# Redefining the β-Phase Stability in Ti-Nb-Zr Alloys for Alloy Design and Microstructural Prediction

## Abstract

β-Phase stability is a key consideration for the design of β-titanium (β-Ti) alloys and subsequent heat treatment and/or thermo-mechanical processing. The concept of the Mo equivalence (Mo-Eq), proposed by Molchanova (Phase diagrams of titanium alloys, 1965), has been commonly used as a general guideline to gauge the stability of a β-Ti alloy. A critical literature review has shown that all four existing Mo-Eq expressions deviate substantially from experimental observations and the well-established *d*-electron theory in predicting the β-phase stability of Ti-Nb-Zr alloys. The reasons are that existing Mo-Eq expressions either completely neglect or significantly overestimate the β-stabilizing effect of Zr. In this study, a new Mo-Eq expression, i.e., (Mo-Eq)_{Ti-Nb-Zr} = 0.238Nb (wt.%) + 0.11Zr (wt.%) + 0.97, has been defined for Ti-Nb-Zr alloys in order to properly address the β-stabilizing effect of Zr. This new Mo-Eq expression showed proven consistency with both experimental observations and the *d*-electron theory in predicting the β-phase stability of various Ti-Nb-Zr alloys.
With necessary modifications, the approach developed is expected to be also applicable to the assessment of the β-phase stability in other Zr-containing Ti alloys.

## Introduction

β-Titanium (β-Ti) alloys that are free from vanadium (V), nickel (Ni) and aluminium (Al) are one of the most promising groups of metallic biomaterials for orthopaedic applications. By definition, β-Ti alloys refer to those which contain sufficient β-stabilizing element(s) to enable the β-phase to be retained in a metastable condition by water quenching.1 They are generally divided into metastable (age-hardenable) and stable β-Ti alloys (not age-hardenable).2 The critical minimum amount of β-stabilizers needed to retain 100% β-phase on water quenching is denoted as β_{c}, while the minimum content of β-stabilizers needed to form stable β-Ti alloys is referred to as β_{s}. Metastable β-Ti alloys are those that fall in between β_{c} and β_{s}, within which α can precipitate upon ageing. As the values of β_{c} and β_{s} depend on alloy chemistry, a parameter known as the Mo equivalent (Mo-Eq) has been defined and commonly used as a general guideline to assess the β-phase stability of a Ti alloy for both alloy design and processing.3^{–}5 It is therefore an important design parameter for Ti alloys. This paper first examines all existing Mo-Eq expressions and then focuses on the development of a new Mo-Eq equation proposed specifically for the Ti-Nb-Zr system. This is followed by a detailed experimental assessment and comparison with Morinaga’s *d*-electron theory for alloy design.

## Existing Mo-Eq Expressions and Their Deficiencies

The concept of the Mo equivalence was proposed by Molchanova [Phase diagrams of titanium alloys (Figs. 193 and 194, p. 158)],3 which was a translation from *Atlas Diagramm Sostoyaniya Titanovykh Splavov* (Israel Program for Scientific Translations, Jerusalem, 1965). Boyer et al.4 presented the data shown in Fig. 193 of Ref. 3 as the Mo equivalence of a titanium alloy in the well-known *Materials properties handbook: titanium alloys* (1994) in the following form:

_{c}value of Mo (i.e., 10 wt.%) to that of respective β-stabilizer. As an α-stabilizer, Al is also included to show its opposing effect when it is present in a β-Ti alloy.

^{–}8 namely:

_{c}value of Mo should be 11 (rather than 10), which was also mentioned in Ref. 3. The slight difference between Eqs. 2, 3 and 4 resulted from experimental and analytical errors. In contrast, Eq. 5 combines both experimental and phase diagram data. For example, the coefficients for Nb and Ta were taken from Eq. 2, while the coefficient for Zr was derived from the Ti-Zr phase diagram.8 As a result, Zr was treated as a β-stabilizer that is more potent than Nb (0.28) and Ta (0.22), but no experimental support was given. Limited evidence suggests9

^{,}10 that the β-stabilizing effect of Zr depends on the quantity of other β-stabilizers present in the alloy.

^{,}11 They provide an experimental basis for a detailed assessment of Eqs. 2 and 5. As can be seen from Table I, despite the high Mo-Eq values predicted by Eq. 5, e.g., 24.12 for Ti-12Nb-30Zr (at.%), α″-martensite still forms on water quenching. In other words, Eq. 5 significantly overestimates the β-stabilizing effect of the alloying elements in these alloys. With regards to Eq. 2, it substantially underestimates the overall β-stabilizing effect of the alloying elements in Ti-Nb-Zr alloys. For example, Eq. 2 predicts Mo-Eq values of 3.64 and 5.43 for Ti-10Nb-45Zr (at.%) and Ti-14Nb-30Zr (at.%), respectively, which are much lower than 10. However, the β-phase was fully retained in both alloys upon water quenching. Equations 2 and 4 offer similar predictions, while Eq. 3 also shows a serious underestimation of the β-stabilizing effect. For instance, the Mo-Eq of Ti-14Nb-30Zr (at.%) is predicted to be 7.57 but full β phase is retained. So is the case for Ti-10Nb-45Zr (at.%), where Mo-Eq = 5.07 but full β phase is retained. On the other hand, Eq. 3 overestimates the β-stabilizing effect in Ti-20.9Nb-9Zr (at.%) (clear formation of α″, although Mo-Eq = 12.36). In summary, none of the existing Mo-Eq equations offers predictions that are consistent with experimental observations for Ti-Nb-Zr alloys. In addition, when all these predictions based on Eqs. 2–5 are compared with the

*d*-electron theory diagram, there is also substantial inconsistency.

Alloy compositions | Phases (water-quenched) | Mo-Eq values by existing expressions (wt.%) | New Mo-Eq by Eq. 7 | ||||
---|---|---|---|---|---|---|---|

at.% | wt.% | Equation 2 | Equation 3 | Equation 4 | Equation 5 | ||

Ti-12Nb-30Zr | Ti-16.82Nb-41.29Zr | α″,β | 4.71 | 6.56 | 5.21 | 24.12 | 9.5 |

Ti-14Nb-30Zr | Ti-19.4Nb-40.7Zr | β | 5.430 | 7.57 | 6.01 | 24.56 | 10.1 |

Ti-8Nb-46Zr | Ti-10.4Nb-58.8Zr | α″,β | 2.91 | 4.06 | 3.22 | 30.55 | 9.9 |

Ti-10Nb-45Zr | Ti-13Nb-57.1Zr | β | 3.64 | 5.07 | 4.03 | 30.48 | 10.1 |

Ti-20.9Nb-9Zr | Ti-31.7Nb-13.4Zr | α″,β | 8.88 | 12.36 | 9.83 | 15.17 | 9.99 |

Ti-20.2Nb-12Zr | Ti-30.18Nb-17.6Zr | β | 8.45 | 11.77 | 9.36 | 16.72 | 10.1 |

Ti-22Nb-4Zr | Ti-34.4Nb-6.13Zr | β | 9.63 | 13.42 | 10.66 | 12.51 | 9.8 |

Ti-22Nb-6Zr | Ti-33.9Nb-9.1Zr | β | 9.49 | 13.22 | 10.51 | 13.77 | 10 |

Ti-22Nb-8Zr | Ti-33.4Nb-11.92Zr | β | 9.35 | 13.03 | 10.35 | 14.95 | 10.2 |

## New Mo-Eq Expression for Ti-Nb-Zr Alloys

_{c}value for Ti-Nb binary alloys is 42 wt.%.12 Consequently, the Mo-Eq coefficient of Nb can be defined as

*ψ*× Nb (wt.%). Figure 1 shows the dependence of the quantity, Δ = 10 −

*ψ*× Nb (wt.%), on the Zr content for all the Ti-Nb-Zr alloys listed in Table I. The resultant significant linear relationship identifies the contribution of Zr in stabilizing the β phase in Ti-Nb-Zr alloys considered in Table I. Accordingly, a new Mo-Eq for Ti-Nb-Zr alloys can be defined from Fig. 1 as

## Applying the New Mo-Eq to the Design of Ti-Nb-Zr Alloys and Validation

_{2}O

_{2}. The samples were subsequently cleaned in an ultrasonic bath using ethanol and etched with Kroll’s reagent. Phase identification was conducted using a Bruker D8 Advance x-ray diffractometer (XRD) with a Cu K

_{α}radiation source at room temperature, and the resultant XRD data was analyzed by MAUD software.

Newly designed metastable Ti-Nb-Zr alloys and corresponding Mo-Eq values

References | Alloy compositions | Mo-Eq values by different expressions | |||||
---|---|---|---|---|---|---|---|

at.% | wt.% | Equation 2 | Equation 3 | Equation 4 | Equation 5 | Equation 7 | |

Zr-based alloys (this study) | Ti-15Nb-40Zr | Ti-19.3Nb-50.7Zr | 5.44 | 7.53 | 5.98 | 29.23 | 11.1 |

Ti-10Nb-55Zr | Ti-12.2Nb-65.8Zr | 3.42 | 4.76 | 3.78 | 34.34 | 11.1 | |

Ti-based alloys (this study) | Ti-10Nb-26Zr | Ti-15Nb-37Zr | 4.2 | 5.85 | 4.65 | 21.59 | 8.6 |

Ti-15Nb-25Zr | Ti-21.3Nb-34.8Zr | 5.96 | 8.31 | 6.6 | 22.32 | 9.9 | |

Ti-15Nb-22Zr | Ti-22Nb-31.3Zr | 6.16 | 8.58 | 6.82 | 20.87 | 9.6 | |

Ti-20Nb-21Zr | Ti-28Nb-29Zr | 7.84 | 10.92 | 8.68 | 21.47 | 10.8 | |

Ti-24.5Nb-18.3Zr | Ti-34Nb-25Zr | 9.52 | 13.26 | 10.54 | 21.27 | 11.8 | |

Ti-22Nb-24Zr | Ti-30Nb-32Zr | 8.4 | 11.7 | 9.3 | 23.44 | 11.6 | |

Ti-20.7Nb-26.7Zr | Ti-28Nb-35.4Zr | 7.84 | 10.92 | 8.68 | 24.48 | 11.5 | |

Ti-18.6Nb-31.1Zr | Ti-24.8Nb-40.7Zr | 6.94 | 9.67 | 7.69 | 26.07 | 11.3 | |

Ti-20Nb-12Zr | Ti-29.9Nb-17.6Zr | 8.4 | 11.7 | 9.3 | 16.86 | 10 | |

Ti-17Nb-21Zr | Ti-24.4Nb-29.6Zr | 6.72 | 9.36 | 7.44 | 20.82 | 10 | |

Ti-11Nb-38Zr | Ti-14.7Nb-50Zr | 4.2 | 5.85 | 4.65 | 27.7 | 10 | |

Ti-6Nb-53Zr | Ti-7.6Nb-65.7Zr | 2.24 | 3.12 | 2.48 | 32.79 | 10 | |

Ti-22.6Nb-4.6Zr | Ti-35Nb-7Zr | 9.8 | 13.65 | 10.85 | 13.09 | 10 |

With regard to the alloys from the literature (Table II), they were all reported to be nearly single β-phase alloys (i.e., may contain a small amount of α″ and/or ω).13^{–}15 By the definition of the Mo-Eq, their Mo-Eq values should be equal to or above 10 wt.%. As can be seen from Table II, the new Mo-Eq (Eq. 7) also shows excellent consistency with experimental observations.

## The Newly Proposed Mo-Eq and the *d*-Electron Theory

*d*-electron theory describes the potential relationship between phase stability and elastic properties of Ti alloys through two electronic properties, bond order,

*B*

_{o}and

*d*-orbital energy level,

*M*

_{d}.12

^{,}16 It was developed by Morinaga16 on the basis of the DV-Xα molecular orbital calculations of electronic structures. The theory has become an essential fundamental tool for the design of β-Ti alloys.1

^{,}12 In this approach,

*B*

_{o}denotes the covalent bond strength between Ti and a specific alloying element and

*M*

_{d}correlates the electronegativity and metallic radius of that element.

*B*

_{o}and

*M*

_{d}values for all the alloying elements of Ti can be found in Ref. 16. Using these

*B*

_{o}and

*M*

_{d}values, the compositional average of

*B*

_{o}and

*M*

_{d}can be calculated based on alloy composition through the following equations.16

*B*

_{o})

_{i}and \( (M_{d} )_{i} \) are the atomic fraction, bond order and

*d*-orbital energy level of the given element, respectively. Then, a \( \overline{{B_{\text{o}} }} \)–\( \overline{{M_{d} }} \) diagram can be plotted by taking the \( \overline{{B_{\text{o}} }} \) and \( \overline{{M_{d} }} \) values of different Ti alloys. Several sets of Ti alloys are used to define the three regions, α, α + β, β, of this \( \overline{{B_{\text{o}} }} \)–\( \overline{{M_{d} }} \) diagram. Along the (α + β)/β boundary in the \( \overline{{B_{\text{o}} }} \)–\( \overline{{M_{d} }} \) diagram, the β-phase is considered metastable.

*B*

_{o}and

*M*

_{d}values for pure Ti, Nb and Zr, which define the regime of the Ti-Nb-Zr alloys, similar to ternary phase diagrams. Each side of this triangle indicates the compositional average values of

*B*

_{o}and

*M*

_{d}of the Ti-Nb, Ti-Zr and Zr-Nb binary alloys. The blue dotted line represents the boundary of critical compositions for metastable β alloys according to the inventor of the

*d*-electron theory.12 This boundary may shift upwards or downwards depending on the exact alloying compositions. In the Ti-Nb-Zr system, adding Zr was found to shift this boundary downwards,14 as shown by the black dashed line in Fig. 3. The Mo-Eq values predicted by Eq. 7 are superimposed onto Fig. 3, which indicates that the newly developed Mo-Eq expression is also in good agreement with the

*d*-electron theory in predicting the β-phase stability. For example, Ti-Nb-Zr alloys having Mo-Eq values higher than 10 wt.% are situated above the black dashed line and vice versa. The new Mo-Eq is thus consistent with the

*d*-electron theory for Ti-Nb-Zr alloys, which was not the case for all the other four Mo-Eq expressions.

## Summary

A new Mo-Eq expression has been formulated for the Ti-Nb-Zr system, i.e., (Mo-Eq)_{Ti-Nb-Zr} = 0.238Nb (wt.%) + 0.11Zr (wt.%) + 0.97, which shows good consistency with experimental observations as well as literature data. In addition, it is highly consistent with the well-established *d*-electron theory in predicting the β-phase stability of various Ti-Nb-Zr alloys. In contrast, none of the existing four Mo-Eq expressions is consistent with the *d*-electron theory, as well as experimental observations. However, caution should be exercised when the alloy composition goes significantly beyond the bounds of the chemistries discussed in this study. It is proposed that the approach developed in this work can also be applied to the assessment of the β-phase stability in other Zr-containing Ti alloys.

## Notes

### Acknowledgements

This work was supported by the Australian Research Council (ARC) through DP150104719. The authors are grateful to Rodney Boyer for drawing our attention to E. K. Molchanova’s original contribution to the concept of the Mo equivalence (see Ref. 3).

## References

- 1.I.J. Polmear, D.H. StJohn, J.F. Nie, and M. Qian,
*Light Alloys: Metallurgy of Light Metals*, 5th ed. (Waltham, MA: Butterworth-Heinemann, 2017).Google Scholar - 2.I. Weiss and S. Semiatin,
*Mater. Sci. Eng. A*243, 46 (1998).CrossRefGoogle Scholar - 3.E.K. Molchanova and S. Glazunov, Phase Diagrams of Titanium Alloys: Transl. by A. Halbreich, N. Kaner, M. Statter, (Israel Program for Scientific Translation, 1965).Google Scholar
- 4.R. Boyer, G. Welsch, and E. Collings,
*Materials properties handbook: titanium alloys*(Metals Park, OH: ASM international, 1994).Google Scholar - 5.P.J. Bania,
*JOM*46, 16 (1994).CrossRefGoogle Scholar - 6.Y.B. Zhou,
*Titanium Alloy Casting*(Beijing: Aviation Industry Press, 2000).Google Scholar - 7.M. Wei, eds.,
*Titanium*(Beijing: China Metallurgical Industry Press, 2008).Google Scholar - 8.Q. Wang, C. Dong, and P.K. Liaw,
*Metall. Trans. A*46, 3440 (2015).CrossRefGoogle Scholar - 9.M. Abdel-Hady, H. Fuwa, K. Hinoshita, H. Kimura, Y. Shinzato, and M. Morinaga,
*Scr. Mater.*57, 1000 (2007).CrossRefGoogle Scholar - 10.Y. Li, Z. Ru, Y. Cui, and K. Luo,
*Int. J. Smart Nano Mater.*2, 272 (2011).Google Scholar - 11.J. Kim, H. Kim, T. Inamura, H. Hosoda, and S. Miyazaki,
*Mater. Sci. Eng. A*403, 334 (2005).CrossRefGoogle Scholar - 12.M. Abdel-Hady, K. Hinoshita, and M. Morinaga,
*Scr. Mater.*55, 477 (2006).CrossRefGoogle Scholar - 13.S. Ozan, J. Lin, Y. Li, R. Ipek, and C. Wen,
*Acta Biomater.*20, 176 (2015).CrossRefGoogle Scholar - 14.L. You and X. Song,
*Scr. Mater.*67, 57 (2012).CrossRefGoogle Scholar - 15.J. Chaves, O. Florêncio, P. Silva, P. Marques, and S. Schneider,
*J. Alloys Compd*616, 420 (2014).CrossRefGoogle Scholar - 16.M. Morinaga and H. Yukawa,
*Bull. Mater. Sci.*20, 805 (1997).CrossRefGoogle Scholar