The impact of Ti and temperature on the stability of Nb5Si3 phases: a first-principles study

Abstract Nb-silicide based alloys could be used at T > 1423 K in future aero-engines. Titanium is an important additive to these new alloys where it improves oxidation, fracture toughness and reduces density. The microstructures of the new alloys consist of an Nb solid solution, and silicides and other intermetallics can be present. Three Nb5Si3 polymorphs are known, namely αNb5Si3 (tI32 Cr5B3-type, D8l), βNb5Si3 (tI32 W5Si3-type, D8m) and γNb5Si3 (hP16 Mn5Si3-type, D88). In these 5–3 silicides Nb atoms can be substituted by Ti atoms. The type of stable Nb5Si3 depends on temperature and concentration of Ti addition and is important for the stability and properties of the alloys. The effect of increasing concentration of Ti on the transition temperature between the polymorphs has not been studied. In this work first-principles calculations were used to predict the stability and physical properties of the various Nb5Si3 silicides alloyed with Ti. Temperature-dependent enthalpies of formation were computed, and the transition temperature between the low (α) and high (β) temperature polymorphs of Nb5Si3 was found to decrease significantly with increasing Ti content. The γNb5Si3 was found to be stable only at high Ti concentrations, above approximately 50 at. % Ti. Calculation of physical properties and the Cauchy pressures, Pugh’s index of ductility and Poisson ratio showed that as the Ti content increased, the bulk moduli of all silicides decreased, while the shear and elastic moduli and the Debye temperature increased for the αNb5Si3 and γNb5Si3 and decreased for βNb5Si3. With the addition of Ti the αNb5Si3 and γNb5Si3 became less ductile, whereas the βNb5Si3 became more ductile. When Ti was added in the αNb5Si3 and βNb5Si3 the linear thermal expansion coefficients of the silicides decreased, but the anisotropy of coefficient of thermal expansion did not change significantly.


Introduction
The development of high-temperature engineering alloys that can operate at temperatures above those of the latest generations of Ni-based superalloys is a priority in current metallurgical research to enable future gas turbine technologies to meet environmental and performance targets [1]. The Nb-silicide based alloys have higher melting temperatures, lower densities and better creep properties and are stable at higher temperatures than the Ni-based superalloys. These new alloys are also known as Nb in situ composites, and their microstructures consist of Nb solid solution that provides toughness and intermetallics that give low-and high-temperature strength and creep resistance [2]. Different alloying additions are used to achieve a balance of properties, in particular room-temperature fracture toughness, low-and high-temperature oxidation resistance and strength and creep [1,2].
The addition of Ti in Nb-silicide based alloys not only reduces their density but also improves their fracture toughness and oxidation resistance [2,4,5]. To achieve a balance of properties, the concentration of Ti in Nb-silicide based alloys must be optimized because Ti (i) does not increase the ductile to brittle transition temperature (DBTT) of bcc Nb for concentrations up to ≈ 24 at. %, (ii) has the weakest effect of all additions X on the yield strength at T = 1095 °C and high-temperature strength at T = 1200 °C of Nb-X solid solution alloys, where X is transition (including refractory) metal [6] and (iii) substitutes for Nb in (Nb,Ti) 5 Si 3 silicides and increases the toughness of unalloyed Nb 5 Si 3 from about 3 MPa√m to about 10 MPa√m at Ti ≈ 25 at. %, but at higher Ti contents the hexagonal (Ti,Nb) 5 Si 3 is stabilized and the toughness drops to values below 3 MPa√m [4]. The stable structure for the fully Ti-substituted end member, i.e. the Ti 5 Si 3 , is hexagonal (hP16 Mn 5 Si 3 -type, D8 8 ). The Ti 5 Si 3 is isomorphous with γNb 5 Si 3 .
Even though Ti is an important addition, there is lack of data in the literature about the effect that Ti has on the stability of the different Nb 5 Si 3 polymorphs. The effect of alloying with Ti on the transformation temperature between the two tetragonal polymorphs has not been reported, nor has the effect of Ti on their coefficient of thermal expansion (CTE). However, it has been shown that high concentrations of Ti in (Ti,Nb) 5 Si 3 stabilized the 5-3 silicide in the hexagonal crystal structure in Nb-silicide based alloys at temperatures below 1500 °C [7,8]. The latter is undesirable because the hexagonal 5-3 silicide is reported to have inferior creep properties than the αNb 5 Si 3 and βNb 5 Si 3 [1,2]. The CTE of Ti 5 Si 3 is also significantly more anisotropic [9].
The early data that were used to construct liquidus projection of the Nb-Ti-Si ternary system did not identify which was the structure of 5-3 compound(s) in the cast alloys (i.e. authors did not clarify which 5-3 polymorph was formed), and the projection gave a primary Nb 5 Si 3 solidification area without specifying whether the primary silicide was the βNb 5 Si 3 or the αNb 5 Si 3 or the hexagonal Ti 5 Si 3 based 5-3 silicide [10]. Geng et al. [11] proposed a liquidus projection for the Nb-Ti-Si ternary system with a large primary αNb 5 Si 3 solidification area. Li et al. [12] revised the Nb-Ti-Si liquidus projection based on a study of ternary alloys in the Nb 5 Si 3 -Ti 5 Si 3 region. The proposed liquidus projection by Li et al. shows that primary βNb 5 Si 3 will form for Ti concentrations up to approximately 40 at. %, the liquidus projection has a very narrow primary αNb 5 Si 3 solidification area and indicates that at higher concentrations primary hexagonal Ti 5 Si 3 will form during solidification. A similar liquidus projection was proposed recently by Jânio Gigolotti et al. [13], with an extended βNb 5 Si 3 region and narrow αNb 5 Si 3 area. No primary αNb 5 Si 3 solidification area is shown in the Nb-Ti-Si liquidus projection by Bulanova and Fartushna [14]. There are no data about the transformation temperature of 5-3 silicides alloyed with Ti below the liquidus.
In this work first-principles calculations are used to study the stability and physical properties of the three polymorphs, αNb 5 Si 3 , βNb 5 Si 3 and γNb 5 Si 3 alloyed with Ti (up to 12.5 at. % Ti for αNb 5 Si 3 and βNb 5 Si 3 and up to 50 at. % Ti for γNb 5 Si 3 ). Density functional theory (DFT) is used to study the enthalpy of formation and properties of the αNb 5 Si 3 , βNb 5 Si 3 and γNb 5 Si 3 compounds with and without Ti additions at T = 0 K. To probe the effect of Ti on the transformation temperatures, the temperature dependence of the heats of formation of the compounds is computed by incorporating phonon calculations. The paper provides new data that advance current understanding of the stability of complex Nb-silicide based alloys and the design and development of new alloys.

Computational details
The CASTEP (Cambridge Serial Total Energy Package) code [15] was used for the calculations, as described by Papadimitriou et al. [16]. The valences for the atomic configurations were Nb-4s 2 4p 6 4d 4 5s 1 , Ti-3s 2 3p 6 3d 2 4s 2 and Si-3s 2 3p 2 . An energy cut-off of 500 eV was sufficient to reduce the error in the total energy to less than 1 meV/atom. A Monkhorst-Pack k-point grid separation of 0.03 Å −1 was used for the integration over the Brillouin zone according to the Monkhorst-Pack scheme [17]. Geometry optimizations of the structures were performed with thresholds for converged structures less than 1 × 10 −7 eV, 1 × 10 −3 eV/Å, 1 × 10 −4 Å and 0.001 GPa, respectively, for energy change per atom, maximum residual force, maximum atomic displacement and maximum stress.
The method of finite displacements was used [16]. The forces on atoms were calculated when slightly perturbing the ionic positions [18]. The supercells used were as follows: 4 × 4 × 4 for Nb, 4 × 4 × 3 for Ti, 3 × 3 × 3 for Si, 2 × 2 × 2 for γNb 5 Si 3 and Ti 5 Si 3 and 2 × 2 × 1 for αNb 5 Si 3 and βNb 5 Si 3 . The vibrational contributions to the enthalpy, entropy, free energy and heat capacity versus temperature and the Debye temperature were obtained using the quasiharmonic approximation [16]. The phonon density of states (DOS) of each element separately was calculated to obtain the finite temperature enthalpy of formation.
The linear thermal expansion coefficients (α) were obtained by generating structures with increasing the ratios a/a 0 and c/c 0 (a 0 and c 0 are the lattice parameters in the ground state) from 0.991 to 1.006 with an increment of 0.003 and conducting a phonon calculation for each volume. The equilibrium lattice parameters a(T,P) and c(T,P) were then calculated at every given temperature using the quasi-harmonic approximation by minimizing the total free energy with respect to volume, thus finding the equilibrium volume at each temperature. After calculating the a(T,P) and c(T,P) the linear thermal expansion coefficients α a and α c were obtained. This procedure was repeated for αNb 5 Si 3 , βNb 5 Si 3 , αNb 16 Ti 4 Si 12 and βNb 16 Ti 4 Si 12 .
The elastic constants and properties were calculated as described in Papadimitriou et al. [16]. The calculation method consisted of applying a given strain and calculating the stress. At each deformation the unit cell was kept fixed, and the internal coordinates were optimized. The matrix of the linear elastic constants was reduced according to the crystal structure of each phase. The maximum number of strain patterns (sets of distortions) for a tetragonal or hexagonal structure is two and one for cubic cells. Six strain steps (varying from -0.003 to 0.003) were used [16].
For the cubic (Nb) and diamond (Si) structures a series of six geometry optimizations were done to evaluate the three independent elastic constants C 11 , C 12 and C 44 , whereas for the tetragonal αNb 5 Si 3 and βNb 5 Si 3 and hexagonal Ti, γNb 5 Si 3 and Ti 5 Si 3 structures the corresponding number was twelve, with the six independent elastic constants being C 11 , C 12 , C 13 , C 33 , C 44 and C 66 for the tetragonal and C 11 , C 12 , C 13 , C 33 and C 44 for the hexagonal. After acquiring the matrix of the elastic constants and confirming that the mechanical stability criteria [19] are satisfied, the bulk (B), Young's (E) and shear (G) moduli, Poisson's ratio (ν) and Debye temperature were obtained as described in Papadimitriou et al. [16].

Site occupancies, lattice constants and densities of states
Twelve structures in total were investigated in the current study, four for each of the αNb 5 Si 3 , βNb 5 Si 3 and γNb 5 Si 3 silicides. In all cases, each of the four structures contained an increasing number of Ti atoms, starting from 1 and increasing to 4. Thus, from the structure with the lowest Ti content to that with the highest, the corresponding percentages were 3.125, 6.25, 9.375 and 12.5 at. % Ti for the αNb 5 Si 3 and βNb 5 Si 3 and 6.25, 12.5, 18.75 and 25 at. % for the γNb 5 Si 3 . Higher Ti concentrations of 37.5 at. % and 50 at. % were considered in order to study the effect of the Ti concentration on the stability of the hexagonal silicide, and provide an estimation of the critical Ti concentration to form γNb 5 Si 3 . Ab initio technique has been used previously to study the effects of alloying on stability and mechanical properties of αNb 5 Si 3 [20][21][22]. In the first-principles study by Chen et al. [21] they considered the effect of the substitution of Nb by Ti on the stability of Nb 5 Si 3 . Chen et al. studied only the substitution of one atom of Nb with Ti (i.e. alloying with 3.125 at. % Ti) on different atomic positions at 0 K. Figure 1 shows the crystal structures of the 5-3 silicide polymorphs. Ti can substitute Nb in all three polymorphs and occupies the more closely packed Nb sites in αNb 5 Si 3 and the less closely packed Nb sites in βNb 5 Si 3 and γNb 5 Si 3 [21,22]. In Figure 1, M and L, respectively, represent the more and the less closely packed sites. In the work presented in this paper, in order to investigate the order of the site occupancies of Ti atoms with increasing Ti concentration, separate geometry optimizations were made, and the enthalpies of formation at 0 Κ were computed (Table 1). In the case of γNb 5 Si 3 the enthalpies of formation for different combinations of occupancies were found to be approximately equal. The enthalpies of the most stable structures are indicated by bold numbers in Table 1. The numbers above each atom show the sequence of site occupation by the Ti atoms, reproduced from chen et al. [21]. reproduced with permission from american Physical Society. . The volume of all the 5-3 silicide polymorphs decreased as the Ti content increased, which is expected as Nb has a larger atomic radius than Ti [23,24]. The partial (PDOS) and total (TDOS) electronic densities of states are shown in Figures 2 to 4 for the α, β and γ 5-3 silicide structures. It can be seen that for all structures the main contribution to the TDOS was the PDOS of d electron states, followed by the p electron states, while the s electron states contribute the least to the TDOS of all structures. The location of the Fermi level is indicative of phase stability. If the Fermi level is located in a deep valley of the TDOS, this indicates phase stability, whereas the opposite is the case if the Fermi level is located near peaks of the TDOS. It is clear that for the unalloyed compounds the valleys near the Fermi levels were deeper in αNb 5 Si 3 (Figure 2(a)) than βNb 5 Si 3 (Figure 2(f)), whereas for the γNb 5 Si 3 (Figure 3(a)) the Fermi level is situated near one of the high peaks of the TDOS. This explains the gradual decrease of phase stability from α to β to γ 5-3 silicide.
The addition of Ti in the αNb 5 Si 3 slightly moves the Fermi level to the bottom of the deepest valley ( Figure  2(b)-(e)), making the silicide even more stable, while in the case of βNb 5 Si 3 the Fermi level moves slightly towards one of the small peaks (Figure 2(g)-(j)) rendering the silicide somewhat less stable. This confirms that the difference between the formation enthalpies of the α and β phases is increased as the aforementioned phases are alloyed (doped) with Ti. In the case of γNb 5 Si 3 , the Ti addition also moves the Fermi level slightly closer to one of the peaks (Figure 3(c)-(f)).
The evolution of the TDOS as the Ti concentration in γNb 5 Si 3 increases shows that the Fermi level would pass the large peak and move towards the valley below, as the Using the enthalpies of formation and equation 1 [21], the impurity formation energies E M f −im were calculated and are shown in Table 2. Negative impurity formation energy means that the impurity-doped (alloyed) phase is more stable than the unalloyed phase, while the lower the impurity formation energy is, the more stable the doped (alloyed) phase. The Ti-doped structures exhibited negative impurity formation energies, which confirmed the study by Chen et al. [21], where the impurity formation energies for Ti in Nb 5 Si 3 at T = 0 K were calculated. It can be seen in Table 2 that all the impurity formation energy values for all polymorphs were negative. The alloyed phase is more stable, the lower the impurity formation energy is. In all cases the impurity formation energy became more negative with each additional Ti atom, indicating increasing stability with increasing Ti substitution for all polymorphs.
The lattice constants and the volumes of the crystal structures of the 5-3 silicide polymorphs in the present study were calculated ( Table 3). The a and c lattice parameters of the αNb 5 Si 3 decreased and increased, respectively, as the Ti concentration increased. In the case of the βNb 5 Si 3 and γNb 5 Si 3 polymorphs both lattice  with the enthalpies of formation of the aforementioned phases (see Table 2), confirms that the hexagonal γNb 5 Table 4. The mechanical stability criteria [19] were met for all phases. The elastic constants for the pure elements were in agreement with the experimental data [25][26][27]. The property data for the un-doped αNb 5 Si 3 , βNb 5 Si 3 and γNb 5 Si 3 from the literature [16] are also given in Table 4. Compared with the VRH scheme, the values obtained by the B-M EOS tend to be larger. There is good agreement between the values from the two calculations. The bulk modulus tends to decrease with increasing Ti concentration in all 5-3 silicides. The calculated values of shear modulus (G) and Young's modulus (E) are given in Table 5. For the αNb 5 Si 3 and γNb 5 Si 3 silicides the shear and Young's moduli tend to increase with increasing Ti addition. In the case of βNb 5 Si 3 the corresponding values decrease.
50 at. %. Li et al. [12] reported that in a cast Nb-25Si-40Ti (at. %) alloy, the βNb 5-x (Ti) x Si 3 was the primary phase, whereas in the cast Nb-30Si-45Ti (at. %), Nb-25Si-45Ti (at. %) and Nb-25Si-60Ti (at. %) alloys the γTi 5-x (Nb) x Si 3 was the primary phase formed during solidification. The microstructures of the alloys studied by Li et al. [12] were not at equilibrium, nevertheless their data suggest that the hexagonal 5-3 silicide becomes stable at high Ti concentrations in excess of 40-45 at. %, which is in agreement with the present study.

Elastic properties
The results of the calculations of the independent elastic constants (C ij ), bulk moduli (B) from elastic constants according to the Voigt-Reuss-Hill (VRH) scheme and bulk moduli and first pressure derivatives of bulk moduli value of Cauchy pressures means respectively a ductile or brittle material [34]. The other two conditions for brittle behavior are G/B > 0.57 and ν < 0.26. The results of the present study would suggest that the most ductile of the unalloyed silicides is the γNb 5 Si 3 , and the least ductile is the αNb 5 Si 3 . The αNb 5 Si 3 and γNb 5 Si 3 silicides become more brittle as the Ti content increases, whereas the βNb 5 Si 3 becomes more ductile.
The Cauchy pressures (C 12 -C 44 for cubic and C 13 -C 44 and C 12 -C 66 for tetragonal and hexagonal structures), Pugh's [33] index of ductility (ratio of shear modulus over bulk modulus (G/B)) and Poisson's ratio (v) were calculated. The values of the aforementioned properties are given in Table 5. These parameters are often used as 'predictors' of the ductile or brittle behavior of intermetallics. For metallic bonding, a positive or negative  shows that all values are significantly lower for the D8 m . This shows that the temperature dependence of the phonon contribution favors the stability of the βNb 5 Si 3 over αNb 5 Si 3 with increasing temperature, which is expected from the binary phase diagram [3] and the experimental data for binary Nb-Si alloys. This trend is also followed by the Ti-alloyed phases, thus indicating that a Ti-alloyed βNb 5 Si 3 should become more stable than the Ti-alloyed αNb 5 Si 3 as the temperature increases.
After acquiring the ΔH f (T) for all unalloyed and alloyed phases, the phase equilibrium at finite temperatures was investigated. Figure 6 shows the enthalpy of formation of the γNb 5 Si 3 for Ti content between 0 and 25 at. % and the enthalpy of formation of Ti 5 Si 3 . The slope of each curve increases as the Ti content increases from 0 at. % to fully Ti-alloyed 5-3 silicide, i.e. Ti 5 Si 3 . In all cases, over the whole temperature range, the Ti 5 Si 3 has the lowest enthalpy of formation.
The enthalpy of formation against temperature of the D8 l , D8 m and D8 8 structures for up to 12.5 at. % Ti is shown in Figure 7. For all phases the enthalpy of formation increases with increasing temperature owing to the phonon contributions. Between 0 and 12.5 at. % Ti the γNb 5 Si 3 is not expected to be stable. This is in agreement with experiments that show that this phase is metastable at low Ti contents. In Figure 7, for 0 at. % Ti, the γNb 5 Si 3 curve does cross the βNb 5 Si 3 curve; however, this occurs at a temperature above the melting temperature of both phases. Here the stable phase would be the liquid.
The elastic moduli for different Ti concentrations in 5-3 silicides are given in Table 5. Elastic moduli reflect the cohesion in a crystal structure. For αNb 5 Si 3 and γNb 5 Si 3 the elastic modulus increases with increasing Ti concentration, whereas for βNb 5 Si 3 the elastic moduli decrease. This suggests that the addition of Ti strengthens atomic bonding in αNb 5 Si 3 and γNb 5 Si 3 , and reduces bond strength in βNb 5 Si 3 .

Enthalpies of formation, transition temperatures and thermal expansion coefficients
The vibrational density of states (DOS) for the elements and silicides of this study were calculated. All the eigenfrequencies were found to be real, hence it was confirmed that the silicides are mechanically stable. After inserting the computed phonon DOS in the relevant formulae the vibrational contribution to free energies per atom (F phon (T)) was calculated for the D8 l , D8 m and D8 8 structures. Data for the pure elements were shown previously in Papadimitriou et al. [16]. The F phonon for both αNb 5 Si 3 and βNb 5 Si 3 silicides decreased faster as the Ti addition increased, whereas for the γNb 5 Si 3 , the F phonon decreased more slowly as the Ti addition increased.
After taking F phonon into account, the phonon contribution to the enthalpy of formation (ΔH f phon (T)) was evaluated for the D8 l , D8 m and D8 8 structures ( Figure 5). For all the silicides the slope increased with increasing Ti addition. Comparison of the D8 l and D8 m structures   melting temperature of unalloyed αNb 5 Si 3 . The results of the present study suggest that a new assessment of the Nb-Ti-Si ternary system is needed.
The linear thermal expansion coefficients of the stable (tetragonal α and β) unalloyed Nb 5 Si 3 and two Ti-alloyed silicides, namely the αNb 16 Ti 4 Si 12 and βNb 16 Ti 4 Si 12 , are shown in Table 6. Also included in Table 6 are experimental values for Ti 5 Si 3 [9]. There is good agreement between the calculated values and the available data in the literature. The CTE of all the silicides is anisotropic. The Ti 5 Si 3 is the most anisotropic, whereas αNb 5 Si 3 is the least. Alloying with 12.5 at. % Ti decreases the thermal expansion coefficients of both αNb 5 Si 3 and βNb 5 Si 3 silicides. However, the addition of Ti does not have a strong effect on the CTE anisotropy of both the α and β Nb 5 Si 3 .

Debye temperatures
The phonon DOS was used to calculate the Debye temperature, as described in Papadimitriou et al. [16]. The calculated values (Table 5) are in good agreement with those calculated using the elastic constants. For the elements the results from the calculations based on phonon DOS and the elastic constants are in good agreement with the literature. Regarding the silicides studied in this paper, the Debye temperatures that were calculated using the two methods are also in good agreement. For the αNb 5 Si 3 and γNb 5 Si 3 silicides the Debye temperature increases with increasing Ti content, but for the βNb 5 Si 3 the opposite is the case, and the Debye temperature decreases slightly as more Nb atoms are substituted by Ti atoms.
Referring to the study by Chen et al. [30], according to which at the same temperature the number of the excited acoustic modes responsible for the stabilization of βNb 5 Si 3 with respect to αNb 5 Si 3 increases with the Ti content, it is the softer shear modulus of the Ti-alloyed βNb 5 Si 3 compared with the Ti-alloyed αNb 5 Si 3 that leads to the stability of this phase. For example, in Table 5 the shear moduli (G) of unalloyed α and β Nb 5 Si 3 , respectively, are 116.8 and 106.4 GPa. Alloying αNb 5 Si 3 with Ti increases the shear modulus from 126.1 to 128.7 GPa when the Ti content increases from 1 to 4 atoms, whereas for βNb 5 Si 3 the shear modulus decreases from 98.5 to 93.1 GPa when the Ti content increases from Comparing the unalloyed αNb 5 Si 3 with the βNb 5 Si 3 silicide, the former is stable up to 2085 K where its heat of formation curve crosses that of βNb 5 Si 3 , which becomes stable above this temperature (Figure 7(a)). This value is in good agreement with the transition temperature reported in the accepted Nb-Si binary phase diagram [3], as discussed in Papadimitriou et al. [16]. After adding Ti to the aforementioned structures this transition temperature decreases significantly to 1431 K for Nb 19 (Figure 7(a-e)). The contribution from the vibrational entropy is much greater for αNb 5 Si 3 with increasing temperature, compared with βNb 5 Si 3 . Hence, the addition of Ti appears to have a larger effect on the phonon contribution of αNb 5 Si 3 , which drives the transition temperature lower. For the Nb-Si-Ti ternary system there are no experimental data with which to compare the calculated transition temperatures given above. In early experimental isothermal sections for similar temperatures [10,35] the prototype of Nb 5 Si 3 was not stated. The error of finite temperature ab initio calculations can be large in some cases due to anharmonicity. Confidence in the above values is justified by the good agreement of the αNb 5 Si 3 → βNb 5 Si 3 transition temperature in the binary Nb-Si system with the literature.
Chen et al. [21] studied the stability of αNb 5 Si 3 and βNb 5 Si 3 when one Nb atom was substituted by a single Ti atom in its preferred site (e.g. the site with the lowest impurity energy) by comparing the differences in the calculated formation energies of the two silicides. They suggested that the larger the difference in formation energy, the higher the temperature of the phase transition. The difference between the enthalpy of formation at 0 K for unalloyed αNb 5 Si 3 and βNb 5 Si 3 and alloyed with 3.25 at. % Ti α and β Nb 5 Si 3 (1 Nb atom replaced by Ti) increases with the Ti addition and is comparable with the results in Chen et al. [21]. Thus, based on the assumption of Chen et al., this would suggest that Ti will stabilize αNb 5 Si 3 over βNb 5 Si 3 , and therefore the transition temperature would be expected to be pushed to higher values. Our results indicate the opposite trend, with Ti addition stabilizing βNb 5 Si 3 and decreasing the transition temperature. For αNb 5 Si 3 alloyed with Ti the temperature dependence of the phonon contribution to the heat of formation is much greater than that for βNb 5 Si 3 alloyed with Ti, and therefore the slope of the ΔH f (T) curve for αNb 5 Si 3 increases more dramatically with increasing temperature than for βNb 5 Si 3 . This indicates the importance of entropic contributions on phase stability that should be accounted for when considering the effect of alloying on transformation temperatures. In a thermodynamic assessment of the Nb-Ti-Si ternary system [11] the model used suggests that the stability of αNb 5 Si 3 increases with increasing Ti content, and that αNb 5 Si 3 alloyed with Ti becomes stable above the Table 6. linear thermal expansion coefficients (α a and α c ) for αnb 5 Si 3 , βnb 5 Si 3 , αnb 16  1 to 4 atoms. Therefore, as the concentration of Ti is increased, the difference in the shear moduli values also increases, and this results in a decrease of the transition temperature.

Conclusions
First-principles calculations were carried out for the D8 l , D8 m and D8 8 polymorphs of Nb 5 Si 3 alloyed with Ti, and the constituent elements. The volume of all structures contracted as the Ti addition increased. Elastic constants, bulk, shear and Young's moduli, Poisson's ratio and Debye temperature were calculated. These calculations showed that as the Ti content increased the bulk moduli of all silicides decreased, while the shear and elastic moduli increased for αNb 5 Si 3 and γNb 5 Si 3 and decreased for βNb 5 Si 3 . The Debye temperatures of αNb 5 Si 3 and γNb 5 Si 3 , and βNb 5 Si 3 , respectively, increased and decreased as the Ti addition increased. The calculations suggested that the γNb 5 Si 3 is the most ductile polymorph. The elastic properties of this silicide are reported in this paper. The alloying with Ti makes the αNb 5 Si 3 , and γNb 5 Si 3 silicides less ductile and βNb 5 Si 3 more ductile.
The transition temperature between the α and β structures decreases as more Ti is added, and at about 50 at. % Ti content the hexagonal silicide becomes stable over its tetragonal polymorphs. The αNb 5 Si 3 and βNb 5 Si 3 exhibit anisotropy of their coefficients of thermal expansion, with the latter being more anisotropic that the former. Alloying the aforementioned compounds with 12.5 at. % Ti decreases their thermal expansion coefficients α a and α c without significantly changing the ratio α a /α c . The results of this study indicate that the Ti-alloyed αNb 5 Si 3 should be the desirable silicide in Nb-silicide based alloys, and that careful consideration must be given to the transition temperature between the two phases. The transition temperatures of the 5-3 silicides alloyed with Ti must be studied experimentally.