Thermal Expansion and Other Thermodynamic Properties of α2-Ti3Al and γ-TiAl Intermetallic Phases from First Principles Methods

Anisotropic thermal expansion coefficients of tetragonal γ-TiAl and hexagonal α2-Ti3Al phases were calculated using first principles methods. Two approaches with different computational costs and degrees of freedom were proposed. The predicted values were compared with available experimental data showing that for γ-TiAl, the more computational demanding method with decoupled impact of volume and temperature effects on the cell shape leads to significantly better results than that with only ground-state optimised unit cell geometry. In the case of the α2-Ti3Al phase, both approaches yielded comparable results. Additionally, heat capacity and bulk modulus were evaluated as functions of temperature for both phases, and were fitted to provide an analytical formula which can be further used.


Introduction
First principles calculations are now a widely used and well-established method for complementing experimental materials science research [1]. Despite the fact that many recent activities have been directed towards big-data and machine learning [2][3][4], there are still many topics which require individualised treatments. An example of such a problem is the discrepancy between the experimentally and theoretically reported stability and chemistry of the Nb 3 Al phase also published in this special issue [5]. Starting from the pioneering works of Grabowski and co-workers [6][7][8][9][10][11], the first-principles thermodynamics by including the vibrational contribution to the free energy within the harmonic approximation have became fairly routine. Among other available tools, phonopy [12] has become widely used thanks to its robustness, openness and flexibility. The there implemented quasi-harmonic approximation (QHA) for calculating thermal properties, such as thermal expansion, bulk modulus or heat capacity, however, does not include effects of temperature-induced changes in the unit cell geometry in terms of c/a or b/a ratios or lattice angles, as may be the case of systems with lower than cubic symmetry.
In this paper we will focus on γ-titanium aluminides, which are a class of intermetallic materials with a broad range of potential high temperature applications [13][14][15][16][17]. They exhibit good specific yield strength and elastic moduli at elevated temperatures, while simultaneously having low density, good oxidation resistance and resistance against Ti-fire [18][19][20][21][22]. Depending on the exact chemical composition, several phases are present in TiAl alloys [23]. The majority phase is the tetragonal γ-TiAl phase (tetragonal L1 0 , space group P4/mmm) after which this material class is named. In addition,

Methods
We used the state-of-the-art program VASP (Vienna Ab-initio Simulation Package) [41] employing Density Functional Theory [42,43] to carry out the first principles calculations. The atomic basis functions were represented by projector augmented wave pseudopotentials with the 3s 2 3p 6 4s 2 3d 2 and 3s 2 2p 1 valence electron configuration for Ti and Al atoms, respectively. The exchange-correlation effects were treated using gradient corrected exchange-correlation functional parametrised by Perdew-Burke-Ernzerhof (GGA-PBE) [44,45] and the plane wave cut-off energy of 500 eV were applied to predict ground state properties of both the γ-TiAl and α 2 -Ti 3 Al phases. The reciprocal unit cell was sampled with 14 × 14 × 14 (γ, 4 atoms) and 12 × 12 × 13 (α 2 , 8 atoms) k-point mesh using the Monkhorst-Pack scheme. These parameters guarantee total energy accuracy better than 1 meV/at.
The structural optimisation includes evaluation of total energies at various volumes. Full relaxation including unit cell shape and internal atomic coordinates optimisation was performed for every volume, yielding lattice parameters a ξ 0 (V) and c ξ 0 (V) (ξ = γ or α 2 ) as functions of volume at 0 K. Thermal properties were evaluated within the quasi-harmonic approximation (QHA) using the phonopy code [12,46] Assuming that the c/a ratio is only a function of volume and not temperature, the resulting temperature dependence of volume V ξ (T) as obtained from the QHA (phonopy-qha package), allows to determine also the temperature dependencies of individual lattice constants x ξ (T), x = a, c and ξ = γ, α 2 , as: This treatment is in the following termed as 'ground state optimised cell shape' (gs-cs).
To probe the validity of the assumption that the c/a is only a function of volume independent of temperature, we have adopted additional scheme. For every volume, we selected 5 c/a ratios around the GGA-PBE equilibrium values ((c/a) GGA-PBE γ = 1.018, (c/a) GGA-PBE α 2 = 0.809). For each of these static configurations, thermodynamic properties within the harmonic approximation (phonopy package) were calculated, hence yielding vibrational Helmholtz free energies F vib (T, V, c/a). The total Helmholtz free energy, F, was constructed by adding the 0 K total energies: The equilibrium geometry at a fixed temperature T was then calculated by a two step fitting. First, we estimated by fitting the F(T = const., V = const, c/a) data with a second order polynomial. Subsequently, the F(T, V) data were fitted with the Birch-Murnaghan equation of state [47] to obtain the equilibrium volume V 0 (T) (in addition to free energy, F(T), bulk modulus, B(T), and pressure derivative of bulk modulus, B (T)). Finally, the (c/a)(T = const., V) data minimising F(T = const., V, c/a) in Equation (3), were linearly interpolated as a function of V, and the equilibrium value at temperature T was estimated from this linear fit at V = V 0 (T). This procedure allows to decouple influence of temperature and pressure (volume) on the cell geometry, and is in the following thus termed 'temperature optimised cell shape' (to-cs) approach. The thermal expansion coefficients were calculated from the estimated lattice parameters as Finally, the heat capacity at constant (ambient) pressure, C p , was estimated from the Helmholtz free energy, F ξ (T), as The latter expressions in Equations (4) and (5) represent numerical derivatives as both, lattice constants and Helmholtz free energy were calculated on a discrete set of temperatures from 0 to 1000 K with a step of 10 K.

Thermal Expansion
We start our analysis by comparing the predicted temperature dependence of specific volumes of the α 2 -Ti 3 Al and γ-TiAl phases using both approaches as described in the Section 2. Figure 1a shows the temperature dependence of specific volume (i.e., volume per atom) for both considered phases as predicted using volume geometries optimised only at 0 K (gs-cs) and at every temperature (to-cs). While these two approaches provide almost identical results for the α 2 -Ti 3 Al phase (blue curves), significant differences are obtained for the γ-TiAl. Namely, the gs-cs method yields larger and faster expanding volumes than the to-cs treatment. The former is also significantly non-linear, suggesting that the resulting coefficient of volume thermal expansion is strongly increasing at higher temperatures and does not reach the usual near-to-linear behaviour for temperatures above room temperature (RT, ∼298 K). γ-TiAl (orange) phases predicted using quasi-harmonic approximation with cell shape optimised at 0 K (dashed, label 'gs-cs' (ground state optimised cell shape)) and at every temperature (solid line, label 'to-cs' (temperature optimised cell shape)).
Importantly, both approaches allow for explicitly estimating a and c lattice constants describing the hexagonal α 2 -Ti 3 Al and tetragonal γ-TiAl structures. Similarly to the specific volume, the c/a ratio for the α 2 structure also does not differ much for both the gs-cs and to-cs approaches. While the absolute values do not differ very much, the qualitative temperature-dependence changes from c/a decreasing with temperature as predicted by the gs-cs method to c/a increasing with raising temperature for the to-cs approach (see Figure 1b). Qualitatively similar behaviour is also predicted for the γ-TiAl phase, although in the opposite sense: the gs-cs and to-cs methods predict slightly increasing and strongly decreasing c/a values, respectively, with increasing temperature.
The specific volume and c/a ratio allow to calculate also the corresponding lattice parameters a and c, and then to further use these to obtain lattice thermal expansion coefficients, α a (Figure 2a) and α c (Figure 2b), according to Equation (4). Regarding the hexagonal α 2 -Ti 3 Al phase, α a is slightly larger than α c for all temperatures. Perhaps the most important difference is that while α a still increases with temperature even above RT and to higher values than 10 × 10 −6 K −1 above ∼600 K, while α c seems to quickly saturate around 10 × 10 −6 K −1 above RT. Importantly, there are no significant differences between the predicted values by gs-cs and to-cs methods. The obtained differences are of the same order as the scatter of the numerical noise imposed by the to-cs method, represented by the individual data points in Figure 2.  and γ-TiAl (orange) phases predicted using quasi-harmonic approximation with cell shape optimised at 0 K (dashed, label 'gs-cs') and at every temperature (solid line, label 'to-cs'). The data points shown by dots are the actual numerically calculated values using Equation (4). The smooth curves are 'guides for the eyes' from interpolation using Bezier curves.
A very different situation is obtained in the γ-TiAl case. The gs-cs case predicts significant temperature dependence of both α a and α c , moreover, both having very similar values. This large increase of TEC with temperature is a consequence of strongly expanding volume of the γ-TiAl (cf. Figure 1a) resulting from the gs-cs method. On the other hand, the to-cs approach predicts large TEC values of ∼15 × 10 −6 K −1 above RT in the a-direction, while 3-fold smaller values of around 5 × 10 −6 K −1 (and basically temperature-independent above RT) are predicted for α c . This behaviour leads to a strong temperature dependence of c/a (cf. Figure 1b).
In summary, while the computationally more demanding to-cs method does not yield too different temperature dependence of structural properties in comparison to the simpler gs-cs approach for the α 2 -Ti 3 Al, non-negligible differences are obtained in the case of the γ-TiAl.

Other Thermodynamic Properties
The calculation of the thermal expansion is based on evaluation of the vibrational entropy term of the Helmholtz free energy, which is the most important contribution, and has been demonstrated several times to be the only important contribution when dealing with non-magnetic materials at temperatures far below melting point [9,11]. The thus obtained thermodynamic potentials, however, offer thermal dependence of other quantities, too.
The heat capacity, C p , at constant (ambient) pressure was evaluated according to Equation (5). The calculated values for the two phases are almost identical, in particular from the to-cs treatment (Figure 3a). This result could be intuitively understood by the fact that the molar heat capacities of Al and Ti are very similar [6]. Such prediction is important, e.g., for the discussion of microstructure evolution upon phase transformations during cooling. The temperature dependence of bulk modulus (i.e., the inverse of compressibility) (Figure 3b) can be estimated from fitting section of the Helmholtz free energy surface at fixed temperature with, e.g., the Birch-Murnaghan equation of state [47]. It turns out that the bulk modulus of the γ-phase is smaller than that of the α 2 -phase in the whole temperature range up to 1000 K. The bulk modulus softens with the increasing temperature by ∼12% between 0 K and 770 K (∼500 • C) and by approximately ∼8.5% between RT and 500 • C for both, γ and α 2 -phases. Significantly different is only the gs-cs temperature dependence for the γ-phase, which yields drop of over 30% between 0 and 770 K, further underlying that this approach is not reasonable for the γ-TiAl, in accordance with other properties discussed so far.
In order to provide the reader with an easy access to our calculated quantities, the trends were fitted with analytical expressions and the resulting fitted parameters as well as the fits themselves are summarised in Appendix A.

Discussion
As mentioned in the introduction, experimental data for comparison are scarce. In fact, data corresponding to exactly ideal conditions of single phases with exact stoichiometric compositions are non-existant at all. Nevertheless, experimental data on single crystal [34], as well as on polycrystalline γ-TiAl [35], suggest that α a > α c , in agreement with our calculations. On the one hand, the differences between a ([1 0 0]) and c ([0 0 1]) directions are not so large in experiments as they are predicted here (cf. Figure 2), on the other hand the experimental data are for Al-rich compositions with 54 and 56 at.% Al. That the composition can play a significant role is demonstrated by the hugely different TEC reported for Ti-46Al-3Nb-1.9Cr (at. %) [38] and Ti-45Al-7.5Nb-0.5C (at. %) [39]. In the light of these hugely scattering experimental data, our predictions are qualitatively correct in implying α a > α c .
The structurally optimised c/a ratio of ∼1.020 is in excellent agreement with the experimentally reported values 1.016 [48] for Al-rich γ-TiAl to 1.02 [17]. The latter two values are higher than the ∼1.012 measured for Ti-45Al-7.5Nb-0.5C (at. %) [39]. Importantly, Li et al. [39] obtained a slightly decreasing c/a ratio with increasing temperature up to 1000 K (followed by a strong decrease for temperatures further increasing up to 1400 K), a result qualitatively well in agreement with our to-cs predictions. Further on, Li et al. [39] reported that the c/a of α 2 -Ti 3 Al stays rather constant, i.e., ∼0.806, in the temperature range from 450 to 1000 K. Despite the fact that the gs-cs and to-cs approaches yield decreasing and increasing c/a with temperature, respectively, the temperature dependence is not very strong (as, e.g., in the case of the γ-TiAl phase) and hence both methods are, in fact, valid for the α 2 -phase. It should also be mentioned that experimental results on polycrystalline specimens may be biased by building up coherency strains between phases with different expansion behaviour [49]. Especially in the case of TiAl alloys, which contain a large volume fraction of lamellar α 2 /γ colonies, this effect may potentially have a significant impact on the obtained experimental data.
Finally, our calculated values of TEC agree with experimental results, but are significantly lower than other DFT-based predictions by Fu et al. [40]. We ascribe this discrepancy to the different methodology used: in the present study, we have explicitly evaluated the vibrational contribution to the Helmholtz free energy by calculating phonon properties, whereas Fu et al. [40] used a semi-classical Debye model. We therefore conclude that explicit evaluation of the phonon frequencies and their contribution to the phonon free energy is essential.

Conclusions
Thermal properties, with a special focus on structural analysis of temperature dependent lattice parameters and coefficients of thermal expansion of tetragonal γ-TiAl and hexagonal α 2 -Ti 3 Al phases of binary Ti-Al system, were calculated using first principles methods. We put our attention on testing whether the c/a ratio is purely a function of volume independent of temperature, or whether temperature and volumetric effects have to be separated. Our calculations show that in the case of the γ-TiAl phase significant differences are obtained, while both approaches yield comparable results for the hexagonal α 2 -Ti 3 Al phase. The predictions were further compared with available experimental data. While this was not straightforward due to lack of single-crystalline data with close-to-perfect stoichiometries, we propose that the to-cs method with decoupled impact of temperature and volume on the cell geometry (c/a ratio) gives better agreement for the γ-TiAl phase. The present paper therefore contributes to advancing first principles thermodynamics beyond systems with cubic symmetry.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Abbreviations
The following abbreviations are used in this manuscript:

DFT
Density Functional Theory GGA Generalised Gradient Approximation gs-cs ground state optimised cell shape HA harmonic approximation QHA quasi-harmonic approximation RT room temperature TEC coefficient of thermal expansion to-cs temperature optimised cell shape

Appendix A. Analytical Fits
The thermodynamic quantities discussed above calculated using the to-cs approach were fitted with a function of the form This function fits accurately all obtained data within the temperature window from 0 to 1000 K. The fitted coefficients for X = F (Helmholtz free energy), C p (molar heat capacity), B (bulk modulus) and α a and α c (TEC in the a and c directions) are summarised in Tables A1 and A2, and the fits are presented in Figure A1. The calculated dependencies provide a consistent set of material constants.