Structural evolution of TiAl during rapid solidification processing revealed by molecular dynamics simulations

Abstract

In this paper, the processes of rapid solidification in TiAl was investigated by molecular dynamics simulations. The structure transformation which occurs during solidification is described by internal energy, radial distribution functions, Honeycutt-Anderson pair analysis technique, mean square displacement and simulation of powder X-ray diffraction patterns. The effects of different cooling rates, 50–0.005 K ps⁻¹, on the amorphous structure formation and crystallization of liquid TiAl are simulated. The results show that an amorphous phase can be obtained when the cooling rate is not less than 0.02 K ps⁻¹, and γ-TiAl + α₂-Ti₃Al mixed phases can be formed when the cooling rate is 0.01 K ps⁻¹.

---

Introduction

Titanium aluminides based on TiAl are considered to be potential candidate structural materials for various applications in the gas turbine and the automotive industry, because they exhibit low density, high melting point, good specific strength, high oxidation resistance and excellent creep properties at elevated temperatures.^1–3 Generally speaking, the properties of the alloys depend on the microstructure of the alloys, and the microstructure is intimately related to the cooling rate during the solidification process.^4–10 Thus, it is urgent to understand the structural transformation of TiAl alloy during the processes of rapid solidification. However, the experimental study corresponding to the solidification of TiAl alloy is difficult to be carried out as it is hard to perform the rapid cooling rate to prepare the TiAl alloys and detect the microstructure transformation during the liquid quenching processes. On the other hand, molecular dynamics simulation provides us with a simple and economical method.^11–19 In 2004, Pei and his co-authors simulated the solidification processes of Ti₃Al alloy with different cooling rates.²⁰ In the following year, they further investigated the crystallization process of Ti₃Al liquid alloy during isothermal annealing.²¹ In 2010, Hou et al. studied the formation and structure order in the rapid quenching AlMg alloy.²² Until recently, rare reports focus on the structural evolutions for amorphous structure formation and crystal nucleation during the solidification processes in liquid TiAl alloy under different cooling rates. Therefore, in this paper, molecular dynamics (MD) simulation apply to investigate the structural evolution of TiAl alloy at atomic level.

Simulation

The simulation is realized by the open code LAMMPS with NPT ensemble at a fixed pressure equalled to an atmosphere.²³ The leapfrog scheme is used for integrating the Newton's equations of motion, and the Maxwell–Boltzmann distribution is used for the initial velocities of the atoms. The time step Δt is 2 fs (1 fs = 10⁻¹⁵ s). The interaction between atoms is described by Zope and Mishin's embedded-atom-method (EAM) potential²⁴ which can accurately reproduce the behaviors for TiAl systems in previous works.^25,26 Visual molecular dynamics (VMD) is used to visualize the atomic structures.²⁷ Original configuration of the simulation was a rectangular block containing 8064 atoms (4032 Ti atoms and 4032 Al atoms denoted by blue ball red one, respectively) with periodic boundary conditions in all directions, as shown in Fig. 1. To get an equilibrium liquid state, the atomic system is run at 2300 K, which is much higher than the melting point of TiAl (about 1800 K), for 3 × 10⁷ time steps. Then, the equilibrium liquid state at 2300 K is set as the starting state during cooling process. The cooling rates are in the range of 50 K ps⁻¹ ∼0.005 K ps⁻¹ (1 ps = 10⁻¹² s). The internal energy, radial distribution function (RDF), mean square displacement (MSD) and the Honeycutt-Anderson (H-A) pair technique²⁸ are employed to monitor the microstructural evolutions of the alloy during cooling processes. In detail of H-A pair analysis technique, if two atoms are within a given separating distance, they are considered to form a bond. The given separating distances are set to be equal to the position of the first minimum in the correlation functions of corresponding radical distribution functions (RDF). And the phase formed at the end of solidification is analyzed by the simulation of powder X-ray diffraction (XRD) patterns.

Fig. 1 The simulation cell for perfect TiAl crystal at 0 K.

Results and discussion

Internal energy

Fig. 2 shows the internal energy as functions of temperature for the ten cooling rates. According to the feature, the curves can be classified into two types. When the cooling rate is not less than 0.02 K ps⁻¹, internal energies persistently decrease with the drop of temperature, showing the characteristics of liquid-to-glass phase transformation. When the cooling rate lowers to 0.01 K ps⁻¹ and 0.005 K ps⁻¹, however, there is a remarkable drop of internal energy at 900 K and 920 K, respectively, which indicates that the crystallization occurs.

Fig. 2 Internal energy as a function of temperature with different cooling rates.

It is easy to understand that the cooling rate influences the microstructure during the rapid solidification process. The slow cooling rate provides a long incubation time, and the atoms in the alloy have more chances to occupy the low-energy position, which leads to the formation of crystal. While the fast cooling rate provides little time for the crystal nucleation, and hinders the energy release of the atomic system. Since the structure and energy of amorphous phase are close to the liquid phase, the continuous decrease of internal energy indicates that amorphous structure can be obtained through the rapid quenching of the liquid structure. The result similar to previous studies.^29,30 The critical cooling rate for crystallization of TiAl alloy is thus estimated about 0.01 K ps⁻¹.

Atomic configurations

Fig. 3 intuitively shows the atomic configuration images at 300 K. In Fig. 3a–h, the atomic arrangement is disordered. This means that fast cooling rates slower atomic migration and result in the formation of the amorphous structure. On the other hand, the slower cooling rate will provide more diffusing time, leading to the atoms have more time to move to the equilibrium position in experimental value. This is explain why the atomic arrangement in Fig. 3i and j is more orderly. Combined with the discussion above, it is to be remarked that the crystalline structure is not a perfectly ordered lattice, due to the cooling rate is so fast, resulting in the atoms do not have enough time to rearrange themselves into the perfect position.

Fig. 3 The structure at 300 K obtained at cooling rate of 50 K ps⁻¹ (a), 10 K ps⁻¹ (b), 5 K ps⁻¹ (c), 1 K ps⁻¹ (d), 0.5 K ps⁻¹ (e), 0.1 K ps⁻¹ (f), 0.05 K ps⁻¹ (g), 0.02 K ps⁻¹ (h), 0.01 K ps⁻¹ (i) and 0.005 K ps⁻¹ (j), respectively.

From the analysis of internal energy and atomic configuration, the rate of 0.02 K ps⁻¹ is threshold, the crystallization will form when the cooling rate is less than 0.02 K ps⁻¹, and the amorphous structure will be obtained when that is higher than 0.02 K ps⁻¹. Thus, the following discussion will be focused on the simulation results of 0.01 K ps⁻¹ and 1 K ps⁻¹, to analysis the influence of the cooling rate on the atomic configuration during the rapid solidification process.

Bond pair analysis

To further describe the microscopic local structure of alloys at two cooling rates of 0.01 and 1 K ps⁻¹ during the rapid solidification, the H-A pair analysis technique is used to identify the different local structures by different indexes.

Seven major types of bond pairs (1421, 1422, 1431, 1441, 1541, 1551 and 1661) are monitored. There are three phases existing in Ti–Al alloy, namely, α₂-Ti₃Al, γ-TiAl and TiAl₃. With H-A pair analysis technique, α₂-Ti₃Al is characterized by bond pairs 1421 and 1422, γ-TiAl is characterized by bond pair 1421, and TiAl₃ is characterized by bond pairs 1421 and 1441. On the other hand, the liquid TiAl alloy are mainly consist of icosahedral cluster and defective icosahedral structures, icosahedral cluster is characterized by bond pairs 1551 and defective icosahedral structures are characterized by bond pairs 1541 and 1431.^21,31 Therefore, the pairs 1421, 1422, 1441 and 1661 are the characterizations of crystal structure, while the pairs 1551, 1541 and 1431 are the characterizations of liquid and glass. Fig. 4a shows the relative fraction of seven bond pairs with temperature dropping at cooling rate of 0.01 K ps⁻¹. The fraction for icosahedral cluster (pair 1551) slightly increase at the temperature of >1100 K. When the temperature dropping from 1100 K to 900 K, the 1551 sharply reduction from 25.6% to 0.4%, and the percentage of the bond pairs indicating disorder (1551, 1541 and 1431) also decrease. Meanwhile the fractions for 1421 and 1422 steady increase. It thus can be inferred that icosahedral cluster sharply decreases when the crystal structures drastically increase. This phenomenon indicates that the icosahedral structure play a key role in the initial nucleation of crystal structures, which has been verified in simple crystal.^32,33 When the temperature is below 700 K, the percentage of the bond pairs for disorder is lower than 10%, and the percent of the bond pairs indicating crystal (1421 and 1422) higher than 90%. It means that the transformation from liquid to crystal occurs at 900 K, which is consistent with the internal energy in Fig. 2. According to the type of bond pairs, the final phases can be predicted as the mixture of γ-TiAl and α₂-Ti₃Al, which is consistent with experiment finding.^34,35 Furthermore, the HCP structure has 50% 1421 bond-type and 50% 1422 bond-type; the FCC structure leads only 1421 bond-type.²² Combined with the analysis above, the ratio of γ-TiAl to α₂-Ti₃Al is 10/13, calculated from the fractions of FCC and HCP are 66% and 26%, respectively.

Fig. 4 Evolution of seven bond pairs at cooling rate of 0.01 K ps⁻¹ (a) and 1 K ps⁻¹ (b).

Fig. 4b shows the change of the percentage of bond pairs at cooling rate of 1 K ps⁻¹. The fraction for bond pair 1551 increases linearly with the decrease of system temperature, which means that icosahedral clusters is the main structure in the final phase. This is because pair 1551 has the lowest energy and energy requirement in disorder bond makes it having the largest population at fast cooling rate.³¹ The percentage of the bond pairs indicating disorder is higher than 80% from 2300 K to 300 K, suggesting that the finally phase is disordered.^36,37

The evolution for the ratio of icosahedral structure in these two rates is entirely different. This is because the slow cooling rate provides a long motion time, and the atoms have more chances to occupy the low-energy position, which ensure the transformation from the icosahedral structure to initial crystal structures. On the other hand, the mobility of atoms in the liquid alloy decreases with cooling rates increases, as shown in Fig. 6. The two factors in the atomistic scale are explained the icosahedral cluster decrease when the temperature dropping from 1100 K to 900 K at slow cooling rate.

Radial distribution functions

The RDF of TiAl at the two cooling rates of 0.01 and 1 K ps⁻¹ during the rapid solidification are shown in Fig. 5a and b, respectively. In both figures, RDF curves present classic liquid state shape at elevated temperature.²⁹ With reduction of temperature, the first peak in the RDF curves grows higher, indicating that the ordering degree in the liquid alloy increases and disordering decreases. When cooling down at slow rate of 0.01 K ps⁻¹, as shown in Fig. 5a, the second peak has been split into three small peaks at 900 K, which shows that the first crystal phase begins to occur. And as the temperature decreases further, the first peak and the other small peaks become more and more intensified. This indicates that the periodic arrangement of the atoms appears in the atomic system. Cooling down at fast rate of 1 K ps⁻¹, as shown in Fig. 5b, a slight splitting of the second peak is observed, which indicates amorphous structure is formed.²⁹ Results of RDF manifest that the phase obtained through the slow cooling process is crystal and through the fast cooling process is amorphous structure, which is consistent with experimental methods.

Fig. 5 RDF versus temperature at the cooling rate of 0.01 K ps⁻¹ (a) and 1 K ps⁻¹ (b).

Mean square displacement

Fig. 6a and b shows the MSD as a function of time and temperature at the cooling rate of 0.01 K ps⁻¹ and 1 K ps⁻¹, respectively. In both figures, the slope of the MSD curves decrease with temperature dropping, which indicates that the diffusion ability of the Al atoms and Ti atoms decreases as temperature drops. In addition, in the temperature range of 800–1000 K, the atoms have almost stopped moving, suggesting that the transformation from liquid to solid. Compared with Fig. 6a and b presents a longer atomic displacement.

Fig. 6 Mean square displacement during the cooling process. (a) The cooling rate is 0.01 K ps⁻¹, (b) the cooling rate is 1 K ps⁻¹.

XRD simulation

XRD is a common and effective method to identify phase. To verify the results of H-A pair analysis, simulation results of XRD pattern of the final phase are performed by powder diffraction function in software Materials Studio 6.1. The final atomic positions in the simulation cell which containing 8064 atoms are import into Materials Studio. To ensure the accuracy of XRD simulation, more atoms are needed. The simulation cell is set as a period, and 50 periods are adopted in all directions. Fig. 7 shows that the phases obtained at 0.01 K ps⁻¹ are γ-TiAl + α₂-Ti₃Al and that obtained at 1 K ps⁻¹ is amorphous structure. The XRD result is consistent with that of H-A pair analysis.

Fig. 7 XRD powder simulation of the final state at cooling rate of 0.01 K ps⁻¹ (a) and 1 K ps⁻¹ (b).

Conclusions

The structural evolution of TiAl during rapid solidification processes are investigated by the molecular dynamics simulation. The results indicate that the phase obtained at fast cooling rates (no less than 0.02 K ps⁻¹) is amorphous structure, and the phases obtained at slow cooling rate (0.01 K ps⁻¹) are γ-TiAl + α₂-Ti₃Al. Analyzing methods, including internal energy, RDF, H-A pair analysis technique, MSD and XRD powder simulation, are adopted, and mutual conforms among them ensure their accuracy.

Acknowledgements

Thanks are given to the financial supports of the Natural Science Foundation of China (no. 51271147, 51201134, 51201135), the Fundamental Research Funds for the Central Universities (3102014JCQ01023) and the Research Fund of the State Key Laboratory of Solidification Processing (NWPU), China (Grant No. 115-QP-2014).

References

1. X. Wu, D. Hu and M. H. Loretto, J. Mater. Sci., 2004, 39, 3935–3940.
2. T. Tetsui, J. Mater. Sci. Eng. A, 2002, 329–331, 582–588.
3. S.-Y. Sung and Y.-J. Kim, Intermetallics, 2006, 14, 1163–1167.
4. V. P. Ermakova, O. Y. Sheshukov and L. A. Marshuk, Met. Sci. Heat Treat., 2010, 52, 349–353.
5. E. S. Gouda, Eur. Phys. J.: Appl. Phys., 2009, 48, 20902.
6. F. Li, X. J. Liu, H. Y. Hou, G. Chen and G. L. Chen, Intermetallics, 2011, 19, 630–635.
7. T. Savaskan, M. Turhal and S. Murphy, Mater. Sci. Technol., 2003, 19, 67–74.
8. Z. G. Liu, L. H. Cai, Y. Y. Chen and F. T. Kong, Acta Metall. Sin., 2008, 44, 569–573.
9. C. Kenel and C. Leinenbach, J. Alloys Compd., 2015, 637, 242–247.
10. J. Fan, J. Liu, S. Tian, S. Wu, S. Wang, H. Gao, J. Guo, X. Wang, Y. Su and H. Fu, J. Alloys Compd., 2015, 650, 8–14.
11. M. P. Allen and D. J. Tildesley, Computer simulation of liquids, Oxford university press, 1989.
12. S. Ozgen and E. Duruk, Mater. Lett., 2004, 58, 1071–1075.
13. M. Shimono and H. Onodera, J. Mater. Sci. Eng. A, 2001, 304, 515–519.
14. J. Lu and J. Szpunar, Acta Metall. Mater., 1993, 41, 2291–2295.
15. D. C. Rapaport, The art of molecular dynamics simulation, Cambridge university press, 2004.
16. Q. Pei, C. Lu and M. Fu, J. Phys.: Condens. Matter, 2004, 16, 4203.
17. P. Jalali and M. Li, Intermetallics, 2004, 12, 1167–1176.
18. L. Qi, H. Zhang and Z. Hu, Intermetallics, 2004, 12, 1191–1195.
19. G. Almyras, D. Papageorgiou, C. E. Lekka, N. Mattern, J. Eckert and G. Evangelakis, Intermetallics, 2011, 19, 657–661.
20. Q. X. Pei, C. Lu and M. W. Fu, J. Phys.: Condens. Matter, 2004, 16, 4203–4210.
21. Q. X. Pei, C. Lu and H. P. Lee, J. Phys.: Condens. Matter, 2005, 17, 1493–1504.
22. Z.-Y. Hou, L.-X. Liu, R.-S. Liu, Z.-A. Tian and J.-G. Wang, J. Non-Cryst. Solids, 2011, 357, 1430–1436.
23. S. Plimpton, J. Comput. Phys., 1995, 117, 1–19.
24. R. R. Zope and Y. Mishin, Phys. Rev. B, 2003, 68, 024102.
25. E. V. Levchenko, A. V. Evteev, T. Lorscheider, I. V. Belova and G. E. Murch, Comput. Mater. Sci., 2013, 79, 316–325.
26. H. Wang, D. S. Xu, R. Yang and P. Veyssière, Acta Mater., 2009, 57, 3725–3737.
27. W. Humphrey, A. Dalke and K. Schulten, J. Mol. Graphics, 1996, 14, 33–38.
28. J. D. Honeycutt and H. C. Andersen, J. Phys. Chem., 1987, 91, 4950–4963.
29. Y. Waseda, The structure of non-crystalline materials: liquids and amorphous solids, McGraw-Hill, New York, 1980.
30. Y. Q. Cheng and E. Ma, Prog. Mater. Sci., 2011, 56, 379–473.
31. D. W. Qi and S. Wang, Phys. Rev. B, 1991, 44, 884–887.
32. Z. Hou, Z. Tian, R. Liu, K. Dong and A. Yu, Comput. Mater. Sci., 2015, 99, 256–261.
33. Y. Shibuta, S. Sakane, T. Takaki and M. Ohno, Acta Mater., 2016, 105, 328–337.
34. E. Hall and S.-C. Huang, Acta Metall. Mater., 1990, 38, 539–549.
35. H.-W. Wang, D.-D. Zhu, C.-M. Zou and Z.-J. Wei, Trans. Nonferrous Met. Soc. China, 2011, 21, s328–s332.
36. C. Briant and J. Burton, Phys. Status Solidi B, 1978, 85, 393–402.
37. Y. Lo, J. Huang, S. Ju and X. Du, Phys. Rev. B, 2007, 76, 024103.

---