Alloy design by tailoring phase stability in commercial Ti alloys.

The mechanical characteristics and the operative deformation mechanisms of a metallic alloy can be optimised by explicitly controlling phase stability. Here an integrated thermoelastic and pseudoelastic model is presented to evaluate the β stability in Ti alloys. The energy landscape of β → α ′ / α ′′ martensitic transformation was expressed in terms of the dilatational and transformational strain energy, the Gibbs free energy change, the external me- chanical work as well as the internal frictional resistance. To test the model, new alloys were developed by tailoring two base alloys, Ti – 6Al – 4V and Ti – 6Al – 7Nb, with the addition of β -stabilising element Mo. The alloys exhibited versatile mechanical behaviours with enhanced plasticity. Martensitic nucleation and growth was fundamentally dominated by the competition between elastic strain energy and chemical driving force, where the latter term tends to lower the transformational energy barrier. The model incorporates thermodynamics and micromechanics to quantitatively investigate the threshold energy for operating transformation-induced plas- ticity and further guides alloy design.

A significant increase in ductility and toughness can result from tailoring phase stability [12,13]. In order to design alloys combining superior mechanical characteristics, the first priority is to understand the microstructure at ambient temperature. When the alloys are cooled rapidly enough from above the β-transus temperature, there is insufficient time for eutectoidal diffusion-controlled decomposition processes to occur, thus the high temperature phase transforms into martensite [14]. For Ti alloys, the high temperature body-centred cubic (bcc) β phase transforms martensitically into a hexagonal close packed (hcp) α ′ phase upon quenching, where α ′ is crystallographically identical to the equilibrium α phase. On the other hand, the solid metastable β phase may transform, under external stress, into a distorted hexagonal structure designated α ′′ which has an orthorhombic unit cell. The α ′′ phase nucleates heterogeneously as thin laths at existing subgrain boundaries with (112) β //(020) α ′′ and [110] β //[001] α ′′ orientation relationships [15]. It may be activated at relatively small strains to accommodate deformation. The phase stability of Ti alloys can be altered by the addition of β-stabilising elements of two types, isomorphous and eutectoid [4]. In general isomorphous β-stabilisers (such as Mo, V, W, Ta and Nb) decompose to form α + β phases with no intermetallic compounds being formed; however, they are relatively high-cost additions. Conversely, eutectoid β-stabilisers (such as Fe, Cr, Cu and Co) have stronger β-stabilising effects, meaning they are more potent in suppressing the β-transus temperature. Yet a limited addition should be controlled in order to avoid the formation of brittle intermetallics. The martensitic transformation is characterised by its displacive nature. It may be regarded as a strain transition with shear and dilatational displacements, respectively parallel and normal to the habit plane [16]. Besides, it has been considered a mode of deformation which competes with dislocation slip when external stresses are applied to the parent phase. A thermodynamics-based interpretation of the thermoelastic equilibrium was introduced by Olson and Cohen [17,18] to explain the thermally induced nucleation and growth of thermoelastic martensite. A key parameter indicating the transformation tendency is the martensite start temperature M s . For the stress-assisted transition, a pseudoelastic force balance was proposed by Green et al. [19] as a mechanical analogue to the thermoelastic force balance. In addition to the elastic strain energy and the chemical driving force terms, a mechanical work contribution was incorporated to provide the extra transformation driving force for the TRIP effect.
The purpose of this work is threefold. First, an integrated thermoelastic and pseudoelastic modelling approach incorporating thermodynamics and micromechanics is proposed for investigating the energy landscape in β→α ′ /α ′′ phase transition. The transformational energy barrier, the elastic strain energy and the chemical driving force were quantitatively evaluated; the atomic frictional stress due to solid solution hardening is, for the first time, introduced. Secondly, a series of new alloys are developed by tailoring two widely used commercial Ti alloys: Ti-6Al-4V and Ti-6Al-7Nb. Mo was added in controlled amounts for altering the microstructure and the phase stability [20]. The new alloys are studied by thermodynamic calculations and examined experimentally. Third, the modelling approach was tested via the proposed alloys and the energy landscape for operating transformation-induced plasticity was calculated. It is shown that the composition-dependent threshold value can further guide new alloy design.

Energy barrier for coherent martensitic nuclei
Martensite nucleation upon rapid cooling results from the competition between elastic strain energy and chemical driving force [21]. A martensitic embryo of a given volume will tend to adopt a shape which minimises the combined interfacial and strain energies. In the case of a thin martensite ellipsoidal inclusion with radius a and semi-thickness c, the overall energy barrier associated to the formation of the martensitic particle is [14,17]: where γ is the interfacial energy per unit area of the product phase. The Gibbs free energy change ΔG chem = Δg chem × 4 3 πa 2 c and the global stored elastic strain energy E str = Δg str × 4 3 πa 2 c, where Δg chem is the volumetric chemical driving force [22] and Δg str is the elastic strain energy per unit volume of martensite; therefore Eq. (1) becomes: The elastic strain energy Δg str of the coherent nucleus is more important than the interfacial energy, since the shear component of the pure Bain strain is as high as 0.32 which produces large strains in the surrounding β matrix [14].

Elastic strain energy & thermoelastic equilibrium
The distortion around a coherent nucleus implies that a crystal containing a product phase is not at its lowest energy state; such extra energy is defined as the nucleus strain energy [23]. The elastic strain energy is computed by defining both martensitic nuclei and β matrix are elastically isotropic. The strain tensor Δg str is divided into deviatoric and hydrostatic components; therefore the coherency strain energy Δg dila str led by a dilatational volume expansion and the transformational strain energy Δg shear str are added to express [21]: The elastic strain energy caused by a dilatational volume change comes from Eshelby's elastic field theory of an ellipsoidal inclusion embedded in an infinite elastically isotropic matrix [24]: where ν is the Poisson's ratio, μ is the shear modulus which is a function of the absolute temperature T [25]. ΔV V is the dilatation and can be expressed by the principal lattice strains [21]: where η 1 , η 2 and η 3 represent the eigenvalues of the strain tensor; they are evaluated with respect to a Cartesian coordinate system with axes parallel to the orthorhombic unit cells of β, α ′ and α ′′ , since these axes remain mutually perpendicular upon transformation [26]. Fig. 1 schematically represents each lattice parameter and the atomic displacement in β→α ′ and β→α ′′ transitions. The principal lattice strains are expressed as [27,28]: where a β is the lattice parameter of the bcc β phase; a M α , b M α and c M α are the lattice parameter of the martensite, where M designates either hexagonal or orthorhombic phase [29].
On the other hand, the nucleus can form by a simple shear parallel to the plane of the thin ellipsoidal nucleus, and complete coherency is maintained at the interface [14]. Therefore the shape change is approximated by pure shear and is described as [24]: where s is the shear component. Therefore the energy barrier ΔG associated with the formation of a coherent martensitic nucleus is derived by combining equations above: (1 + ν) The elastic strain energy and interfacial energy oppose the transition, whereas the chemical driving force by Gibbs free energy change promotes it.
Thereafter the thermoelastic force balance can be established. By defining T 0 as the temperature at which the Gibbs free energies of parent and product phases are equal, thus the martensitic nucleation is prohibited above T 0 and the alloy only displays β phase. A martensitic embryo with semi-thickness c 0 and radius a 0 turns into a nucleus as the temperature is below T 0 . Wollants et al. [30] demonstrated that the martensitic embryo grows radially until it hits a barrier such as a high angle grain boundary. The very thin martensitic plates first form with a very large a/c ratio. From that point, the embryo continues to thicken in  a direction normal to the plate plane. Following the analysis by Bigon et al. [21], thickening stops until it is arrested as the system energy is depleted (∂ΔG/∂c = 0) at the interface between the parent and the product phase. Fig. 2a schematically reproduced the nucleation and growth of a nucleus of martensite [21]. The chemical driving force at M s is given by differentiating Eq. (8) with respect to c and equating to: In this case a is the grain radius and c is the minimum detectable semi-thickness of a martensitic plate. Combining with Eqs. (4) and (7), thus Eq. (9) can be written as: This is an ideally reversible thermoelastic force balance at M s , where all the transformational strain is accommodated elastically.

Stress-assisted martensitic transformation
Pseudoelastic force balance refers to the mechanically reversible behaviour commonly observed in thermoelastic martensitic alloys. The addition of an external mechanical work term lowers the transformational energy barrier for pseudoelasticity: where τ T and γ T is the resolved shear stress and shear strain at the onset of transformation, respectively. A force balance can be derived from an energy expression in Eq. (11) by adopting the lengthening and thickening growth forces in the thermoelastic condition. When the radial growth of a martensitic plate is stopped by obstacles, the plate will continue to thicken in response to the transformational force ∂ΔG/ ∂c in the c direction. In the absence of frictional resistance, thickening will stop when ∂ΔG/∂c = 0 and the plate reaches a mechanical equilibrium under the applied stress [19]: This equation can be interpreted as follows: thermoelasticpseudoelastic equilibrium can be achieved when the force generated by the transformational driving free energy is equal to the force generated by the stored elastic strain energy. For thermoelasticity the driving force is Δg chem , whereas for pseudoelasticity this becomes Δg chem + τ T γ T . Fig. 2b and c illustrate an analogy between thermoelastic and pseudoelastic equilibriums. The volume of a single martensitic plate changes as a function of driving force. The driving force for the thermoelastic transformation is inversely proportional to temperature, whereas the driving force for the pseudoelastic transformation at a given temperature increases with applied stress.
The pseudoelastic force balance represents ideal energy balances in plate growth and reversal, i.e. the transformation strain is elastically accommodated and there is no frictional resistance on the transformational interface [19]. A frictional force can be introduced as a reverse stress τ f acting to oppose the dislocation motion. Thus the net driving force for transformation is effectively reduced by the term τ f γ T and the pseudoelastic force balance becomes: The frictional resistance term becomes positive when the plate is thickening and negative when it reverts. The internal atomic friction stress τ f is generated by the interactions between dislocations and solutes in the solid solution, where dislocation motions are impeded by such interaction. The stress generated by solid solution hardening (SSH) is [31]: where X i is the molar fraction of solute i, λ is a misfit parameter accounting for the solute/solvent lattice parameter misfit and the shear modulus misfit and Z is a temperature-dependent numerical factor whose value can be obtained from a plot of dτ/dX 2/3 i vs. λ 4/3 . The detailed SSH calculation was presented in recent work by authors [32]. Here a threshold parameter ΔΓ is established to describe the energy difference between the transformational promoting terms and the opposing ones: The composition-dependent ΔΓ defines the energy landscape for pseudoelastic martensitic transition to take place. An equilibrium can be achieved by minimising ΔΓ such that the chemical driving force Δg chem and mechanical work τ T γ T approach the magnitude of the elastic strain energy Δg dila str + 2Δg shear str and the frictional resistant energy τ f γ T . To test the model, a series of alloys were developed and investigated in next sections.

Experimental procedure
Commercial Ti64 (Ti-6Al-4V) and Ti67 (Ti-6Al-7Nb) alloy powders supplied by Carpenter Additive were utilised as base materials. The oxygen content in each type of powder is less than 0.06 wt%. Pure Mo powder (99.9 mass %) was added to each grade in controlled amounts for tailoring the microstructure and the phase stability. The alloys were prepared by arc-melting using a nonconsumable tungsten electrode and a water-cooled copper hearth under a Ti gettered argon atmosphere. Each sample with a total mass of 2 g was remelted for 4 times to promote homogeneity and was finally cast into a cylindrical rod with a diameter of 3 mm under a cooling rate of ∼ 10 3 K/s [33,34]. Total weight losses during the arc-melting and casting procedure were between 0.05 and 0.10 wt% and the actual O content in each as-cast alloy is less than 0.1 wt%. The compositions of the alloys as well as the corresponding Mo equivalent Mo eq. [5] are listed in Table 1. In general, a Mo eq. value above of 10 is required to stabilise β phase upon quenching. The two groups of alloys are labelled as Ti64-xMo and Ti67-xMo where x = 10-20, represent the amount of Mo additions. During the rapid cooling, complete solute trapping can be achieved inducing diffusionless solidification [35]. The rapidly solidified alloy systems lead to the supersaturated solid solution with the initial chemical composition of the alloy.
The phase constituents were examined by X-ray diffraction (XRD) with Cu-Kα radiation (D8 Advance from Bruker AXS operated at 40 kV and 30 mA equipped with LynxEye detector). XRD line profiles are analysed to evaluate the peak broadening due to the overlapped peaks. Phase analysis was performed by MAUD software on the XRD patterns measured on the cross-sectional flat surface. Compression tests were carried out on a universal testing machine at room temperature under a constant strain rate ε = 5 × 10 − 3 s − 1 . A strain gauge was used to calibrate and measure the true strain during compression. The tested specimens with a diameter of 3 mm and a length of 6 mm were cut from Table 1 Compositions of the investigated alloys and their corresponding Mo equivalent values Mo eq. .

Alloy label
Composition in mass % Composition in at.% Moeq.

Experimental data
An essential ingredient of the alloy design effort is an accurate description of thermodynamic properties within the composition design space. Fig. 3 displays the equilibrium diagrams of the Ti64-xMo and Ti67-xMo alloys, revealing a change in phase fraction with cooling temperature. Increasing the Mo content leads to a wider temperature widow for the high temperature bcc phase, which further delays the formation of hcp phase and stabilises β phase at ambient temperature. Fig. 4 exhibits the variation of Gibbs free energy in bcc and hcp phases as a function of Mo content at room temperature. The chemical driving force for the transition can be obtained from ΔG bcc→hcp = G bcc − G hcp . The value of ΔG bcc→hcp decreases with the addition of Mo, suggesting the transition is inhibited. The intersection of the two curves represents the composition where the chemical driving force vanishes; alloys with ΔG bcc→hcp < 0 display a β phase fully stabilised at ambient temperature. A quantitative interpretation on the Gibbs free energy change is presented in Section 5. Fig. 5 shows the XRD phase constituents of the investigated alloys.
Both Ti64 and Ti67 base alloys exhibited strong martensitic α ′ phase without visible β-peaks from the diffraction patterns (Fig. 5a). The supersaturated β solid solution become more prominent whilst increasing the Mo additions ( Fig. 5b and d). Full β phase was retained in high Mo containing alloys such as Ti64-15Mo, Ti64-20Mo, Ti67-18Mo and Ti67-20Mo. It is widely acknowledged that bcc to orthorhombic martensitic transformation occurs under deformation, whereas bcc to hcp transition happens upon rapid cooling. In the exceptional cases that hcp α ′ forms from the bcc matrix under external stress, α ′ can only be triggered in alloy compositions of very low phase stabilityeven fully β phase cannot be retained at ambient temperature and the microstructure contains primary α phase from quenching [36]. Besides more evidence would be required to prove that α ′ phase does transform from the β phase. This means α ′′ is the only strain-induced transformation product commonly forming in the present type of metastable Ti alloy, which is well identified in literature [21,37]. A broadening of the main β (110) peak was observed in Ti67-18Mo upon deformation till failure. Peak analysis (Fig. 5f) identified it consists of orthorhombic α ′′ (002) peak, suggesting β→α ′′ transition was activated upon deformation in Ti67-18Mo. Thereafter, the yield stress increases again from Ti64-15Mo to Ti64-20Mo alloy due to the enhanced solid solution strengthening; concomitantly the β alloys exhibited superior ductility. Similar trends were found in the Ti67-xMo series alloys. Fig. 7 exhibits the deformation microstructures of the selected Ti64-10Mo and Ti67-18Mo alloys upon 10% strain. The SEM backscattered electron image of Ti64-10Mo (Fig. 7a) displays parallel slip bands in the β grains and with their propagation being obstructed by grain boundaries. The EBSD pattern quality maps of Ti64-10Mo (Fig. 7b) and Ti67-18Mo (Fig. 7c) clearly identified the grain boundaries as well as the formation of shear bands. Some of the shear bands are thickened to accommodate the inhomogeneously distributed strain The calculation was performed by using the Thermo-Calc database TCTI1: Ti-Alloys. [38]. Although extensive deformation bands were observed, no trace of mechanical twinning was observed in neither alloy. According to our previous work [39], the significant strain-hardening observed in TWIP and TRIP/TWIP Ti alloys mainly stems from the reduced dislocation mean free path cut by the dynamic activation of mechanical twinning, as well as from the kinematic hardening due to the enhanced dislocation pile-up at coherent twin interfaces. The absence of twinning brought moderate strain-hardening, even though strain-induced martensitic transformation was operated. Besides, the role of martensite on strain-hardening is still controversial, as it is classically considered as a soft phase with a modulus lower than that of β matrix; and recent studies reflect orthorhombic martensites may provide relatively limited strain-hardening as they act as less effective obstacles for dislocation glide [37,40]. Fig. 8 presents the phase stability map of Ti alloys by accounting for the composition-dependent electronic structures [41]. Bond order Bo exhibits the electron cloud overlap between solute and solvent, which is a measurement of the covalent bond strength between the alloying elements and the matrix. Metal d-orbital energy level Md correlates with the electronegativity and atomic radii of the solutes. The electronic parameters of each element were calculated by means of the molecular orbital method [42,43]. The average Bo and Md value of each alloy are   well as to aid in distinguishing deformation modes [41]. The Ti64-xMo and Ti67-xMo vectors transited from the α + β domain to the metastable β domain with the increasing of Mo, implying the metastable alloys are likely to trigger TRIP/TWIP effects upon deformation. Nevertheless, one has to keep in mind that the dividing borders were illustrated empirically from early experimental data so that many modern TRIP/TWIP alloys violate the boundary constrains [39], meaning it may not play a determinant role in predicting the operative deformation mechanisms.
Preferably the alloying vectors may provide a visual aid to reflect the effects of different stabilising elements on the transition between deformation modes.

Model application & discussion
The model integrating pseudoelasticity arising from stress-assisted martensitic transformation, is applied to the Ti64-xMo and Ti67-xMo alloys. Volumetric chemical driving force can be obtained from Δg chem = ΔG chem /V φ m , where ΔG chem is the molar Gibbs energy difference between the parent and the product phases. The values were obtained from TCTI1:Ti-Alloys thermodynamic database and listed in Table 2. V φ m is the molar volume of phase φ and its value can be calculated by a linear combination of pure elements plus a regular-solution model for the excess volume [44]: where X i denotes the molar fraction of element i and here X j denotes the   molar fraction of Ti solvent. Ω φ i,j is the molar volume interaction parameter between each solute and the Ti solvent [44]. Applying the compositions of Ti64-xMo and Ti67-xMo alloys to Eq. (16), the calculated mean molar volume is 1.02 × 10 − 5 and 1.03 × 10 − 5 m 3 /mol, respectively. The volume dilatation of the β→α ′ /α ′′ martensitic transformation can be calculated using the current molar-volume assessment [44]: The dilatational strain energy Δg dila str = 2 9 (1+ν) 33 is the Poisson's ratio and μ = 39 GPa the shear modulus of the Ti alloys at room temperature [45].
The transformational strain energy Δg shear c a s 2 μ can be calculated in the following manner. β→α ′′ transition is associated with the {332}〈113〉 twinning [46,47], which is unique as the most operative twinning system in metastable Ti alloys. A fundamental crystallographic model [48] suggests {332}〈113〉 twinning only moves one-half of the atoms from the untwinned lattice sites with an associated shear strain 1 2 ̅̅ 2 √ , then additional shuffling is necessary to transport the rest atoms.
Therefore the transformational shear equals to the corresponding shear strain produced by {332}〈113〉 twinning with s = γ {332}〈113〉 = 1 2 ̅̅ 2 √ . The detectable semi-thickness of a martensitic plate by transmission electron microscopy is c = 50 nm [49]. The average grain size of the samples was 100 μm, thus a = 50 μm. Combining the data above, the shear-induced strain energy Δg shear str = 1.91 × 10 7 J/m 3 . Note that calculated Δg dila str + 2Δg shear str should be a minimum value, since the c/a ratio increases as the martensitic plate grows. The fact that these nuclei remained after stress release implies that the elastic strain energy was reduced via plastic accommodation [19], which prevented their complete reversion.
Referring to the critical resolved shear stress τ T to active transformation, such term can be obtained from the experimental stress-strain curves. τ T is taken as the resolved yield stress corresponding to the first plates to form, where multiple-plate interactions would be minimal [50].  Table 2. The transformational energy landscape ΔΓ vs. molar chemical driving force is illustrated in Fig. 9. β→α ′ /α ′′ transition is operative when ΔΓ⩾0, since the energy promoting phase transformation is larger than the opposing term. ΔΓ decreases as the addition of β-stabilisers, which is associated with the considerably reduced chemical driving force. The positive ΔΓ moving towards to 0 as the decreasing of ΔG chem , meaning that pseudoelastic force balance becomes achievable within appropriate energy threshold. The ΔΓ of Ti67-18Mo alloy is the positively closest towards 0 so that pseudoelastic equilibrium is comparatively favoured. Other alloys become stabilised at ambient temperature as fully β phase when ΔΓ < 0, where martensitic transformation can hardly take place.
The current model treat the orthorhombic α ′′ martensite and the hcp α ′ phase energetically equivalent in thermodynamic calculations. It is reasonable to describe the β→α ′′ transformation as a crystallographically incomplete β→α ′ transformation that should have a similar transformation enthalpy change, given by orthorhombic martensite has a crystal structure intermediate between bcc and hcp lattices [1]. The integrated model was quantitatively tested by experimental data, which adopt composition-dependent input such as the chemical driving force, the lattice dilatation and the elastic strain energy. The frictional resistance was interpreted by the effect of solid solution hardening, where atomic friction was generated by the interactions between the mobile dislocations and the solute atoms. The energy landscape by the calculations can be employed as a criterion to guide plasticity-oriented Ti alloy design.  Fig. 9. The transformational energy landscape ΔΓvs. chemical driving force ΔG chem in each alloy. The positive energy difference ΔΓ moving towards to 0 as the decreasing of ΔG chem , meaning that pseudoelastic force balance becomes achievable within appropriate energy domain.

Conclusions
An integrated thermoelastic and pseudoelastic model was built to evaluate β stability and phase transformations in Ti alloys. The model explicitly calculated the energy landscape to operate β→ α ′ / α ′′ transition, where the competition between chemical driving force and elastic strain energy dominates the transformation equilibrium. Two commercial Ti alloys, Ti64 and Ti67, were tailored with various Mo additions to test the model. The alloys with fully retained β phase exhibited superior ductility, where strain-induced orthorhombic α ′′ martensite was found in Ti67-18Mo alloy. The formation of martensite may accommodate strain yet the contribution to strain-hardening is relatively moderate with the absence of deformation twinning. A composition-dependent phase stability parameter ΔΓ was proposed to define the energy threshold for transformation-induced plasticity. Martensitic transitions perform when ΔΓ⩾0, since the promoting energy is larger than the opposing term. ΔΓ decreases as the addition of β-stabilisers, which is associated with the considerably reduced chemical driving force. The alloy preserves relatively stabilised β phase at ambient temperature when ΔΓ < 0, where martensitic transition is not operative neither upon cooling nor under external stress. A comprehensive modelling approach incorporating thermodynamics and micromechanics is presented; this can be applied to alloy design and to predict strain-induced martensitic transformation in metastable alloys.

Data availability
The data that support the findings of this study are available upon reasonable request.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.