2\times 10^{-4}$ |  0.030
573 | pure Ti | matrix |  2.9610 | $\pm$ | $1\times 10^{-4}$ |  4.6960 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9577 | $\pm$ | $1\times 10^{-4}$ |  4.7045 | $\pm$ | $1\times 10^{-4}$ |  0.013
| Ti-O | compounds |  2.9651 | $\pm$ | $7\times 10^{-4}$ |  4.7132 | $\pm$ | $2\times 10^{-4}$ |  0.025
598 | pure Ti | matrix |  2.9613 | $\pm$ | $1\times 10^{-4}$ |  4.6966 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9594 | $\pm$ | $1\times 10^{-4}$ |  4.7060 | $\pm$ | $1\times 10^{-4}$ |  0.015
| Ti-O | compounds |  2.9624 | $\pm$ | $4\times 10^{-4}$ |  4.7156 | $\pm$ | $2\times 10^{-4}$ |  0.027
623 | pure Ti | matrix |  2.9616 | $\pm$ | $1\times 10^{-4}$ |  4.6977 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9600 | $\pm$ | $1\times 10^{-4}$ |  4.7075 | $\pm$ | $1\times 10^{-4}$ |  0.016
| Ti-O | compounds |  2.9634 | $\pm$ | $3\times 10^{-4}$ |  4.7170 | $\pm$ | $2\times 10^{-4}$ |  0.027
648 | pure Ti | matrix |  2.9621 | $\pm$ | $1\times 10^{-4}$ |  4.6986 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9608 | $\pm$ | $1\times 10^{-4}$ |  4.7087 | $\pm$ | $1\times 10^{-4}$ |  0.017
| Ti-O | compounds |  2.9645 | $\pm$ | $3\times 10^{-4}$ |  4.7169 | $\pm$ | $2\times 10^{-4}$ |  0.026
673 | pure Ti | matrix |  2.9626 | $\pm$ | $1\times 10^{-4}$ |  4.6993 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9607 | $\pm$ | $1\times 10^{-4}$ |  4.7086 | $\pm$ | $1\times 10^{-4}$ |  0.015
| Ti-O | compounds |  2.9640 | $\pm$ | $3\times 10^{-4}$ |  4.7170 | $\pm$ | $2\times 10^{-4}$ |  0.025
698 | pure Ti | matrix |  2.9630 | $\pm$ | $1\times 10^{-4}$ |  4.7002 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9623 | $\pm$ | $1\times 10^{-4}$ |  4.7115 | $\pm$ | $1\times 10^{-4}$ |  0.019
| Ti-O | compounds |  2.9671 | $\pm$ | $5\times 10^{-4}$ |  4.7186 | $\pm$ | $4\times 10^{-4}$ |  0.026
723 | pure Ti | matrix |  2.9636 | $\pm$ | $1\times 10^{-4}$ |  4.7014 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9625 | $\pm$ | $1\times 10^{-4}$ |  4.7128 | $\pm$ | $1\times 10^{-4}$ |  0.019
| Ti-O | compounds |  2.9642 | $\pm$ | $1\times 10^{-4}$ |  4.7224 | $\pm$ | $4\times 10^{-4}$ |  0.029
748 | pure Ti | matrix |  2.9642 | $\pm$ | $1\times 10^{-4}$ |  4.7025 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9632 | $\pm$ | $1\times 10^{-4}$ |  4.7083 | $\pm$ | $1\times 10^{-4}$ |  0.010
| Ti-O | compounds |  2.9643 | $\pm$ | $1\times 10^{-4}$ |  4.7451 | $\pm$ | $7\times 10^{-4}$ |  0.059
773 | pure Ti | matrix |  2.9646 | $\pm$ | $1\times 10^{-4}$ |  4.7036 | $\pm$ | $1\times 10^{-4}$ |
| Ti-O | matrix |  2.9646 | $\pm$ | $1\times 10^{-4}$ |  4.7150 | $\pm$ | $1\times 10^{-4}$ |  0.020
| Ti-O | compounds |  2.9739 | $\pm$ | $4\times 10^{-4}$ |  4.7493 | $\pm$ | $6\times 10^{-4}$ |  0.065

Rietveld refinement of XRD diffractograms allows extracting the lattice
parameters $a$ and $c$ of pure Ti, as well as of both the matrix and the
ordered compounds in the Ti-O alloy (Tab. 5). As expected, the ordered
compounds have larger lattice parameters than the matrix in the Ti-O alloy for
all the considered temperatures. To compare the lattice parameters measured in
the Ti-O alloy with the ones deduced from ab initio calculations, we compute
the differences $\delta a$ and $\delta c$ between the parameters measured in
the alloy and in pure Ti at the same temperature, as these variations are less
sensitive to the choice of the exchange-correlation functional used for ab
initio calculations than absolute values $a$ and $c$ (cf appendix B). We
neglect, for experimental data, the presence of 0.18 at.% oxygen in pure Ti,
as it only has a marginal impact on the obtained $\delta a$ and $\delta c$
values. $\delta a$ and $\delta c$ characterize the increase of lattice
parameters induced by the presence of oxygen. The increase $\delta c$ measured
for the matrix phase is almost constant with the temperature and its value
corresponds to the theoretical one obtained for a random solid solution with
the same nominal concentration as the Ti-O alloy (Fig. 8). This is a strong
indication that there is no real oxygen depletion in the matrix despite the
presence of ordered compounds. The experimental lattice increases $\delta c$
obtained for the compounds are higher than the ones of the matrix but do not
reach the theoretical values expected for ordered compounds with a Ti6O
stoichiometry (Fig. 8). Although these compounds are enriched in oxygen, they
do not have the expected 1/6 stoichiometry. The same conclusions could be
drawn from the obtained $\delta a$ lattice increase although things are less
clear. The uncertainty comes from the smallest expected values for $\delta a$
than for $\delta c$, because the impact of oxygen is smaller on $a$ than $c$
as already shown before (Fig. 5). Besides, because of the sample texture
leading to the absence of the prismatic peak, the experimental precision on
$a$ is not as high as on $c$, explaining why slightly negative values are
obtained for $\delta a$ in the matrix while a lattice increase is expected.

###  IV.3 Discussion

Despite the uncertainty existing on both the theoretical and experimental
variations of Ti lattice parameters with oxygen content and with ordering
state, one can try to determine the composition of the ordered compounds and
of the matrix from the increases $\delta a$ and $\delta c$ of the lattice
parameters, which have been measured by XRD at the different temperatures. A
least square fit of both $a$ and $c$ parameters leads to the following
expression for the occupation of the octahedral interstitial sites

$x_{\rm O}(\delta a,\delta c)=\frac{a_{\rm Ti}\,\alpha_{a}^{\rm O}\,\delta
a+c_{\rm Ti}\,\alpha_{c}^{\rm O}\,\delta c}{\left(a_{\rm Ti}\,\alpha_{a}^{\rm
O}\right)^{2}+\left(c_{\rm Ti}\,\alpha_{c}^{\rm O}\right)^{2}}.$ (4)

Using the lattice parameters of pure Ti, $a_{\rm Ti}$ and $c_{\rm Ti}$,
determined experimentally at each temperature with the theoretical lattice
increase $\alpha_{a}^{\rm O}$ and $\alpha_{c}^{\rm O}$ given in table 4, thus
neglecting their variations with the temperature, one obtains the compositions
$x_{\rm O}$ given in table 5 and shown in Fig. 9. These results confirm that
the ordered compounds have a lower composition than the expected value $x_{\rm
O}=1/6\sim 0.167$ corresponding to Ti6O stoichiometry. Their composition
starts to rise at the two highest temperatures considered in this study, 748
and 773 K, but without reaching the Ti6O composition. However, as noted
before, the intensity of the diffraction peaks assigned to the ordered
compounds is not very intense at these two temperatures (Fig. 12 in appendix
C) and one cannot expect an accurate determination of the corresponding
lattice parameters at these high temperatures where the ordered compounds have
probably started to dissolve. Fig. 9 also illustrates that this approach to
determine the composition of the matrix and of the ordered compounds cannot be
considered fully quantitative: at low temperature, one observes an oxygen
depletion of the matrix and an enrichment of the compounds, as expected, but
at the highest temperatures, both phases are enriched, which violates
conservation law. This uncertainty on the obtained composition arises from the
difficulty to deconvoluate the X-ray diffraction pattern to associate
different lattice parameters to different phases, the closer the lattice
parameters, the more difficult. It is also hard to reach a precision high
enough in the ab initio determination of the increase rates $\alpha_{a}^{\rm
O}$ and $\alpha_{c}^{\rm O}$, especially for such small variations of the
lattice parameters. Finally, when deriving a composition from measured lattice
parameters, we assume that the different phases are free of stress, which is
not true for coherent compounds embedded in a matrix. Nevertheless, despite
all the remaining uncertainties, the approach allows concluding that the
ordered compounds do not have the Ti6O composition and are only slightly
enriched with oxygen compared to the surrounding matrix.

Figure 9: Oxygen content $x_{\rm O}$ of the matrix and of the ordered
compounds determined with Eq. 4 from the experimental lattice parameters (Tab.
5) and from the theoretical increase rates $\alpha_{a}^{\rm O}$ and
$\alpha_{c}^{\rm O}$ (Tab. 4). The horizontal lines correspond to the nominal
composition $x_{\rm O}=0.0181$ of the Ti-O alloy and to the composition of
stoichiometric Ti6O compounds ($x_{\rm O}=1/6\sim 0.167$).

As shown in Ref. [17] with electronic diffraction, these ordered compounds,
which can be found in binary Ti-O alloys, have a crystallographic structure
corresponding to the one known for Ti6O crystal (space group P31c). Although
the binary phase diagram predicts that the equilibrium composition of these
ordered compounds should be close to Ti6O stoichiometry [10], the present work
shows that these precipitates do not have this composition. One possibility is
that oxygen diffusion is not fast enough to reach thermal equilibrium.
Considering the oxygen diffusion coefficient $D_{\rm O}(T)=D^{0}_{\rm
O}\exp{(-Q_{\rm O}/R\,T)}$ with an activation energy $Q_{\rm O}=200$ kJ mol-1
and a diffusion pre-exponential constant $D^{0}_{\rm O}=4.5\times 10^{-5}$ m2
s-1 [35], one can estimate the mean square displacement of oxygen atoms when
diffusing a time laps $\Delta t$, $d_{\rm O}(T,\Delta t)=\sqrt{6\,D_{\rm
O}(T)\,\Delta t}$. For the recrystallization heat treatment ($T=1023$ K and
$\Delta t=600$ s), the oxygen mean square displacement is 3 $\mu$m, thus
showing that equilibrium is reached. But, according to the theoretical phase
diagram [10], this temperature is too high for Ti6O ordered compounds to be
stable. The disappearance of the signature of this ordered phase in the
diffraction patterns above 723 K is also a strong indication that Ti6O
compounds are not stable at the recrystallization temperature. The ordered
compounds should rather appear during the quench, when the temperature becomes
low enough to lead to the decomposition of the initial homogeneous solid
solution in a depleted solid solution and an ordered phase with Ti6O-type
crystallographic structure. If the subsequent quench is fast enough, which
should be true for the water quench used here, one does not expect that the
forming precipitates reach equilibrium, thus explaining why the observed
ordered compounds have a lower composition than the expected Ti6O
stoichiometry.

Considering now the stays ($\Delta t=3600$ s) at different temperatures for
the XRD measurements, the composition of the precipitates remains almost
unchanged up to 723 K (Fig. 9), with the signature of the ordered compounds in
the XRD pattern disappearing at this temperature and above (Fig. 12 in
appendix C). Oxygen diffusion should be too slow below 723 K to allow for a
change of precipitates. This appears reasonable as the oxygen mean square
displacement is only 60 nm at $T=723$ K. At this temperature and above, it
appears high enough to allow for precipitate evolution, in particular their
dissolution, but not below. Better characterization of the kinetic evolution
of the ordered compounds will require additional experimental techniques like
synchrotron XRD with in situ heating to follow more precisely the evolution of
the matrix and compounds lattice parameters, and hence of their composition,
during the different heat treatments. In situ heating experiments in a TEM are
also appealing to image the evolution of these precipitates, in particular of
their size, shape, and density. Once the thermal conditions have been found to
get closer to equilibrium without dissolving the precipitates, it should be
possible to use analytical TEM or atom probe tomography to check that
equilibrium precipitates have a composition close to Ti6O stoichiometry, as
expected from the equilibrium phase diagram [10].

##  V Conclusions

Ab initio calculations correctly predict an increase of titanium lattice
parameters with the oxygen concentration for compositions between pure Ti and
Ti6O. The increase is more important for $c$ than for $a$ parameter, in
agreement with experiments. The ordering state of oxygen atoms in the
octahedral sites of the hcp lattice has an influence on these lattice
parameters, but the impact is less important than the one of oxygen
concentration: for a same oxygen composition, an ordered compound has a
slightly larger (respectively lower) $c$ parameter (respectively $a$
parameter). This is true both at 0 K and at finite temperature, when lattice
expansion is considered in the ab initio approach through the quasi-harmonic
approximation.

Knowing the impact of both oxygen concentration and oxygen ordering on lattice
parameters, it is theoretically possible to relate the lattice mismatches of
ordered compounds existing in Ti-O binary alloys to their composition.
Although the approach cannot be fully quantitative, because of the limited
accuracy of ab initio phonon calculations and their difficulty to predict
small variations of lattice parameters, it nevertheless leads to new
information on the composition of the different phases. Applying the approach
to an alloy containing 6000 wppm O ($x_{\rm O}=1.81$%), one can conclude that
the ordered compounds, which are present after a heat treatment at 1023 K
followed by a water quench, do not have the expected Ti6O stoichiometry but a
composition close to the alloy nominal concentration. This makes these
precipitates undetectable by experimental techniques which are sensitive only
to chemistry, like analytical TEM or atom probe tomography. Oxygen ordering
proceeds therefore at a faster pace than partitioning in this alloy: these
precipitates have a long-range order corresponding to the Ti6O structure, as
has been previously shown with TEM [17], but their composition is lower than
the equilibrium one corresponding to Ti6O stoichiometry. These precipitates,
which are out of equilibrium and probably appear during the quench following
the recrystallization heat treatment, do not evolve until reaching
temperatures high enough to allow for significant diffusion of oxygen, i.e.
above 723 K. It remains to be found in more details how the thermo-mechanical
processing, in particular the temperature and duration of the annealing or the
cooling speed, impact the composition of these ordered compounds.

##  Appendix A Lattice expansion of pure Ti

Figure 10: Lattice expansion of pure titanium as predicted by ab initio
calculations with the quasi-harmonic approximation (lines) and compared to the
experimental data using recommended values of Touloukian et al. [36].

We examine in this appendix how quantitative is our modeling approach relying
on ab initio calculations and the quasi-harmonic approximation to predict the
variations with temperature of the lattice parameters, considering pure Ti for
which experimental data can be found [36]. Calculations have been performed in
a supercell containing 96 Ti atoms which is a $4\times 4\times 3$ replication
of the conventional hcp unit cell. As shown in Fig. 10, the model reproduces
reasonably well the experimental data, in particular the anisotropy of the
linear expansion characterized by a decreasing $c/a$ ratio with temperature
and a contraction of $c$ parameter at very low temperature, in agreement with
previous ab initio calculations [37, 38, 31]. The best agreement is found for
the variations of the volume $V$ with the temperature, as characterized by the
average linear expansion

$\alpha_{L}=\frac{1}{3V}\left.\frac{\partial V}{\partial T}\right|_{P}.$

##  Appendix B Impact of exchange-correlation functional

Figure 11: Variations of $a$ and $c$ lattice parameters with O concentration
for ordered and disordered structures calculated at 0 K with two different
exchange-correlation functionals, GGA-PBE (full symbols continuous lines) and
LDA (empty symbols, dashed lines). Lines are fit of the ab initio
calculations, imposing the lattice parameters of pure Ti for $x_{\rm O}=0$.

Equilibrium lattice parameters obtained from ab initio calculations are known
to depend on the functional used to describe the electronic exchange and
correlation. To test how the exchange-correlation functional impacts the
obtained variations of the lattice parameters, we have performed additional
calculations using the local density approximation (LDA) with the
parameterization of Perdew and Zunger [39]. Except for the exchange-
correlation functional, the ab initio parameters are the same as for GGA
calculations. Calculations have been performed at 0 K, neglecting atomic
vibrations, for pure Ti, Ti12O, and Ti6O, considering a disordered (SQS) and
an ordered state. The structures, which rely on a 72 atoms supercell, are the
sames as the ones described in section III.1. As expected, the LDA functional
leads to slightly smaller lattice parameters than GGA (Fig. 11). But looking
now to relative variations, one sees that LDA and GGA lead to the same linear
variations with the oxygen content. The lines describing these linear
variations are parallel, with only an offset corresponding to the different
lattice parameters predicted by LDA and GGA for pure Ti. This is true both for
the ordered and disordered states, with the increase rate of $a$ (respectively
$c$) lattice parameter being slightly higher (respectively lower) for the
disordered than for the ordered state. As long as one works with the lattice
increases $\delta a$ and $\delta c$ induced by oxygen, there is thus no impact
of the exchange-correlation functional used for ab initio calculations, at
least when considering LDA and GGA-PBE functionals.

##  Appendix C XRD data

Figure 12: X-ray diffraction patterns obtained at different temperatures in
the Ti-O alloy. For each temperature, the experimental diffractogram (black)
is shown with the contributions of the $\alpha$-Ti matrix (green) and the
ordered compounds (purple) obtained from Rietveld refinement.

###### Acknowledgements.

This work was performed using HPC resources from GENCI-IDRIS and -TGCC (Grants
2022-096847). The French National Research Agency (ANR) is acknowledged for
the funding of TiTol Project [ANR-19-CE08-0032]. Y. Millet from the Timet
company is gratefully acknowledged for providing the alloys studied in this
work.

## References

  * Conrad [1981] H. Conrad, Effect of interstitial solutes on the strength and ductility of titanium, Prog. Mater. Sci. 26, 123 (1981).
  * Murray and Wriedt [1987] J. L. Murray and H. A. Wriedt, The O-Ti (oxygen-titanium) system, Bull. Alloy Phase Diagr. 8, 148 (1987).
  * Kornilov and Glazova [1965] I. I. Kornilov and V. V. Glazova, Phase diagram of the system Ti-O2 and some properties of the alloys of this system, in  _Physical metallurgy of titanium_ , edited by I. I. Kornilov (1965) pp. 12–28.
  * Yamaguchi  _et al._ [1966] S. Yamaguchi, M. Koiwa, and M. Hirabayashi, Interstitial superlattice of Ti6O and its transformation, J. Phys. Soc. Jpn. 21, 2096 (1966).
  * Yamaguchi [1969] S. Yamaguchi, Interstitial order-disorder transformation in the Ti-O solid solution. I. Ordered arrangement of oxygen, J. Phys. Soc. Jpn. 27, 155 (1969).
  * Hirabayashi  _et al._ [1974] M. Hirabayashi, S. Yamaguchi, H. Asano, and K. Hiraga, Order-disorder transformations of interstitial solutes in transition metals of IV and V groups, in  _Order-Disorder Transformations in Alloys_ (Springer, Berlin Heidelberg, 1974) pp. 266–302.
  * Banerjee and Mukhopadhyay [2007] S. Banerjee and P. Mukhopadhyay,  _Phase transformations: examples from titanium and zirconium alloys_, edited by R. W. Cahn, Pergamon Materials Series, Vol. 12 (Elsevier Science, 2007).
  * Ruban  _et al._ [2010] A. V. Ruban, V. I. Baykov, B. Johansson, V. V. Dmitriev, and M. S. Blanter, Oxygen and nitrogen interstitial ordering in hcp Ti, Zr, and Hf: An ab initio study, Phys. Rev. B 82, 134110 (2010).
  * Burton and van de Walle [2012] B. P. Burton and A. van de Walle, First principles phase diagram calculations for the octahedral-interstitial system $\alpha$TiOx, 0 $\leq x\leq 1/2$, Calphad 39, 97 (2012).
  * Gunda  _et al._ [2018] N. S. H. Gunda, B. Puchala, and A. V. der Ven, Resolving phase stability in the Ti-O binary with first-principles statistical mechanics methods, Phys. Rev. Mater. 2, 033604 (2018).
  * Yu  _et al._ [2015] Q. Yu, L. Qi, T. Tsuru, R. Traylor, D. Rugg, J. W. Morris, M. Asta, D. C. Chrzan, and A. M. Minor, Origin of dramatic oxygen solute strengthening effect in titanium, Science 347, 635 (2015).
  * Barkia  _et al._ [2015] B. Barkia, V. Doquet, J.-P. Couzinié, I. Guillot, and E. Héripré, In situ monitoring of the deformation mechanisms in titanium with different oxygen contents, Mater. Sci. Eng., A 636, 91 (2015).
  * Barkia  _et al._ [2017] B. Barkia, J.-P. Couzinié, S. Lartigue-Korinek, I. Guillot, and V. Doquet, In situ TEM observations of dislocation dynamics in $\alpha$ titanium: Effect of the oxygen content, Mater. Sci. Eng., A 703, 331 (2017).
  * Chaari  _et al._ [2019] N. Chaari, D. Rodney, and E. Clouet, Oxygen-dislocation interaction in titanium from first principles, Scr. Mater. 162, 200 (2019).
  * Chong  _et al._ [2020] Y. Chong, M. Poschmann, R. Zhang, S. Zhao, M. S. Hooshmand, E. Rothchild, D. L. Olmsted, J. W. Morris, D. C. Chrzan, M. Asta, and A. M. Minor, Mechanistic basis of oxygen sensitivity in titanium, Sci. Adv. 6, eabc4060 (2020).
  * Chong  _et al._ [2023] Y. Chong, R. Gholizadeh, T. Tsuru, R. Zhang, K. Inoue, W. Gao, A. Godfrey, M. Mitsuhara, J. W. Morris, A. M. Minor, and N. Tsuji, Grain refinement in titanium prevents low temperature oxygen embrittlement, Nat. Commun. 14, 404 (2023).
  * Poulain  _et al._ [2022] R. Poulain, S. Delannoy, I. Guillot, F. Amann, R. Guillou, S. Lartigue-Korinek, D. Thiaudière, J.-L. Béchade, E. Clouet, and F. Prima, First experimental evidence of oxygen ordering in dilute titanium–oxygen alloys, Mater. Res. Lett. 10, 481 (2022).
  * Kornilov [1973] I. I. Kornilov, Effect of oxygen on titanium and its alloys, Met. Sci. Heat Treat. 15, 826 (1973).
  * Amann  _et al._ [2023a] F. Amann, R. Poulain, S. Delannoy, J.-P. Couzinié, E. Clouet, I. Guillot, and F. Prima, An improved combination of tensile strength and ductility in titanium alloys via oxygen ordering, Mater. Sci. Eng. A 867, 144720 (2023a).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Blöchl [1994] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  * Monkhorst and Pack [1976] H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).
  * van de Walle  _et al._ [2013] A. van de Walle, P. Tiwary, M. de Jong, D. L. Olmsted, M. Asta, A. Dick, D. Shin, Y. Wang, L.-Q. Chen, and Z.-K. Liu, Efficient stochastic generation of special quasirandom structures, Calphad 42, 13 (2013).
  * Yamaguchi  _et al._ [1970] S. Yamaguchi, K. Hiraga, and M. Hirabayashi, Interstitial order-disorder transformation in the Ti-O solid solution. IV. A neutron diffraction study, J. Phys. Soc. Jpn. 28, 1014 (1970).
  * Wiedemann  _et al._ [1987] K. E. Wiedemann, R. N. Shenoy, and J. Unnam, Microhardness and lattice parameter calibrations of the oxygen solid solutions of unalloyed $\alpha$-titanium and Ti-6Al-2Sn-4Zr-2Mo, Metall. Trans. A 18, 1503 (1987).
  * Togo and Tanaka [2015] A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scripta Mater. 108, 1 (2015).
  * Wolverton and Zunger [1995] C. Wolverton and A. Zunger, First-principles theory of short-range order, electronic excitations, and spin polarization in Ni-V and Pd-V alloys, Phys. Rev. B 52, 8813 (1995).
  * Zhang  _et al._ [2017] X. Zhang, B. Grabowski, F. Körmann, C. Freysoldt, and J. Neugebauer, Accurate electronic free energies of the $3d$, $4d$, and $5d$ transition metals at high temperatures, Phys. Rev. B 95, 165126 (2017).
  * Cottura and Clouet [2018] M. Cottura and E. Clouet, Solubility in Zr-Nb alloys from first-principles, Acta Mater. 144, 21 (2018).
  * Argaman  _et al._ [2016] U. Argaman, E. Eidelstein, O. Levy, and G. Makov, Ab initio study of the phononic origin of negative thermal expansion, Phys. Rev. B 94, 174305 (2016).
  * Poulain [2020] R. Poulain,  _Conception et développement d’une nouvelle famille d’alliages de titane à haute résistance mécanique et biocompatibilité optimisée pour l’implantologie dentaire_ , Ph.D. thesis, Université PSL (2020).
  * Toby [2006] B. H. Toby, ${R}$ factors in Rietveld analysis: How good is good enough?, Powder Diffr. 21, 67 (2006).
  * Amann  _et al._ [2023b] F. Amann, R. Poulain, S. Delannoy, R. Guillou, M. Talla Noutack, Z. Kloenne, S. Nowak, J.-P. Couzinié, F. De Geuser, D. Thiaudière, J.-L. Béchade, E. Clouet, H. Fraser, I. Guillot, and F. Prima, On the oxygen ordering mechanisms in dilute Ti-Zr-O systems, in  _$15^{th}$ World Conference on Titanium_, Vol. in press (2023).
  * Liu and Welsch [1988] Z. Liu and G. Welsch, Literature survey on diffusivities of oxygen, aluminum, and vanadium in alpha titanium, beta titanium, and in rutile, Metall. Trans. A 19, 1121 (1988).
  * Touloukian  _et al._ [1975] Y. S. Touloukian, R. K. Kirby, R. Taylor, and P. D. Desai, Thermal expansion - metallic elements and alloys, in  _Thermophysical Properties of Matter Thermal Expansion_ , Vol. 12 (Plenum, New York, 1975).
  * Souvatzis  _et al._ [2007] P. Souvatzis, O. Eriksson, and M. I. Katsnelson, Anomalous thermal expansion in $\alpha$-titanium, Phys. Rev. Lett. 99, 015901 (2007).
  * Mei  _et al._ [2009] Z.-G. Mei, S.-L. Shang, Y. Wang, and Z.-K. Liu, Density-functional study of the thermodynamic properties and the pressure–temperature phase diagram of Ti, Phys. Rev. B 80, 104116 (2009).
  * Perdew and Zunger [1981] J. P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Phys. Rev. B 23, 5048 (1981).