# Diffusivities and Atomic Mobilities in bcc Ti-Mo-Zr Alloys

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Extraction of Inter-Diffusion Coefficients and Impurity Diffusion Coefficients

_{i}in a ternary system can be expressed as:

#### 2.2. Atomic Mobility and Diffusivity

_{i}by means of the Einstein relation:

_{m}is the molar volume of a phase that was taken to be constant in this work.

## 3. Experiment

## 4. Results and Discussions

#### 4.1. Inter-Diffusion and Impurity Diffusion Coefficients At 1373 K and 1473 K

#### 4.2. Atomic Mobilities in bcc Ti-Zr-Mo System

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Microstructure (BSE) of the diffusion couple M1 annealed at 1373 K for 72 h and the robust error function expansion (ERFEX) representation of the composition-distance profiles of the different couples annealed in 1373 K for 72 h: (

**b**) for M2; (

**c**) for M6; (

**d**) for M8.

**Figure 3.**Diffusion paths of diffusion couples determined using electron probe micro-analysis (EPMA) after annealing process.

**Figure 4.**Three-dimensional (3D) plot of the ternary inter-diffusion coefficients (

**a**) ${\tilde{D}}_{MoMo}^{Ti}$, (

**b**) ${\tilde{D}}_{ZrZr}^{Ti}$, (

**c**) ${\tilde{D}}_{MoZr}^{Ti}$ and (

**d**) ${\tilde{D}}_{ZrMo}^{Ti}$ in the bcc Ti-Mo-Zr alloys at 1373 K, together with the impurity diffusion coefficients ${D}_{Mo\left(Ti-Zr\right)}^{*}$ and ${D}_{Zr\left(Ti-Mo\right)}^{*}$, and binary diffusion coefficients obtained from the literature [33].

**Figure 5.**The variation of ternary inter-diffusion coefficients with the compositions: (

**a**) ${\tilde{D}}_{MoMo}^{Ti}$ with Mo, (

**b**) ${\tilde{D}}_{MoMo}^{Ti}$ with Zr, (

**c**) ${\tilde{D}}_{ZrZr}^{Ti}$ with Mo and (

**d**) ${\tilde{D}}_{ZrZr}^{Ti}$ with Zr at 1373 K.

**Figure 6.**The 3D plot of the ternary inter-diffusion coefficients (

**a**) ${\tilde{D}}_{MoMo}^{Ti}$, (

**b**) ${\tilde{D}}_{ZrZr}^{Ti}$, (

**c**) ${\tilde{D}}_{MoZr}^{Ti}$, and (

**d**) ${\tilde{D}}_{ZrMo}^{Ti}$ in the bcc Ti-Mo-Zr alloys and the variation of ternary inter-diffusion coefficients with the compositions: (

**e**) ${\tilde{D}}_{MoMo}^{Ti}$ with Mo, (

**f**) ${\tilde{D}}_{MoMo}^{Ti}$ with Zr, (

**g**) ${\tilde{D}}_{ZrZr}^{Ti}$ with Mo, and (

**h**) ${\tilde{D}}_{ZrZr}^{Ti}$ with Zr at 1473 K.

**Figure 7.**Calculated inter-diffusion coefficients of the ternary Ti-Mo-Zr system compared with the experimental measurements (in brackets) in this work: (

**a**) ${\tilde{D}}_{MoMo}^{Ti}$, (

**b**) ${\tilde{D}}_{ZrZr}^{Ti}$, (

**c**) ${\tilde{D}}_{MoZr}^{Ti}$ and (

**d**) ${\tilde{D}}_{ZrMo}^{Ti}$ at 1373 K and (

**e**) ${\tilde{D}}_{MoMo}^{Ti}$, (

**f**) ${\tilde{D}}_{ZrZr}^{Ti}$, (

**g**) ${\tilde{D}}_{MoZr}^{Ti}$ and (

**h**) ${\tilde{D}}_{ZrMo}^{Ti}$ at 1473 K.

**Figure 8.**Calculated impurity diffusion coefficients (

**a**) ${D}_{Zr\left(Ti-Mo\right)}^{*}$ and (

**b**) ${D}_{Mo\left(Ti-Zr\right)}^{*}$ as compared with the experimental data.

**Figure 9.**Predicted composition-distance profiles and inter-diffusion fluxes of M3 and N8 as compared with experimental data.

**Figure 10.**Simulated diffusion paths for diffusion couples compared with the experimental measurements: (

**a**) 1373 K and (

**b**) 1473 K.

Temperature (K) | Diffusion Couples | Composition (mole %) |
---|---|---|

1373 | M1 | Ti-2.95Mo/Ti-4.95Zr |

M2 | Ti-4.90Mo/Ti-7.81Zr | |

M3 | Ti-7.83Mo/Ti-11.17Zr | |

M4 | Ti-9.74Mo/Ti-13.64Zr | |

M5 | Ti-11.61Mo/Ti-16.57Zr | |

M6 | Pure Ti/Ti-17.18Mo-5.48Zr | |

M7 | Pure Ti/Ti-11.63Mo-11.51Zr | |

M8 | Pure Ti/Ti-5.91Mo-16.90Zr | |

1473 | N1 | Ti-2.85Mo/Ti-4.85Zr |

N2 | Ti-4.92Mo/Ti-7.39Zr | |

N3 | Ti-7.90Mo/Ti-11.02Zr | |

N4 | Ti-9.90Mo/Ti-13.27Zr | |

N5 | Ti-13.11Mo/Ti-15.15Zr | |

N6 | Pure Ti/Ti-17.14Mo-5.81Zr | |

N7 | Pure Ti/Ti-11.67Mo-11.61Zr | |

N8 | Pure Ti/Ti-5.91Mo-17.32Zr |

Temp. (K) | Diffusion Couple | Intersection Composition (mole %) | Inter-Diffusion Coefficients (m^{2}·s^{−1}) | ||||
---|---|---|---|---|---|---|---|

Mo | Zr | ${\tilde{D}}_{MoMo}^{Ti}$ | ${\tilde{D}}_{MoZr}^{Ti}$ | ${\tilde{D}}_{ZrMo}^{Ti}$ | ${\tilde{D}}_{ZrZr}^{Ti}$ | ||

×10^{−14} | ×10^{−14} | ×10^{−14} | ×10^{−13} | ||||

1373 | M1-M6 | 2.71 | 1.18 | 9.96 ± 1.12 | −7.26 ± 5.24 | 0.96 ± 0.38 | 6.00 ± 0.42 |

M1-M7 | 1.46 | 2.54 | 13.84 ± 0.56 | −2.89 ± 1.72 | 3.57 ± 2.24 | 6.71 ± 0.37 | |

M1-M8 | 0.45 | 3.45 | 17.08 ± 1.81 | 0.31 ± 0.25 | 12.63 ± 5.54 | 8.29 ± 0.31 | |

M2-M6 | 4.73 | 1.38 | 6.67 ± 0.60 | −2.95 ± 1.93 | 1.55 ± 0.20 | 4.52 ± 0.11 | |

M2-M7 | 3.32 | 3.33 | 9.79 ± 0.25 | −2.93 ± 0.27 | 2.52 ± 1.47 | 5.24 ± 0.10 | |

M2-M8 | 1.38 | 4.93 | 15.41 ± 0.56 | −0.49 ± 0.33 | 11.30 ± 1.09 | 7.14 ± 0.10 | |

M3-M6 | 7.72 | 1.56 | 3.56 ± 0.41 | −1.18 ± 0.41 | 1.07 ± 0.13 | 2.89 ± 0.10 | |

M3-M7 | 5.99 | 4.24 | 6.45 ± 0.11 | −2.50 ± 0.36 | 1.25 ± 0.60 | 3.44 ± 0.15 | |

M3-M8 | 2.90 | 6.74 | 14.48 ± 0.28 | −1.25 ± 0.48 | 7.11 ± 0.99 | 5.49 ± 0.15 | |

M4-M6 | 9.49 | 1.68 | 2.93 ± 0.17 | −2.94 ± 1.04 | 0.50 ± 0.15 | 2.36 ± 0.06 | |

M4-M7 | 7.58 | 4.76 | 5.92 ± 0.28 | −3.06 ± 0.58 | 0.34 ± 0.71 | 2.75 ± 0.14 | |

M4-M8 | 3.88 | 8.07 | 14.69 ± 0.93 | −2.30 ± 0.84 | 5.98 ± 1.07 | 4.52 ± 0.04 | |

M5-M6 | 11.45 | 1.83 | 2.31 ± 0.29 | −1.40 ± 1.31 | 0.16 ± 0.46 | 1.62 ± 0.06 | |

M5-M7 | 9.24 | 5.50 | 4.84 ± 0.60 | −1.86 ± 0.68 | 0.72 ± 0.78 | 1.97 ± 0.02 | |

M5-M8 | 4.80 | 9.73 | 14.49 ± 1.04 | −2.08 ± 0.50 | 4.76 ± 3.54 | 3.82 ± 0.16 | |

×10^{−13} | ×10^{−14} | ×10^{−13} | ×10^{−12} | ||||

1473 | N1-N6 | 1.524 | 3.605 | 3.37 ± 0.18 | −4.00 ± 2.19 | 0.45 ± 0.22 | 1.59 ± 0.06 |

N1-N7 | 2.597 | 1.898 | 4.43 ± 0.01 | −2.46 ± 1.08 | 0.68 ± 0.40 | 1.69 ± 0.02 | |

N1-N8 | 3.365 | 0.81 | 5.42 ± 0.13 | −1.39 ± 1.30 | 1.97 ± 1.16 | 1.83 ± 0.05 | |

N2-N6 | 1.707 | 4.399 | 2.39 ± 0.19 | −13.24 ± 4.86 | 0.55 ± 0.06 | 1.23 ± 0.01 | |

N2-N7 | 3.589 | 3.386 | 3.95 ± 0.12 | −3.33 ± 2.17 | 1.20 ± 0.23 | 1.43 ± 0.03 | |

N2-N8 | 5.701 | 1.951 | 5.19 ± 0.12 | −2.16 ± 1.40 | 2.38 ± 0.21 | 1.62 ± 0.02 | |

N3-N6 | 2.43 | 8.144 | 1.68 ± 0.04 | −2.24 ± 2.12 | 0.52 ± 0.09 | 0.82 ± 0.02 | |

N3-N7 | 4.724 | 5.243 | 2.80 ± 0.05 | −3.75 ± 0.51 | 1.16 ± 0.15 | 1.06 ± 0.02 | |

N3-N8 | 6.881 | 2.508 | 4.91 ± 0.08 | −2.11 ± 0.76 | 2.04 ± 0.51 | 1.36 ± 0.03 | |

N4-N6 | 2.552 | 8.725 | 1.50 ± 0.09 | −13.40 ± 4.90 | 0.36 ± 0.10 | 0.71 ± 0.01 | |

N4-N7 | 5.489 | 6.461 | 2.72 ± 0.17 | −11.73 ± 2.86 | 0.84 ± 0.06 | 0.93 ± 0.01 | |

N4-N8 | 8.913 | 3.466 | 6.04 ± 0.46 | −8.67 ± 2.82 | 1.18 ± 0.19 | 1.27 ± 0.02 | |

N5-N6 | 3.438 | 12.795 | 0.85 ± 0.03 | −3.56 ± 1.06 | 0.36 ± 0.06 | 0.40 ± 0.01 | |

N5-N7 | 6.723 | 8.091 | 2.03 ± 0.07 | −1.85 ± 0.97 | 1.26 ± 0.26 | 0.68 ± 0.01 | |

N5-N8 | 9.993 | 3.932 | 5.20 ± 0.44 | −3.24 ± 2.66 | 1.88 ± 0.66 | 1.12 ± 0.03 |

**Table 3.**Impurity diffusion coefficients of Zr in Ti-Mo and Mo in Ti-Zr alloys at 1373 K and 1473 K.

Temperature/K | Composition | Impurity Diffusion Coefficients (×10^{−13} m^{2}·s^{−1}) | Composition | Impurity Diffusion Coefficients (×10^{−13} m^{2}·s^{−1}) |
---|---|---|---|---|

1373 | ${D}_{Zr\left(Ti-2.95Mo\right)}^{*}$ | 5.85 ± 0.58 | ${D}_{Mo\left(Ti-4.95Zr\right)}^{*}$ | 1.54 ± 0.09 |

${D}_{Zr\left(Ti-4.90Mo\right)}^{*}$ | 4.51 ± 0.88 | ${D}_{Mo\left(Ti-7.81Zr\right)}^{*}$ | 1.96 ± 0.62 | |

${D}_{Zr\left(Ti-7.83Mo\right)}^{*}$ | 2.55 ± 0.05 | ${D}_{Mo\left(Ti-11.17Zr\right)}^{*}$ | 2.86 ± 0.71 | |

${D}_{Zr\left(Ti-9.74Mo\right)}^{*}$ | 2.13 ± 0.14 | ${D}_{Mo\left(Ti-13.64Zr\right)}^{*}$ | 4.37 ± 1.68 | |

${D}_{Zr\left(Ti-11.61Mo\right)}^{*}$ | 1.39 ± 0.15 | ${D}_{Mo\left(Ti-16.57Zr\right)}^{*}$ | 5.07 ± 1.45 | |

1473 | ${D}_{Zr\left(Ti-2.89Mo\right)}^{*}$ | 15.45 ± 1.22 | ${D}_{Mo\left(Ti-4.85Zr\right)}^{*}$ | 5.04 ± 1.24 |

${D}_{Zr\left(Ti-4.92Mo\right)}^{*}$ | 11.31 ± 0.74 | ${D}_{Mo\left(Ti-7.39Zr\right)}^{*}$ | 6.42 ± 2.38 | |

${D}_{Zr\left(Ti-7.90Mo\right)}^{*}$ | 7.24 ± 0.07 | ${D}_{Mo\left(Ti-11.02Zr\right)}^{*}$ | 9.78 ± 1.82 | |

${D}_{Zr\left(Ti-9.90Mo\right)}^{*}$ | 6.08 ± 0.26 | ${D}_{Mo\left(Ti-13.27Zr\right)}^{*}$ | 12.61 ± 3.46 | |

${D}_{Zr\left(Ti-13.11Mo\right)}^{*}$ | 3.04 ± 0.17 | ${D}_{Mo\left(Ti-15.15Zr\right)}^{*}$ | 14.66 ± 2.39 |

Mobility | Parameter, J/mole | Reference |
---|---|---|

Mobility of Mo | ||

${Q}_{Mo}^{Mo}$ | −479740.87 − 63.98·T | [33] |

${Q}_{Mo}^{Ti}$ | −196255.40 − 105.21·T | [33] |

${Q}_{Mo}^{Zr}$ | −154895.63 − 140.91·T (T ≤ 1450 K) −214913.77 − 114.17·T (T ≥ 1450 K) | [34] |

${}^{0}{Q}_{Mo}^{Mo,Ti}$ | −24153.22 − 45.32·T | [33] |

${}^{1}{Q}_{Mo}^{Mo,Ti}$ | −61804.04 | [33] |

${Q}_{Mo}^{Mo,Zr}$ | 150325.48 + 10.03·T | [34] |

${Q}_{Mo}^{Ti,Zr}$ | −268357.13 + 283.18·T | This work |

${}^{0}{Q}_{Mo}^{Mo,Ti,Zr}$ | 53485.75 | This work |

${}^{1}{Q}_{Mo}^{Mo,Ti,Zr}$ | 2707991.87 | This work |

${}^{2}{Q}_{Mo}^{Mo,Ti,Zr}$ | −719018.41 | This work |

Mobility of Ti | ||

${Q}_{Ti}^{Mo}$ | −435701.23 − 72.67·T | [33] |

${Q}_{Ti}^{Ti}$ | −151989.95 − 127.37·T | [33] |

${Q}_{Ti}^{Zr}$ | −140356.54 − 138.12·T | [33] |

${}^{0}{Q}_{Ti}^{Mo,Ti}$ | −91728.48 + 64.56·T | [33] |

${}^{1}{Q}_{Ti}^{Mo,Ti}$ | −96300.05 | [33] |

${}^{0}{Q}_{Ti}^{Ti,Zr}$ | −15826.04 + 62.55·T | [33] |

${}^{1}{Q}_{Ti}^{Ti,Zr}$ | 8243.54 | [33] |

${}^{0}{Q}_{Ti}^{Mo,Ti,Zr}$ | −1304851.67 | This work |

${}^{1}{Q}_{Ti}^{Mo,Ti,Zr}$ | −6019149.28 | This work |

${}^{2}{Q}_{Ti}^{Mo,Ti,Zr}$ | −1128202.89 | This work |

Mobility of Zr | ||

${Q}_{Zr}^{Mo}$ | −464587.32 − 64.72·T | [34] |

${Q}_{Zr}^{Ti}$ | −131670.56 − 133.36·T | [33] |

${Q}_{Zr}^{Zr}$ | −104624.81 − 163.15·T (T≤1573 K) −161543.53 − 126.10·T (T≥1573 K) | [33] |

${Q}_{Zr}^{Mo,Ti}$ | −81189.65 + 88.51·T | This work |

${Q}_{Zr}^{Mo,Zr}$ | 210325.67 + 15.19·T | [34] |

${}^{0}{Q}_{Zr}^{Ti,Zr}$ | −12581.03 + 33.38·T | [33] |

${}^{1}{Q}_{Zr}^{Ti,Zr}$ | 2898.60 | [33] |

${}^{0}{Q}_{Zr}^{Mo,Ti,Zr}$ | 2864755.37 | This work |

${}^{1}{Q}_{Zr}^{Mo,Ti,Zr}$ | −340701.76 | This work |

${}^{2}{Q}_{Zr}^{Mo,Ti,Zr}$ | −609499.74 | This work |

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**MDPI and ACS Style**

Bai, W.; Xu, G.; Tan, M.; Yang, Z.; Zeng, L.; Wu, D.; Liu, L.; Zhang, L.
Diffusivities and Atomic Mobilities in bcc Ti-Mo-Zr Alloys. *Materials* **2018**, *11*, 1909.
https://doi.org/10.3390/ma11101909

**AMA Style**

Bai W, Xu G, Tan M, Yang Z, Zeng L, Wu D, Liu L, Zhang L.
Diffusivities and Atomic Mobilities in bcc Ti-Mo-Zr Alloys. *Materials*. 2018; 11(10):1909.
https://doi.org/10.3390/ma11101909

**Chicago/Turabian Style**

Bai, Weimin, Guanglong Xu, Mingyue Tan, Zhijie Yang, Lijun Zeng, Di Wu, Libin Liu, and Ligang Zhang.
2018. "Diffusivities and Atomic Mobilities in bcc Ti-Mo-Zr Alloys" *Materials* 11, no. 10: 1909.
https://doi.org/10.3390/ma11101909