# Theory-Guided Materials Design of Multi-Phase Ti-Nb Alloys with Bone-Matching Elastic Properties

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## Abstract

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## 1. Introduction

#### Theory-Guided Materials Design

**Figure 1.**Schematic overview of the multi-scale materials-design strategy combining analysis of the thermodynamic phase-stability; and calculation of single-crystalline elasticity data obtained at atomic level by first-principles calculations with self-consistent homogenization techniques in order to bridge scale differences.

## 2. Methodology

#### 2.1. Ab Initio Calculations

**Figure 2.**(

**a**) The 2 × 2 × 2 16-atomic supercells used in the calculations of the cubic phase alloys; and (

**b**) the hexagonal close-packed ones, with one half or the atoms (numbered 1–8) located in odd atomic layers and the second half (numbered 1’–8’) in even atomic layers (for sake of clarity depicted by larger red spheres) in the [001]${}_{\mathrm{bcc}}$ and [0001]${}_{\mathrm{hcp}}$ directions. The β-Ti-37.5at.% Nb alloys were modeled by two different ordered compounds with the Nb atoms located in positions marked in the figure by either numbers {124678} (below referred to as β-Ti-Nb${}^{I}$) or the set {1${}^{\prime}$24577${}^{\prime}$} (further referred to as β-Ti-Nb${}^{II}$).

#### 2.2. Analytic Homogenization Scheme

#### 2.3. Single-Phase Aggregate

#### 2.4. Multi-Phase Composite

#### 2.5. Homogenized Young’s Modulus and Poisson’s Ratio

#### 2.6. Experimental Methods

## 3. Results and Discussion

**Figure 3.**(

**a**) Compositional dependence of the ab initio free energies of formation ${E}_{f}$ at T = 881 ${}^{\circ}$C for binary Ti-Nb alloys. The diamonds/circles mark ${E}_{f}$ for the hcp/bcc phase. The dotted line in part (a) shows the Gibbs construction that determined composition of both structural components of the composite (vertical arrows); (

**b**) The compositional dependence of the thus predicted β-phase volume faction is shown in by full circles together with the experimental volumetric fractions (empty circles).

**Table 1.**Theoretically predicted structural parameters and elastic constants of pure elements (Ti, Nb) and the cubic Ti-Nb compounds (β-Ti-37.5%Nb${}^{I}$, β-Ti-37.5%Nb${}^{II}$, for details see Figure 2) compared with available experimental data.

Material | a | $c/a$ | ${C}_{11}$ | ${C}_{12}$ | ${C}_{13}$ | ${C}_{33}$ | ${C}_{44}$ |
---|---|---|---|---|---|---|---|

Ti theory | 2.921 | 1.585 | 200 | 72 | 90 | 191 | 40 |

Ti theory[49] | 2.946 | 1.584 | 172 | 87 | 73 | 191 | 41 |

Ti experiment[52] | 2.951 | 1.587 | 162 | 92 | 69 | 181 | 47 |

Nb theory | 3.335 | - | 227 | 129 | - | - | 22 |

Nb theory[49] | 3.325 | - | 247 | 134 | - | - | 15.6 |

Nb experiment[53] | 3.301 | - | 246 | 133 | - | - | 28 |

β-Ti-37.5%Nb${}^{I}$ theory | 3.261 | - | 156 | 121 | - | - | 10 |

β-Ti-37.5%Nb${}^{II}$ theory | 3.264 | - | 168 | 118 | - | - | 29 |

**Table 2.**Theoretically predicted polycrystalline integral elastic parameters and phase-composition of Ti-Nb composites with selected Nb concentrations (of actually cast samples) together with the experimental data.

Material | ${v}_{\beta}^{\mathrm{theory}}$ | ${v}_{\beta}^{\mathrm{exp}.}$ | ${B}_{\alpha /\beta}^{*}$ | ${\mu}_{\alpha /\beta}^{*}$ | ${Y}_{\alpha /\beta}^{*}$ | ${Y}_{\alpha /\beta}^{\mathrm{exp}}$ |
---|---|---|---|---|---|---|

α-Ti | 0 | 0 | 122 | 115 | 132 | - |

Ti-10at.% Nb | 0.17 | 0.06 | 124 | 0.43 | 115 | 91 |

Ti-20at.% Nb | 0.49 | 0.60 | 127 | 0.32 | 89 | 75 |

Ti-25at.% Nb | 0.60 | 0.81 | 129 | 0.28 | 78 | 74 |

Ti-30at.% Nb | 0.75 | 0.90 | 130 | 0.24 | 69 | 72 |

β-Ti-37.5at.%Nb | 1 | 1 | 133 | 19 | 54 | - |

**Figure 4.**The theoretically predicted dependence of the (

**a**) homogenized shear modulus ${\mu}_{\alpha /\beta}^{*}$; (

**b**) Young’s modulus ${Y}_{\alpha /\beta}^{*}$; and (

**c**) Poisson’s ratio ${\nu}^{{*}_{\alpha /\beta}}$ of the Ti-Nb composite as a function of the volumetric β-phase content numerically determined with 0.1 compositional step. The values are compared with the values obtained from a linear interpolation (dashed lines) between the α and β components.

**Figure 5.**Predicted and experimentally obtained Young’s moduli. The theoretical single-crystal Young’s moduli for the soft [001] crystal direction of the cubic lattice cell are shown by empty squares. The homogenized Young moduli of hypothetical β-phase polycrystals with varying Nb content (not from the Gibb’s construction) are shown by full diamonds connected by a line. Full circles stand for the predicted ${Y}_{\alpha /\beta}^{*}$ and experimental data are visualized by empty circles.

## 4. Summary and Conclusions

## Acknowledgment

## 5. Appendix

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**MDPI and ACS Style**

Friák, M.; Counts, W.A.; Ma, D.; Sander, B.; Holec, D.; Raabe, D.; Neugebauer, J.
Theory-Guided Materials Design of Multi-Phase Ti-Nb Alloys with Bone-Matching Elastic Properties. *Materials* **2012**, *5*, 1853-1872.
https://doi.org/10.3390/ma5101853

**AMA Style**

Friák M, Counts WA, Ma D, Sander B, Holec D, Raabe D, Neugebauer J.
Theory-Guided Materials Design of Multi-Phase Ti-Nb Alloys with Bone-Matching Elastic Properties. *Materials*. 2012; 5(10):1853-1872.
https://doi.org/10.3390/ma5101853

**Chicago/Turabian Style**

Friák, Martin, William Art Counts, Duancheng Ma, Benedikt Sander, David Holec, Dierk Raabe, and Jörg Neugebauer.
2012. "Theory-Guided Materials Design of Multi-Phase Ti-Nb Alloys with Bone-Matching Elastic Properties" *Materials* 5, no. 10: 1853-1872.
https://doi.org/10.3390/ma5101853