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Acta Metall Sin  2020, Vol. 56 Issue (9): 1304-1312    DOI: 10.11900/0412.1961.2020.00015
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First-Principles Study on Free Energy and Elastic Properties of Disordered β-Ti1-xNbx Alloy: Comparison Between SQS and CPA
ZHANG Haijun1,2, QIU Shi3, SUN Zhimei3, HU Qingmiao1(), YANG Rui1
1 Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China
2 School of Materials Science and Engineering, University of Science and Technology of China, Shenyang 110016, China
3 School of Materials Science and Engineering, Beihang University, Beijing 100191, China
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Elastic modulus is one of the key properties for the application of biomedical β titanium alloy as human bone replacement because the elastic modulus of the alloy has to match that of the bone so as to avoid the stress shielding effect. Alloying of Nb is commonly used in biomedical β titanium alloys. In the present work, the lattice parameter, free energy and elastic modulus of β-Ti1-xNbx alloy were investigated by using first-principles method based on density functional theory. The random distribution of Nb atoms in the alloy were described by using both special quasirandom structure (SQS) and the coherent potential approximation (CPA) techniques, in combination with first principles plane-wave pseudopotential (VASP) and exact muffin-tin orbital (EMTO) methods, respectively. The results showed that the lattice constants from both VASP-SQS and EMTO-CPA calculations increase linearly with Nb content x, while the influence of the local lattice distortion is negligible. The calculations of the free energies demonstrated that EMTO-CPA predicts reasonably the phase decomposition of β-Ti1-xNbx at relatively low temperature whereas VASP-SQS does not, which might be ascribed to the fact that the free energy depends strongly on the detailed SQS structures. The elastic constants C11 and C12 calculated by using EMTO-CPA and VASP-SQS without atomic relaxation increase with Nb content whereas C44 decreases. EMTO-CPA overestimates the elastic stability of β-Ti1-xNbx. At low Nb content, the local lattice distortion is abnormally large due to the lattice instability of the β-Ti1-xNbx, making the free energy and elastic constant against x from VASP-SQS calculations with atomic relaxation deviate significantly from the general trend.

Key words:  titanium alloy      random solid solution      special quasirandom structure      coherent potential approximation      elastic constant      first-principles calculation     
Received:  10 January 2020     
ZTFLH:  TG146.2  
Fund: National Key Research and Development Program of China(2016YFB0701301);National Natural Science Foundation of China(91860107);National Basic Research Program of China(2014CB644001)
Corresponding Authors:  HU Qingmiao     E-mail:

Cite this article: 

ZHANG Haijun, QIU Shi, SUN Zhimei, HU Qingmiao, YANG Rui. First-Principles Study on Free Energy and Elastic Properties of Disordered β-Ti1-xNbx Alloy: Comparison Between SQS and CPA. Acta Metall Sin, 2020, 56(9): 1304-1312.

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Fig.1  Lattice constants of β-Ti1-xNbx alloys plotted as functions of composition (For comparison, experimental measurements[44~46] are plotted by black squares; SQS—special quasirandom structure, CPA—coherent potential approximation; x—atomic fraction of Nb)
Fig.2  Formation enthalpy of β-Ti1-xNbx alloys plotted as functions of composition
Fig.3  Gibbs free energy (ΔG) (a) and phase diagram (b) of β-Ti1-xNbx alloys as functions of compos-ition and temperature
Fig.4  Single crystal elastic constants C11 (a), C12 (b) and C44 (c) of β-Ti1-xNbx alloys as functions of composition (The blue scattered symbols are from experimental measurements[52~54])
Fig.5  Shear moduli (a~c) and Young's moduli (d~f) of Ti30Nb24 (a, d), Ti24Nb30 (b, e) and Ti18Nb36 (c, f) as a function of crystal orientation from unrelaxed SQS calculations (The Zener ratio A=2C44/(C11-C12) is a measure of the elastic anisotropy)
Fig. 6  Mechanical stability criteria of β-Ti1-xNbx alloys as functions of composition
Fig.7  Local lattice distortion (Δd) and distortion energy (ΔE) of β-Ti1-xNbxalloys as functions of composition (Results are obtained from relaxed and unrelax SQS calculations)
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