Alloying Effect on Transformation Strain and Martensitic Transformation Temperature of Ti-Based Alloys from Ab Initio Calculations

The accurate prediction of alloying effects on the martensitic transition temperature (Ms) is still a big challenge. To investigate the composition-dependent lattice deformation strain and the Ms upon the β to α″ phase transition, we calculate the total energies and transformation strains for two selected Ti−Nb−Al and Ti−Nb−Ta ternaries employing a first-principles method. The adopted approach accurately estimates the alloying effect on lattice strain and the Ms by comparing it with the available measurements. The largest elongation and the largest compression due to the lattice strain occur along ±[011]β and ±[100]β, respectively. As compared to the overestimation of the Ms from existing empirical relationships, an improved Ms estimation can be realized using our proposed empirical relation by associating the measured Ms with the energy difference between the β and α″ phases. There is a satisfactory agreement between the predicted and measured Ms, implying that the proposed empirical relation could accurately describe the coupling alloying effect on Ms. Both Al and Ta strongly decrease the Ms, which is in line with the available observations. A correlation between the Ms and elastic modulus, C44, is found, implying that elastic moduli may be regarded as a prefactor of composition-dependent Ms. This work sheds deep light on precisely and directly predicting the Ms of Ti-containing alloys from the first-principles method.

The phase transformation temperature of pure Ti from the α phase (hexagonal closepacked (hcp)) to β phase (body centered-cubic (bcc)) is 1154 K [3].When the content of βstabilizing elements is low, the hcp martensite (α ) and orthorhombic martensite (α ) can be created from the β austenite phase by high-speed cooling [33].Additionally, the hexagonal ω phase can be generated from the β phase via severe plastic deformation [34][35][36] and from the α phase under the drive of high-temperature torsion [33,35,[37][38][39][40].The ω phase is detrimental to the shape memory effect and superelasticity of martensite.In this work, we effect on the lattice deformation strain, stereographic projections of the lattice strains, and comparisons between the predicted and measured M s for Ti-based alloys are shown in Section 3. We put forward a conclusion in Section 4.

Methodology
The total energies were calculated using the first-principles exact muffin-tin orbitals (EMTO) [55] method.The self-consistent calculations were performed using the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation [56].The Kohn-Sham equations were solved with scalar-relativistic approximation and soft-core approximation.To integrate the valence states below the Fermi level, Green's function was calculated for 16 complex energy points.The basis sets included the s, p, d, and f orbitals in EMTO.The alloys considered here were nonmagnetic.The random distribution in a solid solution was described by the coherent-potential approximation (CPA) [55], implying that the degree of the atomic order of a solid solution is treated in a completely disordered way.To process the electrostatic correction to the CPA, the screened impurity model [57] was used with a screening parameter of 0.6.The k point meshes were carefully chosen to describe the tiny changes in energy for different phases.Hence, the used k point meshes were 25 × 25 × 25 and 11 × 11 × 11 for the β and α phases in the irreducible Brillouin zone.The theoretical equilibrium lattice constant was determined by fitting the total energies of nine different atomic volumes based on a Morse equation of states.
Based on the crystallographic relationship among the β, α , and α phases [9,41,48], the α phase is equivalent to the β phase if b/a = c/a = √ 2 and shuffle y = 0, while the α phase turns into the α phase if b/a = √ 3 and y/b = 1/6.Consequently, four variable parameters, including the Winger-Seitz radius (w, in Bohr), the axis ratios of c/a and b/a, and shuffle y, dominate the total energy upon the β → α phase transition.It is found that a small volume difference produces a relatively tiny energy difference of less than 0.2 mRy for a given phase [58].Therefore, to reduce the complexity of structural optimization, we ignore the influence induced by volume on different phases when calculating the total energy and consider total energy as functions of b/a, c/a, and y.In the present study, the range of the b/a is 1.40-1.75,and the interval of the c/a is 1.40-1.70.The interval of y is set from 0 to 1/6b for every b/a and c/a.Spline interpolation is chosen to find the equilibrium shuffle y in each group of b/a and c/a.Then, two-dimensional cubic interpolation is used to determine the equilibrium b/a and c/a.

Results and Discussion
Transformation strain principally affects shape recovery strain.The martensitic transformation strain depends on lattice strain and lattice correspondence [9,41,48,53,66].The lattice correspondence between the β austenite phase and α orthorhombic martensite phase is displayed in Figure S1 and can be described, as below: The lattice deformation strains η 1 , η 2 , and η 3 [9,41,48,53,66] along the three principal axes of [100] β , [011] β , and [011] β are written as follows: where a , b , and c represent the lattice constants of the α phase and a β is the lattice parameter of β phase.
Figure 1 shows the present theoretical equilibrium lattice constants of Ti−22Nb−xX (x = 0-10, X = Al, Ta) in the β and α phases, compared with the available experimental data [4,9,11] in the β phase.Theoretical lattice constants from our static calculations are generally smaller than the experimental values.The partial reason may come from ignoring the thermal expansion caused by temperature effects.Alternatively, the deviation between calculation and measurement partially contributes to the different alloy compositions in our selected Ti−22Nb−Al and the measured Ti−24Nb−Al.However, the same composition dependence of the lattice constant appears for both theoretical calculations and available measurements.For example, the lattice constant a β of Ti−Nb−Al ternaries decreases about 1.07 × 10 −3 Å/1 at.% with an increase in Al, which is consistent with the available experimental decrement of 1.71 × 10 −3 Å/1 at.% [4] and 1.9 × 10 −3 Å/1 at.% [6].For Ti−Nb−Ta ternaries, a β keeps almost constant at around 3.26 Å, which is in line with the previous first-principles calculations [67].( The lattice deformation strains η1, η2, and η3 [9,41,48,53,66] along the three principal axes of [100]β, [011]β, and [01 ̅ 1]β are written as follows: where a′, b′, and c′ represent the lattice constants of the α″ phase and aβ is the lattice parameter of β phase. Figure 1 shows the present theoretical equilibrium lattice constants of Ti−22Nb−xX (x = 0−10, X = Al, Ta) in the β and α″ phases, compared with the available experimental data [4,9,11] in the β phase.Theoretical lattice constants from our static calculations are generally smaller than the experimental values.The partial reason may come from ignoring the thermal expansion caused by temperature effects.Alternatively, the deviation between calculation and measurement partially contributes to the different alloy compositions in our selected Ti−22Nb−Al and the measured Ti−24Nb−Al.However, the same composition dependence of the lattice constant appears for both theoretical calculations and available measurements.For example, the lattice constant aβ of Ti−Nb−Al ternaries decreases about 1.07 × 10 −3 Å/1 at.% with an increase in Al, which is consistent with the available experimental decrement of 1.71 × 10 −3 Å/1 at.% [4] and 1.9 × 10 −3 Å/1 at.% [6].For Ti−Nb−Ta ternaries, aβ keeps almost constant at around 3.26 Å, which is in line with the previous first-principles calculations [67].Like the decreasing a β of Ti−Nb−Al, the lattice constants a , b , and c of Ti−Nb−Al in the α phase also linearly decrease with increasing Al content.The a , b , and c of Ti−Nb−Al reduce by 1.37 × 10 −3 Å, 6.76 × 10 −4 Å, and 1.85 × 10 −3 Å with a 1 at.% increase in Al, respectively.The situation becomes complex for α Ti−Nb−Ta ternaries.The a of Ti−Nb−Ta is insensitive to Ta content, but the b' first decreases and then increases with increasing Ta content, while the c shows a linear increasing trend.Such complicated composition dependence agrees with the available measurement [9] and first-principles calculations [67], although the changes in a , b , and c for a given Ti−Nb−Ta system are somehow scattered [9,67].For example, the a , b , and c for Ti−37.5Nb−(12.5,18.75)Ta alloys [67] increase as Ta content increases, while the a (b and c ) increases (decrease) with alloying Ta into Ti−(14-18)Nb−(0-10)Ta alloys [9], differing from the theoretical trends in Figure 1b-d.This deviation may come from different Nb and Ta contents and different experimental processes.
In Figure 2a-c, we display the lattice deformation strains η 1 , η 2 , and η 3 of Ti−22Nb−xX (x = 0-10, in at.%; X = Al, Ta) using Equation (2).It is generally accepted that there is a positive correlation between lattice strain and recoverable strain in SMAs [32,48].From Figure 2, it can be observed that η 1 is negative, while η 2 is positive, indicating that the martensitic contracts (expands) the lattice along the [100] β ([011] β ) direction.This finding is consistent with the available measurements in Ti−Nb−Ta [9].Note that η 1 (η 3 ) is the largest (smallest) among all three deformation strains in Ti−Nb−Al and Ti−Nb−Ta ternaries.For Ti−Nb−Al alloys, the absolute magnitudes of η 1 and η 2 (η 3 ) increase (decreases) with increasing Al content, relative to the increase (decrease) in lattice strain.Additionally, the doping of Al produces different variations in lattice strains, which is different from previous theoretical [48] and measured [41,43] observations in Ti−Nb binary alloys.Unlike Ti−Nb−Al, the composition dependence of η 1 , η 2 , and η 3 of Ti−Nb−Ta ternary alloys is opposite to that of Ti−Nb−Al.The absolute magnitudes of both η 1 and η 2 (η 3 ) reduce (increases) with increasing Ta content, implying a decrease (increase) in lattice strain.It is found that the present predicted η 2 increases (decreases) with alloying Al (Ta), which is in line with former first-principles calculations [47].Note that theoretical a β in the β phase is smaller than the available experimental one by an overall error of 1.2%, as shown in Figure 1.The calculated a in the α phase is also underestimated, but theoretical b and c are rather close to the measurements.Therefore, the absolute magnitudes of η 1 , η 2 , and η 3 calculated by Equation ( 2) are all larger than the available measurements [6,9,11].The measured η 1 , η 2 , and η 3 in Ti−24Nb−3Al [6] are −2.96%,2.98%, and −0.04%, respectively.The measured η 1 , η 2 , and η 3 in Ti−17Nb−10Ta [9] are −2.28%,2.56%, and −0.38%, respectively, and the measured η 1 , η 2 , and η 3 in Ti−22Nb−6Ta [11] are −2.07%,2.47%, and −0.44%, respectively.Kim et al. [41] proposed an approach to calculate the maximum transformation strain (ε i M ) along a certain orientation and the average maximum transformation strain (ε M ) for a polycrystal with randomly distributed grains.Following Kim's approach [41], the lattice distortion matrix (T) during the β → α phase transformation relative to the coordinates of the β phase can be illustrated, as in Equation (3): Supposing a stochastic vector, x, in the β phase is transformed to x in the α phase due to martensitic transition, the maximum transformation strain, ε i M , along every orientation, can be evaluated, as in Equation (4): For comparison, the available experimental maximum recovery strains [4,7] are also displayed.
Kim et al. [41] proposed an approach to calculate the maximum transformation strain ( M  ) along a certain orientation and the average maximum transformation strain ( M ) for a polycrystal with randomly distributed grains.Following Kim's approach [41], the lattice distortion matrix (T) during the β → α″ phase transformation relative to the coordinates of the β phase can be illustrated, as in Equation (3): Supposing a stochastic vector,  ⃗, in the β phase is transformed to ′ ⃗⃗⃗⃗ in the α″ phase due to martensitic transition, the maximum transformation strain,  M  , along every orientation, can be evaluated, as in Equation ( 4): Kim's approach has been successfully applied to predict the  M  and  M of Ti-Nb binaries [41,68] and TiNb-based ternaries [9,66].To distinguish different strains, Figure S2 in the Supplementary Materials displays the relationship of the lattice deformation strains (η1, η2, and η3), the maximum transformation strain ( M  ), and the average maximum transformation strain ( M ).According to Equations ( 3) and ( 4), 57 representative orientations (i.e., the vertex and the midpoint of the edge at each standard stereographic triangle, as shown in Figure S3a) located in the standard stereographic circle are chosen to exhibit the stereographic projections of lattice strains along (100)β and (001)β, respectively.For comparison, the available experimental maximum recovery strains [4,7] are also displayed.
Kim's approach has been successfully applied to predict the ε i M and εM of Ti-Nb binaries [41,68] and TiNb-based ternaries [9,66].To distinguish different strains, Figure S2 in the Supplementary Materials displays the relationship of the lattice deformation strains (η 1 , η 2 , and η 3 ), the maximum transformation strain (ε i M ), and the average maximum transformation strain (ε M ).According to Equations ( 3) and ( 4), 57 representative orientations (i.e., the vertex and the midpoint of the edge at each standard stereographic triangle, as shown in Figure S3a) located in the standard stereographic circle are chosen to exhibit the stereographic projections of lattice strains along (100) β and (001) β , respectively.
Based on Kim's [41] approach, as shown in Equations ( 3) and ( 4), we first choose 13 representative orientations located in the [001]−[011]−[111] standard stereographic triangle (shown in the Figure S3b) and calculate the ε i M along these orientations.Consequently, the predicted εM can be obtained by spline interpolation of the ε i M .Thus, we compare the calculated εM for both Ti−Nb−Al and Ti−Nb−Ta alloys in Figure 2d, along with the available measured maximum recovery strains [4,7] for comparison.According to Figure 2d, the εM predicted by Equations ( 3) and ( 4) for Ti−Nb−Al and Ti−Nb−Ta ternaries are higher than the experimental values.This is partially from the underestimation of theoretical a β and a in the β and α phase.Furthermore, the measured recovered strains also depend on the tensile strains, which are limited due to the increasing remaining plastic strain.Despite the fact that the predicted εM for both alloys are somehow overestimated compared to those of the experimental counterparts, the composition dependence of theoretical εM reproduces the measurements [4,7].From Figure 2d, it can be observed that the theoretical εM first weakly increases and then decreases with an increase in Al and Ta contents, and reaches a maximum εM of 5.34% (5.25%) in Ti−22Nb−4Ta (Ti−22Nb−2Al).It can be observed that the maximum recovery strain is about 3.89% in Ti−24Nb−3Al [4] and 3.20% in both Ti−22Nb−4Ta and Ti−22Nb−5Ta [7] at RT, respectively.For lower Al and Ta contents, the effect of solid solution strengthening plays a leading role in the increase in the recovered strain [4,44,69].With increasing Al and Ta contents, the critical stress may become the dominant factor in the previous measurements for TiNb-based alloys [4,7,54,66,70,71].From Figure 2d, it can be observed that both the predicted and measured εM of Ti−Nb−Al (except for Ti−22Nb−4Al) are higher than those of Ti−Nb−Ta, despite the different magnitudes of εM that appear in our 0 K calculations and RT measurements [4,7].The deviation for Ti−Nb−Al may originate from the different compositions used in our calculations and available measurements [4] and intermetallic compounds or second-phase particles in the experiments [4,7].
In Figure 3, we demonstrate the contour plots of the lattice strain (Equation ( 4)) using stereographic projections on the (100) β and (001) β of the β unit cell in Ti−Nb−Al and Ti−Nb−Ta ternary alloys.For the sake of simplicity, only 17 orientations are marked in Figure 3. Deviations from uniform coloring easily illustrate the direction and degree of deviatoric behavior.The red (blue) color of the contour plots denotes the maximum negative (positive) strain.The contour plots indicate the maximum transformation strains of the martensitic transformations in Ti−Nb−Al and Ti−Nb−Ta ternary alloys.From Figure 3, it can be distinctly observed that the largest elongations are along ±[011] β and that the largest compressions occur along ±[100] β for Ti−Nb−Al and Ti−Nb−Ta alloys, agreeing with the observations on Ti−Nb binary alloys [68].As shown in Figure 3a,b, an increase in elongated lattice strain ranges from 6.59% to 6.80% as Al content increases, while an increase in contracted lattice strain ranges from 8.81% to 8.93% for Ti−Nb−Al alloys.Namely, the largest contraction and the largest elongation in Ti−Nb−Al alloys linearly increases by 0.02 and 0.07%/at.%, respectively.The situation becomes different for the Ti−Nb−Ta system.From Figure 3c,d, it can be observed that the largest contraction in the Ti−Nb−Ta alloys remains almost constant at around 8.80%, while the largest elongation decreases by 0.03%/at.% with increasing Ta content.
We calculate the total energies, E, at the corresponding equilibrium volume (Figure 1a) of the β phase in each composition.After fixing the shuffle y, the total energy contours of the β to α phase transformation for Ti−22Nb−xX (x = 0-10, in at.%; X = Al, Ta) are plotted in Figure 4 as a function of the ratios of b/a and c/a.From Figure 4, it can be seen that the most stable phase in the Ti−22Nb binary alloy appears to be the α phase (c/a = 1.60, b/a = 1.65), which is in line with the available experimental results [72][73][74] on Ti−Nb binaries.The c/a and b/a of the α phase (as shown in Table S1) for the Al-containing and Ta-containing ternaries remain almost unchanged, agreeing with the available measurement on Ti−Nb−Ta alloys [9].Additionally, the predicted shuffle y (as shown in Table S1) for Ti−Nb−Ta ternary alloys is almost constant and is around 1.50, while the calculated y for Ti−Nb−Al alloys declines from 1.50 to 1.43 with increasing Al content.This finding indicates that Al has a greater ability to lower shuffle y than Ta, suggesting greater capacity on the lattice distortion induced by Al.
The energy difference, ∆E β→α (∆E β→α = E α − E β , in mRy), between the β and α phases indicates the relative stability of the β and α phases.The ∆E β→α < 0 shows that the α phase is more stable than the β phase.If the absolute value of the ∆E β→α becomes smaller with increasing alloying elements, the relative stability of the α phase is regularly weakened and the ability to generate the β phase is gradually promoted.From Figure 4, it can be seen that the change in ∆E β→α in the Ti−Nb−Al system is from −1.12 to −0.94 mRy with increasing Al content, revealing that the relative stability of the α phase weakly decreases.The ∆E β→α of Ti−Nb−Ta alloys varies from −1.12 to −0.39 mRy with increasing Ta content, implying that the relative stability of the α phase strongly decreases.This finding demonstrates that Ta [7,75] is a much stronger β stabilizer in Ti alloys than Al [26], implying that Ta distinctly promotes the formation of the β phase when compared to doping Al.The available measurements have shown that both Nb and Al can act as β stabilizers in TiNb-based alloys [26,54,70,71].From Figure 4, it can be observed that both Al and Ta can reduce energy differences in different magnitudes, but Ta shows a much stronger ability to stabilize the β phase than Al.Moreover, Al can reduce the energy difference between the β and α phases in Ti−Ta−Al ternary alloys [27].We calculate the total energies, E, at the corresponding equilibrium volume (Figure 1a) of the β phase in each composition.After fixing the shuffle y, the total energy contours of the β to α″ phase transformation for Ti−22Nb−xX (x = 0−10, in at.%; X = Al, Ta) are plotted in Figure 4 as a function of the ratios of b/a and c/a.From Figure 4, it can be seen that the most stable phase in the Ti−22Nb binary alloy appears to be the α″ phase (c/a = 1.60, b/a = 1.65), which is in line with the available experimental results [72][73][74] on Ti−Nb binaries.The c/a and b/a of the α″ phase (as shown in Table S1) for the Al-containing and Ta-containing ternaries remain almost unchanged, agreeing with the available measurement on Ti−Nb−Ta alloys [9].Additionally, the predicted shuffle y (as shown in Table S1) for Ti−Nb−Ta ternary alloys is almost constant and is around 1.50, while the calculated y for Ti−Nb−Al alloys declines from 1.50 to 1.43 with increasing Al content.This finding indicates that Al has a greater ability to lower shuffle y than Ta, suggesting greater capacity on the lattice distortion induced by Al.
The energy difference, ΔEβ→α″ (ΔEβ→α″ = Eα″ − Eβ, in mRy), between the β and α″ phases indicates the relative stability of the β and α″ phases.The ΔEβ→α″ < 0 shows that the α″ phase is more stable than the β phase.If the absolute value of the ΔEβ→α″ becomes smaller with increasing alloying elements, the relative stability of the α″ phase is regularly weakened and the ability to generate the β phase is gradually promoted.From Figure 4, it can be seen that the change in ΔEβ→α″ in the Ti−Nb−Al system is from −1.12 to −0.94 mRy with increasing Al content, revealing that the relative stability of the α″ phase weakly decreases.The ΔEβ→α″ of Ti−Nb−Ta alloys varies from −1.12 to −0.39 mRy with increasing Ta content, implying that the relative stability of the α″ phase strongly decreases.This finding demonstrates that Ta [7,75] is a much stronger β stabilizer in Ti alloys than Al [26], implying that Ta distinctly promotes the formation of the β phase when compared to doping Al.The available measurements have shown that both Nb and Al can act as β stabilizers in TiNbbased alloys [26,54,70,71].From Figure 4, it can be observed that both Al and Ta can reduce energy differences in different magnitudes, but Ta shows a much stronger ability to stabilize the β phase than Al.Moreover, Al can reduce the energy difference between the β and α″ phases in Ti−Ta−Al ternary alloys [27].For the sake of convenience, the martensitic transformation temperature is investigated based on a proposed hypothesis.In this work, we approximate the two Ti−Nb−Al and Ti−Nb−Ta ternaries into individual Ti−(Nb + Al) and Ti−(Nb + Ta) pseudobinaries, respectively.Since the measured M s decreases by 40, 40, and 30 K with an increase of 1 at.% Nb, Al, and Ta, respectively, it is indicated that the doping of Nb [41], Al [4][5][6]54], and Ta [7] shows a similar magnitude order on the M s .Alternatively, it is found that the calculated ∆E β→α for different Ti−Nb−Al alloys having the same (Nb + Al) content is almost the same (as shown in Figure S4).Namely, our calculated ∆E β→α is insensitive to specific alloy components.Therefore, it is assumed that different Ti−Nb−Al alloys approximately possess the same ∆E β→α if Ti−Nb−Al alloys contain the same (Nb + Al) content.Like Ti−Nb−Al, the Ti−Nb−Ta system having the same (Nb + Ta) content exhibits the same ∆E β→α .Consequently, Table 1 shows that Ti−Nb−X alloys containing the same (Nb + X) (X = Al, Ta) content have the same predicted M s due to the same ∆E β→α based on our pseudobinary hypothesis.For example, the ∆E β→α of both Ti−23Nb−3Al and Ti−24Nb−2Al is the same as that of Ti−22Nb−4Al since these three alloys contain the same (Nb + Al) content.Namely, they have the same ∆E β→α of −1.141 mRy and then possess the same predicted M Al s of 335.2 K. Figure 5a,b plot the available measured M s [4,6,7] and the present theoretical ∆E β→α for Ti−22Nb−xX (x = 0-10, X = Al, Ta) ternary alloys as functions of the (Nb + X) (X = Al, Ta) content.Note that the values of the ∆E β→α are all negative, indicating that the α phase is more stable than the β phase.The absolute value of ∆E β→α decreases with increasing Al and Ta contents, signifying that the relative stability of the α phase is gradually weakened and the tendency to generate the β phase is enhanced.This finding agrees with the available observations on Ti−Nb−Al [4] and Ti−Nb−Ta [7].
It is still a challenge to directly predict the M s using a first-principles method.Based on former first-principles calculations [47,48], the lower the absolute ∆E β→α , the lower the M s .Furthermore, Minami et al. [47] and Sun et al. [48] correlated the M s and the ∆E β→α between the β and α phase for Ti−Nb binaries.Their correlations can qualitatively predict the composition dependence of the M s .However, the evaluated M s derived from their empirical relationships [47,48] greatly overestimated the measurements overall.Despite the fact that the alloying effect on TiNb-based ternaries [47,48] and high-entropy alloys [48] has been qualitatively investigated, there is no quantitative research on TiNb-based ternary systems.Furthermore, extensive experimental observations have used different functions, such as linear [41,43,47], 1.5 degrees [45], and cubic polynomial [46], to fit the M s for different Ti−Nb binary alloys.Therefore, these functions [41,43,[45][46][47] used in binary systems may lower the accuracy of Ti-based ternary and multicomponent alloys due to the ignorance of the coupling effect of alloying elements.In this work, the coupling effect of alloying elements is considered by adopting a pseudobinary hypothesis on Ti−(Nb + Al) and Ti−(Nb + Ta) systems.For the sake of convenience, the martensitic transformation temperature is investigated based on a proposed hypothesis.In this work, we approximate the two Ti−Nb−Al and Ti−Nb−Ta ternaries into individual Ti−(Nb + Al) and Ti−(Nb + Ta) pseudobinaries, respectively.Since the measured Ms decreases by 40, 40, and 30 K with an increase of 1 at.% Nb, Al, and Ta, respectively, it is indicated that the doping of Nb [41], Al [4][5][6]54], and Ta [7] shows a similar magnitude order on the Ms.Alternatively, it is found that the calculated ΔEβ→α″ for different Ti−Nb−Al alloys having the same (Nb + Al) content is almost the  Table 1.Theoretically calculated energy difference (∆E β→α , in mRy) between the β and α phases and estimated martensitic transformation temperature (M s , in K) for Ti−Nb−Al and Ti−Nb−Ta (in at.%) alloys.Note that the predicted M Al s and M Ta s are derived from Equations ( 5) and ( 6), respectively.For comparison, we show the available experimental M Expt,Al s [4,6] and M Expt,Ta s [7] and the evaluated M 1 s from former empirical relationships [48].
Here, we construct the relationships between the calculated composition-dependent ∆E β→α for Ti−(Nb + Al) and Ti−(Nb + Ta) pseudobinaries with the measured M s .In this way, one may accurately determine the M s employing first-principles calculations.
Since the alloying elements Al and Ta have different influences on the energy difference between the β and α phase, we separately fit two empirical relationships by connecting our theoretical ∆E β→α with the measured M s for Ti−Nb−Al [4,6] and Ti−Nb−Ta [7] alloys.For the Ti−Nb−Al system, an empirical relationship derived from Figure 5a can be expressed, as in Equation ( 5): For the Ti−Nb−Ta system, another empirical relationship draw from Figure 5b can be fitted, as in Equation ( 6): where the unit of M Al s and M Ta s is K, and the unit of ∆E β→α is mRy.Although Al and Ta have similar alloying effects on the M s , their influences on the energy difference, ∆E β→α , are different.As shown in Equations ( 5) and ( 6), the different coefficients of ∆E β→α for Al-containing and Ta-containing systems are 1.13 and 4.36, respectively.Based on the ∆E β→α calculated from first-principles calculations, the theoretical M s for Ti−Nb−Al and Ti−Nb−Ta alloys can be quickly predicted from Equations ( 5) and ( 6), respectively.
To assess the reliability of the predicted M Al s by Equation ( 5) and M Ta s by Equation ( 6), Table 1 displays the present predicted M Al s and M Ta s , the available M Expt,Al s and M Expt,Ta s [4,6,7], and the estimated M 1 s from former empirical relationships [48] for Ti−Nb−Al and Ti−Nb−Ta alloys.The composition dependence of the present predicted M s for Ti−Nb−Al and Ti−Nb−Ta alloys reproduces their experimental counterparts.For Ti−Nb−Ta, the average error between the predicted M Ta s by Equation ( 6) and available M Expt,Ta s [7] is about 4%.Compared to the current M Al s and M Ta s , the predicted M 1 s by former empirical relationships [48] shows the unreasonable composition dependence of Ti−Nb−Al and Ti−Nb−Ta alloys and is greatly overestimated relative to the experimental counterparts [4,6,7].When compared to the measurements [4,6], an opposite alloying effect of Al on the M s can be estimated by fitting an empirical equation [45].Therefore, the present empirical relationships of Equations ( 5) and ( 6) accurately predict the M s , corresponding to former empirical relationships [45,48].As shown in .The deviation in Equation ( 5) for Ti−Nb−Al may result from the different alloy compositions used in our calculated ∆E β→α and the measured M Expt,Al s [4,6].The prediction of Equation (5) may further deteriorate for higher (Nb + Al) contents, such as the predicted M Al s of −85.5 K for Ti−22Nb−10Al.As shown in Table 1 and Figure 6, both the present predictions and available measurements [4,6,7] qualitatively predict the similar composition dependence of M s , despite the fact that the predicted M Al s and M Ta s are somehow higher than the relative M Expt,Al s and M Expt,Ta s . The predicted M s decreases by 28 and 30 K with an increase of 1 at.% Al and Ta, corresponding to a decrease in the measured M s by 13 [4] [4,6] are below RT.However, the predicted M Al s decreases by 46 K/1 at.% Nb for Ti−(23-24)Nb−3Al alloys, which is consistent with the measured decline of 40 K and 40 K for Ti−Nb binary alloys [41] and Ti−Nb−Al ternary alloys [6], respectively.This finding suggests that the coupling effect of alloying elements are appropriately described based on our pseudobinary hypothesis.
To further directly compare the discrepancy between our predicted and measured M s , in Figure 6, we plot the predicted M s for Ti−Nb−Al by Equation ( 5) and for Ti−Nb−Ta and Ti−Nb−Zr by Equation ( 6), along with the measured M s for Ti−Nb−Al [4,6], Ti−Nb−Ta [7,13], and Ti−Nb−Zr [12,13,44] ternary alloys.Since Zr and Ta have similar alloying effects on the M s , we assume that the energy difference, ∆E β→α , in the Ti−Nb−Zr alloy is approximate to the ∆E β→α in the Ti−Nb−Ta alloy when the (Nb + Zr) content is equal to the (Nb + Ta) content.As shown in Figure 6, there are average errors of about 28%, 4%, and 13% between our predicted and measured M s for Ti−Nb−Al [4,6], Ti−Nb−Ta [7,13], and Ti−Nb−Zr [12,13,44], respectively.It can be concluded that there is a general agreement between the prediction and measurements.above RT, while the  s Expt,Al [4,6] are below RT.However, the predicted  s Al decreases by 46 K/1 at.% Nb for Ti−(23−24)Nb−3Al alloys, which is consistent with the measured decline of 40 K and 40 K for Ti−Nb binary alloys [41] and Ti−Nb−Al ternary alloys [6], respectively.This finding suggests that the coupling effect of alloying elements are appropriately described based on our pseudobinary hypothesis.To further directly compare the discrepancy between our predicted and measured Ms, in Figure 6, we plot the predicted Ms for Ti−Nb−Al by Equation ( 5) and for Ti−Nb−Ta and Ti−Nb−Zr by Equation ( 6), along with the measured Ms for Ti−Nb−Al [4,6], Ti−Nb−Ta [7,13], and Ti−Nb−Zr [12,13,44] ternary alloys.Since Zr and Ta have similar alloying effects on the Ms, we assume that the energy difference, ΔEβ→α″, in the Ti−Nb−Zr alloy is approximate to the ΔEβ→α″ in the Ti−Nb−Ta alloy when the (Nb + Zr) content is equal to the (Nb + Ta) content.As shown in Figure 6, there are average errors of about 28%, 4%, and 13% between our predicted and measured Ms for Ti−Nb−Al [4,6], Ti−Nb−Ta [7,13], and Ti−Nb−Zr [12,13,44], respectively.It can be concluded that there is a general agreement between the prediction and measurements.
Ren and Otsuka [76] explained the compositional dependence of Ms using the Landau-type model.In the process of martensitic transformation, elastic modulus decreases gradually with cooling and reaches a critical value before martensitic transformation [76].Ren and Otsuka [76] explained the compositional dependence of M s using the Landautype model.In the process of martensitic transformation, elastic modulus decreases gradually with cooling and reaches a critical value before martensitic transformation [76].Therefore, if the elastic constants C and C 44 of the β phase increase, the cooling should continue to lower temperatures before a critical elastic constant and M s decreases.The critical elastic constraint of martensite alloys is temperature-independent.Therefore, when the elastic modulus changes, the M s must also change due to critical elastic constraints.The relationship [76] between the M s and elastic modulus (C) can be approximately expressed as follows: where the M s is the martensitic transformation temperature, C is the elastic modulus (the C can be either C 44 , C , or some other elastic modulus), and γ is the temperature coefficient of elastic modulus.Therefore, the increase in the martensite temperature is consistent with the decrease in elastic modulus.The relationship proposed by Ren and Otsuka [76] [49,50], implying a correlation between the elastic moduli (C and C 44 ) and the M s .
In Figure 7, we display the calculated C and C 44 and the predicted M Al s and M Ta s , as well as the available experimental M Expt,Al s [4,6] and M Expt,Ta s [7]

Conclusions
Using first-principles EMTO-CPA calculations, we systematically calculated the total energy contours, lattice deformation strains (η1, η2, and η3), maximum transformation strains ( M  ), and the martensitic transition temperature (Ms) during the β → α″ phase transformation for two selected Ti−Nb−Al and Ti−Nb−Ta ternary alloys.The present theoretical calculations and the available experiments gave the same composition dependence on the lattice strains and Ms.As for the calculated stereographic projections of lattice strains alongside phase transformation along (100)β and (001)β, the largest elongation and the largest contraction due to the lattice strain occurred along ±[011]β and ±[100]β, respectively.The addition of Al and Ta increased and decreased the transformation strain by 0.07 and 0.03%/at.%, respectively.
The effect of either Al or Ta additions on the energy difference (ΔEβ→α″) between the β and α″ phases was also studied, suggesting that both Al and Ta can lower ΔEβ→α″.The relative phase stability of α″ gradually weakened but the tendency to generate the β phase became stronger as Al and Ta contents increased.Aiming to directly assess the Ms from first-principles calculations, two empirical relationships were fitted by associating the measured Ms with the calculated ΔEβ→α″.When compared to the overestimation by the existing relationships, there was a satisfactory agreement between the predicted and measured Ms, implying that the proposed relationships could accurately describe the coupling effect of alloying elements on the Ms.In this work, the theoretically predicted Ms were reduced by around 46, 28, and 30 K with an increase of 1 at.% Nb, Al, and Ta, respectively, corresponding to measured declines in Ms by 40, 40, and 30 K, respectively.Moreover, there was a correlation between Ms and C44, implying that an elastic modulus  [4,6] and Ti-Nb-Ta [7] ternary alloys are shown for comparison.Note that the available measured alloy compositions are Ti− (16,18,23,24)Nb−3Al [6] and Ti−24Nb−(1-4)Al [4] for Ti-Nb-Al alloys and Ti−22Nb−(4-8)Ta [7] for Ti−Nb−Ta alloys.For completeness, the M s of Ti−22Nb [7] and Ti−24Nb [4] are also displayed.Note that different colored arrows in the figure mark the M s , C 44 , or C , respectively.

Conclusions
Using first-principles EMTO-CPA calculations, we systematically calculated the total energy contours, lattice deformation strains (η 1 , η 2 , and η 3 ), maximum transformation strains (ε i M ), and the martensitic transition temperature (M s ) during the β → α phase transformation for two selected Ti−Nb−Al and Ti−Nb−Ta ternary alloys.The present theoretical calculations and the available experiments gave the same composition dependence on the lattice strains and M s .As for the calculated stereographic projections of lattice strains alongside phase transformation along (100) β and (001) β , the largest elongation and the largest contraction due to the lattice strain occurred along ±[011] β and ±[100] β , respectively.The addition of Al and Ta increased and decreased the transformation strain by 0.07 and 0.03%/at.%, respectively.
The effect of either Al or Ta additions on the energy difference (∆E β→α ) between the β and α phases was also studied, suggesting that both Al and Ta can lower ∆E β→α .The relative phase stability of α gradually weakened but the tendency to generate the β phase became stronger as Al and Ta contents increased.Aiming to directly assess the M s from first-principles calculations, two empirical relationships were fitted by associating the measured M s with the calculated ∆E β→α .When compared to the overestimation by the existing relationships, there was a satisfactory agreement between the predicted and measured M s , implying that the proposed relationships could accurately describe the coupling effect of alloying elements on the M s .In this work, the theoretically predicted M s were reduced by around 46, 28, and 30 K with an increase of 1 at.% Nb, Al, and Ta, respectively, corresponding to measured declines in M s by 40, 40, and 30 K, respectively.Moreover, there was a correlation between M s and C 44 , implying that an elastic modulus can be used as a prefactor to evaluate composition-dependent M s .This work can contribute to accurately estimating the M s of Ti-based alloys.

Figure 4 .
Figure 4. Total energy contours (in mRy) of the β and α″ phase transformation for (a) Ti−22Nb−xAl (x = 0−10, in at.%) and (b) Ti−22Nb−xTa (x = 0−10, in at.%) ternary alloys as a function of the ratios of b/a and c/a from first-principles calculations.All energies are plotted relative to the corresponding β phase minima.The pink solid circles and pink open circles represent the β and α″ phases, respectively.

Figure 4 .
Figure 4. Total energy contours (in mRy) of the β and α phase transformation for (a) Ti−22Nb−xAl (x = 0-10, in at.%) and (b) Ti−22Nb−xTa (x = 0-10, in at.%) ternary alloys as a function of the ratios of b/a and c/a from first-principles calculations.All energies are plotted relative to the corresponding β phase minima.The pink solid circles and pink open circles represent the β and α phases, respectively.

Figure 5 .Figure 5 .
Figure 5. Theoretically calculated energy difference (ΔEβ→α″, in mRy) for (a) Ti−22Nb−xAl (x = 0−10, in at.%) and (b) Ti−22Nb−xTa (x = 0−10, in at.%) as a function of alloying elements, as well as the available measured martensitic transformation temperature (Ms, in K) [4,6,7].The measured alloys and 30 K [7], respectively.Both M Ta s and M Expt,Ta s begin to fall below RT when x > 4 at.% Ta.The case is quite complex for Ti−Nb−Al.The predicted M Al s for Ti−22Nb−xAl starts to fall below RT if x > 6 at.% Al.However, the M Al s for Ti−23Nb−xAl and Ti−24Nb−xAl (except for Ti−24Nb−4Al) are above RT, while the M Expt,Al s plex for Ti−Nb−Al.The predicted  s Al for Ti−22Nb−xAl starts to fall below RT if x > 6 at.% Al.However, the  s Al for Ti−23Nb−xAl and Ti−24Nb−xAl (except for Ti−24Nb−4Al) are

Table 1 .
Theoretically calculated energy difference (ΔEβ→α″, in mRy) between the β and α″ phases and estimated martensitic transformation temperature (Ms, in K) for Ti−Nb−Al and Ti−Nb−Ta (in at.%) alloys.Note that the predicted  s Al and  s Ta are derived from Equations (