The role of molybdenum in suppressing cold dwell fatigue in titanium alloys

Abstract

We test a hypothesis to explain why Ti-6242 is susceptible to cold dwell fatigue (CDF), whereas Ti-6246 is not. The hypothesis is that, in Ti-6246, substitutional Mo-atoms in α-Ti grains trap vacancies, thereby limiting creep relaxation. In Ti-6242, this creep relaxation enhances the loading of grains unfavourably oriented for slip and they subsequently fracture. Using density functional theory to calculate formation and binding energies between Mo-atoms and vacancies, we find no support for the hypothesis. In the light of this result, and experimental observations of the microstructures in these alloys, we agree with the recent suggestion (Qiu et al. 2014 Metall. Mater. Trans. A 45, 6075–6087. (doi:10.1007/s11661-014-2541-5)) that Ti-6246 has a much smaller susceptibility to CDF because it has a smaller grain size and a more homogeneous distribution of grain orientations. We propose that the reduction of the susceptibility to CDF of Ti-6242 at temperatures above about 200°C is due to the activation of 〈c+a〉 slip in ‘hard’ grains, which reduces the loading of grain boundaries.

1. Introduction

Titanium alloys are used in the front end of gas turbine engines for fan and compressor discs and blades. Cold dwell fatigue (CDF) of Ti-alloys is believed to be responsible for some failures of discs and blades in aircraft after much shorter times in service than expected.

The ‘dwell’ comprises the period during take-off when the relevant engine components are relatively cold and they are exposed to large stresses. CDF can reduce the time to failure of susceptible Ti-alloys by more than an order of magnitude [1].

The mechanical properties of Ti-alloys can be optimized by alloying elements which stabilize one of the two Ti allotropes: the hexagonal close-packed (hcp) α-phase or the body-centred cubic (bcc) β-phase. Ti-alloys for turbines are mainly near-α- or α/β-alloys, with the amount of β-content determined by the concentration of β-stabilizers such as Mo [2].

In discussing CDF, it is usual to distinguish soft and hard grains in the α-phase. The critical resolved shear stress for slip in Ti is lowest for dislocations with 〈a〉-type Burgers vectors on prism and basal planes. The critical resolved shear stress for dislocations with Burgers vectors 〈c+a〉 on pyramidal planes is approximately four times higher at room temperature [3] than that for prism or basal slip in Ti-6.6Al single crystals. An α-grain loaded in tension with its c-axis inclined to the stress axis will be able to deform by slip on prism planes, basal planes, or both because those planes experience shear stresses. Such a grain is termed ‘soft’. If the tensile axis is parallel to the c-axis, the grain is termed ‘hard’ because deformation by slip may involve only 〈c+a〉 dislocations on pyramidal planes, for which the critical resolved shear stress is substantially greater.

Load redistribution from soft to hard grains during the dwell period is central to CDF of α-rich Ti-alloys [4]. This load redistribution is effected by creep at stresses as low as 60% of the yield stress and at temperatures as low as room temperature [5,6]. The mechanism of this ‘cold creep’ appears [4] to involve the planar glide of dislocations with 〈a〉-type Burgers vectors in soft grains, which gradually become exhausted as load is transferred to hard grains. The planar nature of slip in Ti-alloys containing 6 wt.% Al during room temperature creep was shown [7] to be a result of short range order of Ti and Al atoms.

Bache et al. [8,9] proposed a mechanism for crack nucleation during CDF based on an adaptation of the pile-up model proposed by Stroh [10,11]. The mechanism involves the pile-up of dislocations at a grain boundary between a soft α-grain and a hard α-grain (figure 1). The stress concentration developed ahead of the pile-up in the hard grain cannot be relieved by slip and a crack is nucleated. Failure occurs once the crack grows to a critical length.

Figure 1.

The Stroh model, adapted by Evans & Bache’s group [8,9]. Slip accumulates in the soft grain at the boundary with a hard grain which results in a stress concentration in the hard grain and eventual crack nucleation.

At temperatures above approximately 200°C CDF ceases to be a failure mechanism [3]. Thus, although the creep that loads hard grains during the dwell period occurs at low temperatures, there must be some thermally activated deformation process at temperatures above 200°C that results in a more homogeneous stress distribution and grain boundaries are not subjected to such high loads.

Zhang et al. [12] developed an empirically based crystal plasticity model that reproduces this temperature dependence of CDF. The model assumes dislocations become pinned and released by some unspecified thermally activated process with an activation energy ΔH. Their model incorporates experimentally determined temperature dependencies of the critical resolved shear stresses on the prismatic, basal and pyramidal planes, and of the anisotropic elastic constants. The calculated strain rates on the different slip systems include a common Boltzmann factor with activation energy ΔH and an attempt frequency ν. The model was shown to reproduce experimental creep data on single crystals, and polycrystals at different strain rates. At temperatures above 230°C, the model predicts that the loading of grain boundaries is reduced so that the CDF mechanism is eliminated. To obtain these results, Zhang et al. set the activation energy ΔH to 0.62 eV and ν to 10¹¹ s⁻¹.

The alloys Ti-6Al-2Sn-4Zr-2Mo (Ti-6242) and Ti-6Al-2Sn-4Zr-6Mo (Ti-6246), where the digits denote weight percentages, have different susceptibilities to CDF: Ti-6242 is highly susceptible to CDF, whereas Ti-6246 is not [1]. The principal difference between them is that Ti-6246 has 4 wt.% more Mo than Ti-6242. Despite extensive research [13], the role Mo plays in affecting CDF susceptibility is not yet known. In an illuminating set of experiments, Qiu et al. [1] studied the susceptibility of Ti-624X alloys to CDF, where X=2,3,4,5,6. Mo is a β-phase stabilizer and the volume fraction of β-phase increased uniformly from 10% in Ti-6242 to 40% in Ti-6246. Following a thermo-mechanical treatment, normalized to the β-transus temperature of each alloy, all the grain structures were nearly equiaxed. Mo is a slow diffuser in the α-phase, and its rejection by the α-phase into the β-phase resulted in a uniformly decreasing grain size with increasing Mo-content from ≈13.6 μm in Ti-6242 to ≈5.3 μm in Ti-6246. The orientation distributions of the grains changed significantly with increasing Mo-content. Ti-6242 displayed regions where there are clusters of similarly oriented α-grains. As the Mo-content increased, these clusters became smaller and included differently oriented α-grains. In Ti-6246, the orientation distribution was almost homogeneous [1], with no clusters of similarly oriented grains.

Qiu et al. [1] showed that CDF cracks formed in the low Mo-content alloys where there were clusters of hard α-grains. Relief of the load through slip in these grains would have to involve 〈c+a〉 dislocations on pyramidal planes for which the critical resolved shear stress was too high. Cracks formed in these hard grains, possibly through the mechanism illustrated in figure 1. By contrast, any cracks that nucleated in hard grains in Ti-6246 were much shorter, partly because the grains were smaller and partly because the orientations of neighbouring grains were generally different so that stress could be relieved through slip on prism planes, basal planes, or both. Thus, one explanation [1] for the reduction in the susceptibility to CDF associated with higher Mo-content alloys is microstructural: the smaller, more homogeneously oriented α-grains in the higher Mo-content alloys inhibits the formation and growth of CDF cracks. However, this does not explain the disappearance of CDF at temperatures above approximately 200°C, a point we return to in §4.

Qiu et al. [1] found the Mo-content of α-grains increased from 0.25 wt.% to 0.65 wt.% as the overall Mo-content in the alloys increased from 2 wt.% to 6 wt.%. Since CDF cracking does not occur in the β-phase [14], this raises the possibility that the reduction in the susceptibility to CDF with increasing Mo-content in the alloy is a consequence of the higher Mo-content in the α-grains. For example, if the loading of the hard grains that occurs during the dwell period is a result of creep involving mass transport by a vacancy mechanism, then the additional Mo in α-grains in Ti-6246 may be trapping the vacancies, thereby suppressing creep. In addition, since the smaller grain structure of Ti-6246 is at least partly a result of the rejection of Mo from the α-phase, it is relevant to investigate why it is not only energetically favourable for Mo to occupy the β-phase, but also why it stabilizes the β-phase, resulting in a larger volume fraction of the β phase with increasing Mo-content.

Huang et al. [15] computed the formation and stabilization energies of transition metal elements, including Mo in the α-, β- and ω-phases. The focus of their work was to understand the influence of transition metal elements on phase diagrams. Tegner et al. [16] studied the effect of alloying elements, including Mo, on the phase stability of Ti-alloys and showed that the stabilization of the β-phase was directly correlated with the filling of the d-band.

To investigate whether the additional Mo present in α-grains of Ti-6246 alloys affects their intrinsic creep properties, we need several ingredients that can be assessed with computations using density functional theory (DFT). These include the formation energies of the vacancy, Mo substitutional and interstitial defects, and the binding energy between a Mo-atom and a vacancy. Some of these quantities have been calculated before but for consistency and completeness of the results, and as a check on our own computations, we have calculated all these quantities in this study.

2. Material and methods

(a). Density functional theory parameters

All DFT results in this paper were obtained using VASP [17–20]. We used the projector augmented wave (PAW) method [21] and a Ti PAW potential with 10 valence electrons and a Mo PAW potential with six valence electrons, as included in the VASP database [22–24]. For the exchange–correlation functional, we used two parametrizations of the generalized gradient approximation: for most calculations we used that of Perdew–Burke–Ernzerhof (PBE) [25], and where stated we used the parameterization of Perdew and Wang (PW91) [26].

A plane wave energy cut-off of 350 eV resulted in an error in the total energy arising from the incomplete basis set of 0.5 meV atom⁻¹ for α-Ti. Convergence to 0.3 meV atom⁻¹ was reached with a 23×23×15 Γ-centred Monkhorst–Pack (MP) k-point mesh for the primitive unit cell. In this case, the product of the number of k-points and the number of atoms in the unit cell (2) is about 16 000. We ensured all other samplings of k-points are equivalent to or larger than this product. We used the Methfessel-Paxton smearing scheme [27] with a smearing width of 0.1 eV. The error introduced by this smearing scheme is about 0.3 meV atom⁻¹.

For calculations of defects, the relaxations were deemed complete when the maximum force on any atom was less than 10 meV Å⁻¹, and the total electronic energy was converged to within 0.005 meV atom⁻¹.

(b). The perfect α- and β-phases

We computed energy volume curves for α- and β-Ti and fitted them to the Birch–Murnaghan equation of state [28,29] to determine the lattice parameters a=2.94 Å for α-Ti and a=3.25 Å for β-Ti. A value of 1.586 for the c/a ratio of α-Ti was then determined by minimizing the total energy with respect to c. The α-Ti lattice parameter values are close to the experimental values of 2.95 Å and 1.587, respectively [30].

Using the equilibrium lattice parameters for α- and β-Ti obtained with the above procedure, we found the cohesive energies of the two bulk phases to be −5.273 eV and −5.165 eV, respectively, with an energy difference of 0.107 eV. These energies are in good agreement with those of Huang et al. [31].

(c). Supercell size effects in point defect calculations

Owing to the use of periodic boundary conditions, there are elastic and electronic interactions between a point defect in the computational cell and its periodic images. The influence of these interactions on the computed energy of the point defect can be reduced to an acceptable level by increasing the supercell size and by suitable choices of the k-point mesh [32,33]. We used Γ-centred MP k-point grids with an odd number of k-points along each reciprocal direction.

We have carried out a systematic study to investigate the influence of the supercell size on the formation energy of the vacancy in α-Ti (table 1). The supercell vectors were kept fixed during the energy minimization. The notation for the supercell vectors is m×n×p, where m, n and p indicate the numbers of hcp primitive vectors a₁, a₂ and c, where a₁ and a₂ are of length a in the basal plane and are at 60° to each other, and c has length c and is perpendicular to the basal plane. As seen in table 1, we found that a 2×2×2 supercell containing 15 atoms was sufficient to converge the formation energy to within 0.03 eV. Comparison with literature values is made in §3a.

Table 1.

Vacancy formation energies (E^f) in α-Ti and cohesive energies (E^coh) of the perfect hcp crystal in eV computed for different supercell sizes. MP mesh is the Monkhorst–Pack mesh of k-points. kp×atom is the number of k-points multiplied by the number of atoms in the supercell to facilitate comparisons between the k-point sampling in different supercells.

supercell	no. atoms	MP mesh	kp×atom	Ef	Ecoh
2×2×2	15	13×13×9	22 815	2.13	7.8897
3×3×2	35	9×9×9	25 515	2.13	7.8899
3×3×3	53	9×9×7	30 051	2.13	7.8901
4×4×2	63	7×7×9	27 783	2.15	7.8900
4×4×3	95	7×7×7	32 585	2.16	7.8901

As a further test of the convergence with respect to supercell size, we calculated the formation energy of the octahedral self-interstitial in α-Ti and compared our results with those of Vérité et al. [34]. It is seen in table 2 that our results differ from those of Vérité et al. [34] by less than 3%. Also in agreement with Vérité et al. [34], we found the 6×6×4 supercell was sufficient to achieve satisfactory convergence of the formation energy of the self-interstitial defect. In the light of this result, we used a 6×6×4 supercell to calculate the formation energy of interstitial and substitutional Mo-atoms in α-Ti and the binding energy between the Mo substitutional atom and a vacancy in α-Ti.

Table 2.

Self-interstitial formation energies in α-Ti computed here compared with Vérité et al. [34].

supercell	no. atoms	MP mesh	kp×atom	Ef	ref. [34]
5×5×4	201	5×5×3	15 075	2.58	—
6×6×4	289	5×5×3	21 675	2.46	2.40
6×6×5	361	5×5×3	27 075	2.47	2.40

(d). The formation energies of a Mo-atom in α- and β-phases of Ti

Let E(N_Ti,Mo) be the total energy of a supercell containing N_Ti titanium atoms in the α- or β-phases and a Mo-atom either in a substitutional or interstitial site. We define the energy of formation of a Mo point defect as follows [35]:

$$E^{\mathrm{f}}=E(N_{\mathrm{Ti}},\mathrm{Mo})-N_{\mathrm{Ti}}{\mu }_{\mathrm{Ti}}-{\mu }_{\mathrm{Mo}},$$ 2.1

where μ_Ti and μ_Mo are the chemical potentials of Ti and Mo, respectively. We equate μ_Mo to the total energy per atom of Mo in its equilibrium bcc crystal structure. We equate μ_Ti to the total energy per atom of a supercell of pure α- or β-Ti with the same supercell vectors as the supercell containing the defect, and using the same k-point sampling.

The definition of μ_Mo as the total energy per atom of bcc-Mo would be appropriate if the alloy coexisted in equilibrium with bcc-Mo. However, that is not the case in Ti-6242 and Ti-6246 alloys where Mo exists only in solid solution in α- and β-Ti-phases. But we may still calculate whether Mo has a lower energy in the α- or β-phases of Ti by subtracting the formation energies defined by equation (2.1) because the chemical potential of Mo cancels in the subtraction.

(e). Mo-vacancy binding energy in α-Ti

We define the binding energy between a Mo substitutional atom and a vacancy in α-Ti as the total energy when the defects are infinitely far apart minus the total energy when they occupy neighbouring sites. When the vacancy is attracted to the Mo-atom, the binding energy is positive. In practice, it is calculated as the difference in total energy between a 6×6×4 supercell in which the vacancy and the substitutional Mo-atom are as far apart as possible, and the same supercell in which the vacancy occupies a site neighbouring the Mo-atom. A 5×5×3 MP k-point mesh was used for these calculations.

Since the six neighbouring sites in the basal plane are not equivalent to the six neighbouring sites above and below the basal plane, there are two distinct configurations of the Mo-vacancy pair, either in the same basal plane or straddling two basal planes. In the following, we calculate the binding energies of both configurations in α-Ti.

3. Results

(a). Vacancy in α-Ti

Our vacancy formation energies computed with the PBE and PW91 exchange–correlation (XC) functionals (table 3) are within 10% of those obtained by others using these functionals. The PBEsol results of Medasani, Shang and colleagues [37,39,40] have been included because the functional was optimized to give improved bond lengths for solids and surfaces when compared with the PBE XC functional [46]. With a calculated formation energy of around 2 eV, the equilibrium concentration of vacancies in α-Ti is virtually zero at room temperature.

Table 3.

Vacancy formation energies in Ti from DFT using different exchange–correlation functionals (PBE, PBEsol, PW91 and LDA), tight-binding (TB) and experiment. All formation energies are quoted in eV. The estimate deduced from the results of Köppers et al. [36] and Shang et al. [37] is discussed in §3a.

		formation energy, eV
PBE	this work	2.15
	Scotti & Mottura [38]	1.97
	Medasani et al.[39]	2.08
	Shang et al.[40]	2.09
PBEsol	Medasani et al.[39]	2.15
	Shang et al.[37]	2.14
	Shang et al.[40]	2.16
PW91	This work	1.99
	Medasani et al.[39]	1.99
	Raji et al.[41]	1.97
LDA	Medasani et al.[39]	2.08
	Shang et al.[40]	2.07
TB	Trinkle et al.[42]	2.03
Expt.	Estimate from [36,37]	2.74
	Hashimoto et al.[43]	1.27
	(bcc) Kraftmakher [44], Shestopal [45]	1.55

In addition to the formation energy, we consider the changes in the lengths of bonds between nearest neighbours of the vacancy. Figure 2 shows that bond lengths vary by as much as 3% from the ideal bond lengths of 2.94 Åin the basal plane and by 1% of 2.88 Å out of the basal plane. The changes in the bond lengths maintain the local C_3v point symmetry of an atomic site in the hcp crystal.

Figure 2.

Distortions of the bonds between atoms neighbouring the vacancy in α-Ti. Bond lengths are expressed as percentages of the corresponding equilibrium bond lengths 2.94 Å in the basal plane and 2.88 Å out of the basal plane. The vacancy occupies a site of C_3v symmetry. (Online version in colour.)

Experimental results for the vacancy formation energy are much less consistent among each other than the theoretical values, varying from 1.27 eV [43] to 2.74 eV [36,37]. In the studies of Köppers et al. [36] on ultra-pure samples of α-Ti, the authors deduced an activation energy for self-diffusion parallel to the basal plane of 3.14±0.02 eV from measurements in the temperature range 600–860°C. Assuming self-diffusion is by a vacancy mechanism, and assuming a vacancy migration energy parallel to the basal plane of 0.40 eV, as computed by Shang et al. [37], we obtain a vacancy formation energy of 2.74 eV. This energy is significantly higher than the DFT results given in table 3.

Glensk et al. [47] showed that in Cu the entropy of formation of a vacancy varies linearly with temperature, contrary to the assumption underpinning the usual Arrhenius relation of a temperature-independent entropy. The consequence is that the enthalpy of formation at absolute zero is overestimated if one applies the Arrhenius relation to data obtained at high temperatures, such as the data obtained by Köppers et al. [36]. For a vacancy in Cu, it was shown [47] that the overestimate of the enthalpy of formation at absolute zero was about 20%. If the enthalpy of formation of the vacancy in α-Ti decreases in a similar way, then the corrected experimental enthalpy of formation will be significantly closer to the DFT results presented in table 3.

The vacancy formation energy obtained by Hashimoto et al. [43], based on positron annihilation measurements, uses an empirical relation [48] obtained for fcc metals E^f_v (in eV)=1.46×10⁻³T_C, where $E_{\mathrm{v}}^{\mathrm{f}}$ is the vacancy formation energy and T_C is the critical temperature at which the temperature dependence of the normalized peak positron count deviates from a linear relation [43] due to thermal expansion. In the experiment of Hashimoto et al., T_C=597°C, so this is a high temperature measurement. It is unclear whether the empirical relation [48] is applicable to hcp metals not least because there are two independent thermal expansion coefficients.

The vacancy formation energy of Shestopal [45] was obtained by measuring the specific heat as a function of temperature and subtracting a linear contribution not caused by vacancy formation that follows a trend observed in other metals. The measurement was made over a range a temperatures above the α–β transition at 882°C, and it is therefore a measurement of the vacancy formation energy in the β-phase.

(b). Interstitial Mo in α-Ti

To search for the minimum energy configuration of the Mo-interstitial, we carried out relaxations of the Mo-atom in six interstitial positions in the hcp lattice. We adopted the naming convention of Johnson & Beeler [49] for the relaxed configurations. Six distinct metastable relaxed configurations were found for the Mo-interstitial, and the formation energies are listed in table 4. There are three variants of the split dumbbell configuration in the basal plane, BS, related by the three-fold rotational symmetry along the c-axis at each atomic site in α-Ti. The BS-dumbbell is identified in figure 3a, and it has the smallest energy of the interstitial configurations we surveyed. It is seen in figure 3a that most of the distortion occurred in the basal plane containing the Mo-interstitial. The lengths of nearest neighbour bonds in the BS-configuration are shown in figure 3b. The split dumbbell in the basal plane has been found to be a metastable configuration of self-interstitials in Ti, Zr and Hf with formation energies within 0.2 eV of the lowest energy structures [34,50,51].

Table 4.

Relaxed formation energies of six Mo-interstitial configurations in α-Ti, and of Mo substitutional atoms in α- and β-Ti. Binding energies between a Mo substitutional atom and a vacancy in α-Ti (Mo-Vac in α-Ti) in the same basal plane and adjacent basal planes are in the last two rows. (A positive binding energy means the defects are attracted to each other).

Mo defect configuration	formation energy, eV
interstitial
octahedral	2.32
basal octahedral	2.67
basal split dumbell (BS)	2.17
BS 90° rotation	2.23
split dumbell	2.54
tetragonal	3.37
substitutional
α-Ti	0.57
β-Ti	−0.71
Mo-Vac in α-Ti	binding energy, eV
same basal	0.22
adjacent basal	0.23

Figure 3.

(a) Projection along the c-axis of two adjacent basal planes in the 6×6×4 supercell showing the minimum energy dumbbell structure of the Mo-interstitial (red in colour or dark grey in black and white) and Ti atoms (blue in colour or light grey in black and white). The dumbbell is outlined. There are two equivalent structures related by the three-fold rotational symmetry normal to the page. The distortion introduced by the Mo-atom is highlighted by lines between atoms that are parallel $[1\overline{2}10]$ in the perfect crystal. (b) A close-up of the Mo-interstitial and its nearest neighbour bond lengths. (Online version in colour.)

(c). Substitutional Mo in α- and β-Ti

The formation energy of the relaxed substitutional Mo-atom in α-Ti was found to be +0.57 eV. Its local relaxed atomic structure is shown in figure 4. If we use a free, non-spin polarized atom as the reference state for the chemical potential of Mo, we can compare our calculation of the formation energy of the substitutional impurity with that calculated by Huang et al. [15]. Spin polarization does not significantly affect bulk energies in bcc-Mo, but it does affect the energy of an isolated Mo-atom, as shown by Huang et al. [15]. Using this chemical potential, we obtained a formation energy of −9.83 eV, which compares well with the value of −9.88 eV obtained by LF Huang (2016, personal communication). These energies are very large in magnitude and they indicate that the crystalline state of Mo is a more appropriate choice of reference state for the chemical potential.

Figure 4.

(a) The Mo substitutional atom (red in colour or dark grey in black and white) in α-Ti (blue in colour or light grey in black and white) showing bond lengths in the basal plane, expressed as percentages of the ideal bond length in the basal plane of α-Ti. The relaxed defect occupies a site of C_3v symmetry, as highlighted by greenand brown shading (light and dark grey shading respectively in black and white) the same as a Ti-atom in α-Ti. (b) Projected view along $[1\overline{1}00]$ of the Mo substitutional point defect showing bond lengths expressed as a percentage of corresponding bonds in pure α-Ti. The smaller circles are on adjacent planes along $[1\overline{1}00]$. (Online version in colour.)

To investigate why Mo has a higher solubility in the β-phase than the α-phase, we calculated the relaxed configuration of a substitutional Mo-atom in a supercell made up of 6×6×6 conventional unit cells of β-Ti. From this calculation, we obtained a formation energy of −0.71 eV. Thus, whereas the dissolution of Mo in the α-phase is endothermic, it is exothermic in the β-phase.

We have computed the partial densities of electronic states in the α- and β-phases of pure Ti, and found the results to be in good agreement with those by Huang et al. [31]. The s and p bands are broad and their densities of states are relatively featureless. By contrast, the d-band has a width of about 10 eV and it varies much more rapidly with energy. Mo has more d-electrons than Ti. Therefore, following the argument of Tegner et al. [16] based on the virtual crystal approximation, increasing the occupation of the d-band in α-Ti through the substitution of Ti by Mo raises the electronic energy because the Fermi energy is in a narrow minimum of the density of states of the d-band in pure α-Ti. On the other hand, the substitution of a Ti atom by Mo in the β-Ti structure reduces the electronic energy because the Fermi energy is at a local maximum of the density of states of the d-band in pure β-Ti. This explains not only the positive and negative energies of formation of Mo substitutional defects in the α- and β-phases, respectively, but it also explains why Mo stabilizes the β-phase.

(d). Binding energy between a Mo substitutional atom and a vacancy in α-Ti

The Mo substitutional defect and vacancy bind with an energy of 0.22 eV when both defects are in the same basal plane, and 0.23 eV when they are in adjacent basal planes. Our results agree fairly well with those of Xu et al. [52], with good agreement for the binding energy when the defects are in the same basal plane, and a difference of about 0.1 eV between the binding energies on adjacent basal planes. The larger difference in the latter case could be due to the smaller supercell size of 4×4×2 used by Xu et al. [52]. The changes in the bond lengths associated with the Mo-vacancy complex are large, varying between −10% and +7% of the ideal bond length in the basal plane. The supercell size parallel to the c lattice vector used by Xu et al. [52] is half that used in this study and that may be insufficient to allow the distortions we see in figure 5.

Figure 5.

(a) Basal plane in α-Ti containing a relaxed Mo substitutional atom (red in colour, dark grey in black and white) adjacent to a vacancy. (b) Atoms represented by larger circles in a basal plane containing a relaxed Mo substitutional atom (red in colour, dark grey in black and white)and atoms represented by smaller circles in an adjacent basal plane containing a vacancy at a nearest neighbour site of the Mo-atom. All bond lengths are expressed as percentages of the equilibrium lattice parameter a=2.94 Å . (Online version in colour.)

Although the binding energy is positive, it is insufficient to bind a Mo-atom and a vacancy at room temperature for a significant period of time. With a typical vibrational attempt frequency of 10¹² Hz the defects will take less than 10 ns to unbind. The vacancies are therefore almost immediately liberated from the Mo-atom by thermal fluctuations even at room temperature. Therefore, if there were sufficient vacancies present to enable creep to occur by a mechanism involving mass transport, the presence of the additional Mo in the α grains of Ti-6246 would not prevent mass transport from occurring by a vacancy mechanism. This is the central result of the paper. It indicates that the reduction in the susceptibility to CDF in Ti-6246, with its higher Mo-content, is not due to suppression of load transfer to hard grains during the dwell period as a result of creep involving mass transport. Put another way, we see no reason for the intrinsic creep properties of α-grains in Ti-6242 and Ti-6246 to differ. By ‘intrinsic’ we mean excluding differences that may arise from microstructural features such as their different grain sizes and local grain orientation distributions.

4. Discussion

With a formation energy of 2.15 eV, the equilibrium concentration of vacancies at room temperature is 7.6×10⁻³⁷, i.e. zero. At 200°C it rises to 1.2×10⁻²³, still completely negligible. At these temperatures, we may conclude that there are no intrinsic vacancies in thermal equilibrium. However, it is well known [36,53] that vacancies may also be produced in Ti by an extrinsic mechanism where trace amounts of residual Fe, Co and Ni substitutional impurities become interstitial defects leaving behind vacancies. Based on our computed DFT energies, we can test such a scenario for Mo. The energy required for a substitutional Mo-atom in α-Ti to become interstitial and leave behind a vacancy is 2.17−0.57+2.15=3.75 eV. We may conclude that Mo is not an extrinsic source of vacancies in α-Ti.

In this paper, we have tested the hypothesis that the cause of the smaller susceptibility to CDF of Ti-6246 is a reduction in the load transfer to hard grains during the dwell period of CDF as a result of vacancy trapping by the higher concentration of Mo-atoms in the α-phase seen experimentally by Qiu et al. [1]. The assumption here is that the load transfer to hard grains is effected by creep in the soft grains involving mass transport by a vacancy mechanism, such as dislocation climb. The principal result of this paper is that such vacancy trapping by Mo-atoms does not occur at room temperature. In §3d, we found that the binding energy between a substitutional Mo-atom and a Ti-vacancy in α-Ti is only 0.23 eV. At room temperature, a vacancy would escape from a Mo-atom in less than 10 ns through thermal excitations.

It appears that the more likely explanation for the smaller susceptibility of Ti-6246 to CDF than Ti-6242 is microstructural in origin: the smaller grains and more homogeneous distribution of grain orientations in Ti-6246 observed by Qiu et al. [1] reduce the probability of fatigue cracks in hard grains in Ti-6246 from becoming critical. But this does not explain the elimination of the CDF susceptibility in Ti-6242 at temperatures above about 200°C [3].

We suggest that the origin of the observed temperature dependence of CDF is the large reduction in the critical resolved shear stress for 〈c+a〉 slip at temperatures between 0°C and 300°C observed [3] in single crystal experiments of Ti-6.6Al. We suggest that this reduction enables sufficient plastic deformation to take place in hard grains by 〈c+a〉 slip leading to smaller loads on grain boundaries. After this research was completed we became aware that a similar suggestion has recently been made by Somnath Ghosh and co-workers [54,55].

If the intrinsic lattice resistance to the motion of 〈c+a〉 dislocations is large, they are likely to move by the nucleation and migration of double kinks, which are thermally activated processes. The dislocation velocity, v_d, then becomes

$$v_{\mathrm{d}}=4h\nu \exp (-\frac{\mathrm{\Delta }G}{k_{\mathrm{B}}T})\sinh \left(\frac{|\tau |bhp}{k_{\mathrm{B}}T}\right)\exp \left(\frac{|\tau |bh{l}_c}{k_{\mathrm{B}}T}\right),$$ 4.1

where ΔG is the sum of the free energy of formation of a double kink and the free energy of migration of either kink, |τ| is the absolute value of the resolved shear stress on the pyramidal plane in the direction of 〈c+a〉, ν is the attempt frequency, b is the magnitude of c+a, h is the height of each kink, p is the crystal period along the straight 〈c+a〉 dislocation line, l_c is the critical separation of the double kink for it to continue to separate into two kinks, k_B is Boltzmann’s constant and T is the temperature. Orowan’s equation, $\dot{\gamma }={\rho }_{\mathrm{M}}b{v}_d$, provides the link to the contribution to the strain rate $\dot{\gamma }$, where ρ_M is the density of mobile 〈c+a〉 dislocations. To test this suggestion, we need reliable calculations of the formation energy of double kinks and the migration energy of single kinks on 〈c+a〉 dislocations—a formidable challenge for DFT.

We can estimate ΔG as follows. The Debye temperature in Ti is 147°C [56]. At room temperature, the attempt frequency is, therefore, less than the Debye frequency, ν_D, whereas at temperatures above 200°C the attempt frequency may be equated to ν_D=8.75×10¹² s⁻¹. Consider a screw c+a dislocation for which b=p=5.51 Å . The kink height, h, is the spacing of the Peierls valleys parallel to c+a, which is approximately the lattice constant a=2.94 Å . Setting T=200°C we obtain |τ|bhp/(k_BT)=8.20 and $\sinh (|\tau |bhp/(k_{\mathrm{B}}T))$ is very well approximated by $\frac{1}{2}\exp (|\tau |bhp/(k_{\mathrm{B}}T))$. Assuming l_c≈p, equation (4.1) becomes

$$v_{\mathrm{d}}=2h\nu \exp (-\frac{\mathrm{\Delta }G}{k_{\mathrm{B}}T})\exp \left(\frac{2|\tau |bhp}{k_{\mathrm{B}}T}\right).$$ 4.2

We find $\exp ((2|\tau |bhp)/(k_{\mathrm{B}}T))=1.32\times {10}^7$. To achieve a typical strain rate of 10⁻⁴ s⁻¹ with a mobile 〈c+a〉 dislocation density of ≈10¹⁰ m⁻², the average dislocation velocity needs to be about 1.81×10⁻⁵ ms⁻¹. Substituting these numbers into equation (4.2), we deduce ΔG≈1.46 eV. To check this estimate, we may use this value of ΔG in equation (4.1) to evaluate the critical resolved shear stress for slip of screw 〈c+a〉 dislocations at 27°C required to generate a strain rate of 10⁻⁴ s⁻¹. We obtain 860 MPa, in fair agreement with the experimental measurement of [3], p. 21 of ≈800 MPa, given the uncertainties in the assumed density of mobile 〈c+a〉 dislocations, the assumed strain rate and in l_c. Combining equation (4.2) with the Orowan equation, we obtain the following relation between the critical resolved shear stress and the strain rate

$$\tau =\frac{\mathrm{\Delta }G}{\mathrm{\Omega }}+k_{\mathrm{B}}T\ln \left(\frac{\dot{\gamma }p}{\nu \mathrm{\Omega }{\rho }_{\mathrm{M}}}\right),$$ 4.3

where Ω=2bhp is the activation volume for creating a double kink and moving a single kink.

5. Conclusion

We tested a hypothesis that vacancies in α-Ti may be trapped by Mo substitutional impurity atoms. The hypothesis was motivated by the observation [1] that the Mo-content of α-grains in Ti-6246 is larger than that in Ti-6242 grains. The former alloy has a much smaller susceptibility to CDF than the latter. Load is transferred to hard grains during the dwell period of CDF by creep relaxation in soft grains. If this load transfer could be suppressed, then that would reduce the susceptibility of the alloy to CDF. If the creep relaxation involved mass transport effected by vacancy diffusion, then trapping the vacancies would reduce the creep, and hence the susceptibility to CDF. However, it was found that the binding energy between vacancies and substitutional Mo-atoms was insufficient to trap the vacancies for long enough at room temperature. Therefore, the hypothesis was not supported by our calculations.

In the light of this result and of the experimental observations by Qiu et al. [1], we agree with Qiu et al. that the reason for the smaller susceptibility of Ti-6246 to CDF than Ti-6242 is the smaller grain size and more homogeneous distribution of grain orientations in the former than in the latter. These microstructural features prevent fatigue cracks in hard α-grains from becoming critical. It is suggested that the susceptibility of Ti-6242 to CDF is almost eliminated at temperatures above 200°C because 〈c+a〉 slip is activated in the hard grains so that stress concentrations are much less likely to arise at grain boundaries. It is suggested that the temperature dependence of 〈c+a〉 slip, which implies a strain rate dependence of 〈c+a〉 slip, is due to the thermally activated formation and migration of kinks on 〈c+a〉 dislocations.

Data accessibility

Supplemental data for this article can be accessed https://doi.org/10.6084/m9.figshare.4754038.

Authors' contributions

A.P.S., D.R. and P.D.H. conceived and designed the project. A.J.R. carried out the D.F.T. simulations and A.J.R., A.P.S., P.D.H. and B.G. analysed the resulting data. A.P.S. derived the model of 〈c+a〉 dislocations moving via the double kink mechanism. P.D.H. and A.P.S. supervised simulations carried out at Imperial College and B.G. supervised simulations carried out at the Max-Planck-Institut für Eisenforschung. A.J.R. and A.P.S. wrote the paper, and all authors checked and approved it for publication.

Competing interests

We declare we have no competing interests.

Funding

A.J.R. was supported through a studentship in the Centre for Doctoral Training on Theory and Simulation of Materials at Imperial College funded by EPSRC under grant no. EP/G036888/1. Additional support was provided by Rolls-Royce plc.