K, Rajamallu and Dey, Suhash Ranjan
(2018)
Design and Development of β Ti Alloys Through First Principles Calculations and Powder Metallurgy Route for Biomedical Applications.
PhD thesis, Indian Institute of Technology Hyderabad.
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Abstract
With the increase in old age population due to better healthcare facilities, the demand for biomaterials has been continuously on the rise. United Nations reports that the age group 80 or above is growing faster than any younger generation irrespective of their geographic location or developmental stage. The percentage of the world’s aged population above 60 years is increased from 8% (1950) to 12% (2013). It is going to increase even at faster rates in the coming decades to reach 21% by 2050. The individuals in the age of 50 and above are more prone towards bone joint degeneration and inflammatory problems. On the other hand, people having physical injuries because of sports or excessive/incorrect exercise habits or accidents need biomaterials for a longer period. This requires a must increase in durability of biomaterials to reduce the revision of surgery. Hence, the lifespan of the implants must be served for a prolonged period so as to avoid revision surgeries.
The metallic biomaterials show relatively high Young's modulus when it is compared to that of human bone (~30 GPa). Due to their mismatch in stiffness, maximum stress is taken by the implant and this causes inhomogeneous stress distribution along the implant. Inhomogeneous stress distribution caused by the difference in Young's modulus of the implant and bone give rise to ‘stress shielding effect’ which causes loosening of the bone during longterm usage. Mechanical and biological compatibilities are essential to design and fabricate suitable biomaterial. In particular, the β (bcc) titanium alloys containing alloying elements of nontoxic and allergyfree have been found as potential candidates owing to their low elastic modulus near to human bone which in turn improves the mechanical biocompatibility.
In recent years, computational studies have become an important tool to predict and understand the properties and behavior of materials at different conditions. The computational materials science is significantly improved in terms of accuracy, reliability, time and efficiency with the combination of effective algorithms and highperformance computing facilities. The development of novel advanced materials can be accelerated manifolds if the computational methods and experimental studies are combined properly. Though the study of phase stability of equilibrium α (hcp) and β (bcc) phase of binary TiNb is already investigated, random solid solution effects and configurational entropy calculations are not included in those studies. The metastable orthorhombic phase α" is also not studied in the binary TiNb till 50at.% of Nb addition using random solid solution and configurational terms. Moreover, there is a lack of systematic study on effects of ternary alloying (Zr and Sn) additions in TiNb system. Finally, asdesigned alloys are fabricated using powder metallurgy route and characterized in order to validate our calculated results and also compared with the previously reported results.
The abinitio calculations are carried out within density functional framework as implemented in the VASP code. The electronion interaction is approximated using the projected augmented wave method. The PerdewBurkeErnzerhoff form of the generalized gradient approximation is utilized for exchangecorrelation energy. A standard plane wave basis set with a kinetic energy cutoff of 400eV is used to expand the electron wave functions. The electron wave functions are expanded in a standard plane wave basis set with a kinetic energy cutoff of 400eV. The Brillouin zone of hcp α, bcc β, and orthorhombic α" structure is sampled using 10x10x9, 9x9x9, and 5x3x6 MonkhorstPack kpoint mesh, respectively. The selfconsistency in calculations is obtained with the energy converged up to 10−6eV/cell and the structural relaxations are calculated until the largest force on each ion becomes less than 10−3eV/Aº. All the calculations for hcp (α) and bcc (β) are carried out using 2x2x2 supercells whereas for orthorhombic (α") are performed using 2x2x1 having sixteen atoms. The hcp (α) unit cell of pure Ti, bcc (β) unit cell of pure Nb, and the orthorhombic (α") unit cell of pure Ti have P63/mmc (No: 194), Im3m (No: 229), and Cmcm (No: 63) space group symmetry, respectively. The elastic constants are obtained by the stressstrain approach as implemented in the VASP code wherein the elastic tensor is obtained by performing finite distortions of the unit cell. In the elastic moduli tensor, contributions for distortions with rigid ions and the ionic relaxations are included. The strain value up to 1% is chosen for the deformation of the lattice.
Thermodynamic stability is estimated from the computed formation energies of different phases in TixNb compositions, where x is varied up to 50at.% (x = 6.26, 12.5, 18.75, 25, 31.25, 37.5, and 50at.%). In order to minimize the error in the calculations of phase stability as well as elastic properties as a result of random solid solution, maximum possible configurations are considered for the calculations. In the present work, 49, 58, and 63 configurations are used for α, β, and α" phases, respectively. The configurations differ from each other in terms of NbNb distances i.e. a number of first NbNb neighbor (FN), second NbNb neighbor (SN), third NbNb neighbor (TN) etc. Our results suggest that first NbNb neighbors energetically favor β phase for all compositions. The stability of configurations of the given β phase of the binary TiNb composition increases with increasing number of first NbNb neighbors. The bcc β phase is stabilized over the hcp α phase beyond 25at.% of Nb as a result of lowering of formation energy of the bcc β phase than the hcp α phase. It indicates that critical Nb content in TiNb alloy system is 25at.% for it to remain stable in the bcc β phase. These theoretical results are consistent with reported experimental results. However, the orthorhombic phase is the most stable phase over the range of 12.5 to 37.5at.% of Nb in TiNb alloys. At higher temperatures, α" phase becomes unstable and the crossover from α" to β phase takes place at a low concentration of Nb i.e. 25at.% of Nb. The calculated Young’s modulus of bcc (β) phase increases with increasing Nb content and above 25at.% of Nb, it varies within the range of 6080 GPa. On the other hand, there is no simple trend seen in the calculated Young’s modulus for orthorhombic α" in binary TiNb alloys. In intermediate range (12.537.5at.%), the elastic modulus of stable orthorhombic α" phase is higher than that for corresponding bcc β phase composition. This indicates α" is harder phase than the β phase in binary TiNb alloys.
The effect of ternary alloying additions (Zr and Sn) on β phase stability is investigated in Ti18.75at.%Nb and Ti25at.%Nb systems. The formation energy of bcc β phase is low for all ternary alloying elements additions which indicate that Zr and Sn act as β stabilizers in the TiNb alloys. When β phase stability with respect to α phase is considered, Sn behaves as a strong β stabilizer than to Zr, which in turn is stronger β stabilizing element than Nb for Ti18.75at.%Nb. In Ti25at.%Nb system, the effect of β phase stability by the addition of ternary elements is also found in the order as Sn>Nb>Zr. The lattice constant of the binary TixNb system in β phase increases at the rate of 0.06x103 nm per 1at.%Nb addition. On the other hand, the lattice parameter of ternary alloys increases by 0.3x103nm and 0.4x103nm with 1at.% addition of Zr and Sn to Ti25at.%Nb system, respectively. The density of bcc β Ti system is increased with the addition of elements Nb, Zr, and Sn. However, the calculated densities of Zr added ternary alloys exhibited relatively lower values than that of its corresponding ternary β TiNbSn alloys. Zr addition is found to be advantageous over Sn addition as Ti18.75at.%Nb6.25at.%Zr composition and Ti25at.%NbxZr system retain the stable β phase in addition to having low Young’s modulus (lowest value of 54 GPa for Ti25at.%Nb6.25at.%Zr) as compared with Ti18.75at.%Nb6.25at.%Sn composition and Ti25at.%NbxSn system. This elastic modulus is further decreased along the crystallographic [100] direction (≈ 45 GPa) which is close to the human bone.
The binary TixNb (x = 18.75, 25, and 31.25at.%) alloys are prepared using simple mixing of pure elemental powders in appropriate proportions. Two techniques for sintering are adopted (i) Spark plasma sintering (pressureassisted sintering) and (ii) Conventional powder metallurgy (P/M) (pressureless sintering). Synthesis parameters such as sintering temperature and holding time etc. are optimized in both techniques in order to get homogenous microstructure. In spark plasma sintering (SPS), complete homogeneous β phase is achieved in Ti25at.%Nb using 1300ºC sintering temperature with 60min holding time under 50MPa pressure. On the other hand, complete β phase is obtained in Ti25at.%Nb through conventional P/M route using sintering temperature of 1400ºC for 120min holding time which is considered from the dilatometry studies. These optimized synthesis parameters are used to fabricate all other binary TiNb, ternary Ti18.75at.%Nb6.25Zr/Sn, and Ti25at.%Nb6.25Zr/Sn alloys. The binary Ti18.75at.%Nb consists of α+β phase in both techniques. The Ti25at.%Nb and Ti31.25at.%Nb alloys consist of almost complete β with minor orthorhombic phase. All the ternary alloying systems such as Ti18.75at.%Nb6.25at.%Zr/Sn and Ti25at.%Nb6.25at.%Zr/Sn show complete β phase with minute orthorhombic phase. These results are consistent with our predicted results. The volume fraction of orthorhombic α" phase for SPS sample is always noticed higher than its corresponding conventional P/M samples. This fact is attributed to the applied pressure, high cooling, and heating rates during the SPS synthesis. The measured elastic properties of binary TixNb are fallen within the range of 8090 GPa. The ternary alloys also exhibit elastic properties ranging from 80100 GPa. However, the ternary TiNbZr alloys always show lower Young’s modulus than the corresponding TiNbSn alloys.
In summary, theoretical results are very useful to understand phase stability of Ti alloys and their elastic properties in order to obtain the suitable orthopedic implant applications. It is found that the binary Ti25at.%Nb (80 GPa) and ternary Ti25at.%Nb6.25at.%Zr (54 GPa) alloys provide stable β phase with low elastic modulus and thus are suitable for biomedical applications. The elastic modulus is further decreased to 45 GPa in the [100] crystallographic direction. As designed binary TiNb, ternary TiNbZr and TiNbSn alloys are experimentally fabricated using powder metallurgy route. All these alloys show complete β phase which is in good agreement with the calculated results. However, there are discrepancies between calculated and experimental Young’s modulus of Ti alloys. These are attributed to the differences in volume fraction of various phases, compositional variations (contamination during synthesis) and temperature effects.
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