Simulating the Glass Transition Temperature of Bulk Metallic Glasses using Density Functional Theory


Introduction

The ability to model the properties and characteristics of a many-body microscopic system is of interest in several fields – computational physics, chemistry, material science, to name a few. Based on the principle of quantum mechanics, Density Functional Theory (DFT) is a promising tool used to investigate the properties of many-body systems like atoms and molecules. According to quantum mechanics, the wavefunction of a system, as described by Schrodinger’s Equation, is sufficient to derive all the information about this system. However, it is impossible to solve Schrodinger’s equation for a many-body system. DFT enables one to approximate the solution of this equation using functionals of spatially dependent electron density of a system. Since any many-body system can be described by a functional of its spatially dependent electron density, DFT is one of the most versatile and accurate methods applied in computational physics and chemistry.


Phaseshift uses DFT to model the Glass Transition Temperature (Tg) of Bulk Metallic Glasses (BMGs), a class of alloys known for their high durability in taxing environments and other unique material properties due to its unique non-crystalline (amorphous) atomic structure. Tg is an important characteristic temperature of BMGs, at which they transition from hard solid compounds into a relatively soft and viscous liquid, enabling them to be molded into any shape without causing any major changes in its atomic arrangement. This study demonstrates a DFT simulation of near-eutectic Zr-Cu BMG compositions to model its Tg. The calculated Tg values were also compared to the measurements and estimates reported in the literature.


Methodology

Plane-wave DFT calculations were carried out using Vienna ab initio Simulation Package (VASP) version 6. Plane-wave DFT has shown to be very effective in modeling the properties of metallic systems. The initial structure of the Zr-Cu binary compositions was generated by Born-Oppenheimer Molecular Dynamics (BOMD) by taking a crystalline system with appropriate composition to 2000 K and methodically quenching it to room temperature. This process will retain the amorphous structure of the melt. A constant Temperature-Pressure ensemble (NPT) was applied during this. Residual stresses and forces on the cell and atoms, if any, were relieved using a variable cell geometry optimization. A conjugate gradient algorithm was used for this. Once the ground state of the structure has minimized to the required accuracy, self-consistent calculations were carried out. Tg of the BMG is calculated by determining the change of slope in the Volume vs Temperature plot during the quenching process mentioned above. PBE-GGA formalism was used as the exchange-correlation functional and Methfessel-Paxton method for smearing while keeping the electron temperatures at acceptable levels.


As described above, the process was applied to 5 near-eutectic Zr-Cu binary compositions: Zr40Cu60, Zr45Cu55, Zr50Cu50, Zr55Cu45, and Zr60Cu40 (at. %). The Zr-Cu binary system was chosen because of its wide reporting in literature which makes it convenient to find the experimental measurements of its Tg. Compositions with Cu ranging from 40% - 60% atomic fraction were chosen to test whether if DFT can effectively capture minor deviations (at least 5%) in the compositions.


Results


Table 1: Tg of CuZr binary compositions obtained from DFT.

Figure 1: Comparison of Tg value obtained from DFT against those reported in literature.

The estimated values of Tg for the chosen binary compositions are listed in table 1 and a comparison against values reported in the literature is shown in figure 1. Kwon (1), Wang (2), Mei-Bao (3), and Mattern (4) report experimental measurements, while MSK and MKOSYP are values simulated using Molecular Dynamics, as reported by Mendelev (5).


As observed, the Tg increases linearly with increasing Cu content of the Zr-Cu binary composition. The experimental values, reported by Kwon (1), Wang (2), Mei-Bao (3), and Mattern (4), all support this observation and as is confirmed by the trend shown by DFT calculations between Cu 50% - 60% range with a general over-estimation by DFT. However, there is a disagreement between the experimental and simulated Tg values of compositions with Cu < 50%. According to Mattern (4), the Tg values should continue to decrease, but the MD simulation (the more accurate MKOSYP (5) amongst the two) and DFT calculations show a rise in Tg values. One possible reason could be that the simulations are unable to capture the structural changes that occur in the Cu < 50% composition range, or that the effective heating rates during the simulations and experimental measurements have that effect on the Tg values. More on the effect of heating rate is discussed later.

Figure 2: Zr-Cu Binary Phase Chart (6)

It has been demonstrated that BMG compositions with deep and wide eutectic point are good glass formers, with near-eutectic compositions being the best glass formers. In the Zr-Cu binary system, the eutectic point lies around Zr50Cu50 (as shown by the phase chart in figure 2 (6)). The eutectic point is the lowest point in the phase chart of a binary system at which the material can still exist as a liquid. Therefore, it can be observed that the compositions with Cu < 40% or Cu > 65% are worse glass formers as the liquidus temperature starts to rise drastically compared to near-eutectic compositions. This could also be a reason as to why the simulations are able to demonstrate a higher accuracy in the near-eutectic range (as shown in figure 3) as any composition out of this range might start to crystallize in experimental settings, the effect of which the current simulations setups are unable to capture causing a mismatch in Tg values.

Figure 3: Tg of near-eutectic CuZr BMGs

The heating rate at which the measurements of Tg are made can also have a drastic effect on the final measurement. Mei-Bao (3) measure the effect of heating rate on the Tg of Zr50Cu50 composition. As shown by the ln ϕ (where ϕ is the heating rate) vs Tg graph in figure 4, the Tg value can climb as much as 20 K with increasing heating rates. While the highest reported experimental heating rate was ~120 K/min, the effective heating/cooling rates during DFT could be as high as 10^8 K/min. Therefore, an overestimation by DFT of Tg could be an artifact of extremely high effective cooling rates at which DFT operates as compared to real-world scenarios in which the heating rates available during measurements (such as that in a Differential Scanning Calorimeter) are limited to ~100 K/min.


Figure 4: Tg vs Heating Rate

Conclusion

As demonstrated in this article, DFT can serve as a powerful tool to model the characteristics of alloys. In this study, the Glass Transition Temperature (Tg) was modeled using DFT and compared against values reported in the literature. DFT calculations were found to be in confirmation of the trends demonstrated by experimental measurements as well as simulations performed by other groups using Molecular Dynamics (MD) (5). There is a general overestimation of Tg values by DFT which could be attributed to the extremely high effective heating/cooling rates present in DFT simulations. The compositions with Cu < 50% were observed to have an increase in simulated Tg values as opposed to a linearly decreasing trend observed in experimental measurements. However, as shown by the binary phase charts, those compositions are generally considered bad glass formers due to higher liquidus temperatures than the near-eutectic compositions and the simulations might be unsuccessful in capturing any crystallization that could be happening in an experimental setting due to a lower heating rate. Overall, DFT serves the purpose of modeling reasonably realistic material compositions and capturing their characteristic material properties.


About Phaseshift Technologies

Phaseshift Technologies is a materials development company that develops novel high-performance alloys using Machine Learning and Computational Chemistry. As demonstrated in this study, DFT serves as one of several tools Phaseshift uses to develop the chemistry of alloys and guide the experimental efforts. Constantly iterating and developing new and unique computational workflows, Phaseshift aims to reduce the time requirement and cost of materials research and development. The company currently comprises a team of Materials Scientists, Machine Learning Engineers, and Physicists, with a highly specialized set of domain expertise in high-performance alloys.


References

1. Formation of Amorphous Phase in the Binary Cu-Zr Alloy System. O. J. Kwon, Y. C. Kim, K. B. Kim, Y. K. Lee, and E. Fleury. 3, 2006, Metals and Materials International, Vol. 12, p. 208.


2. Bulk metallic glass formation in the binary Cu–Zr system. D. Wang, Y. Li, B. B. Sun, M. L. Sui, K. Lu, and E. Ma. 2204, Applied Physics Letters, p. 4030.


3. Binary Cu–Zr Bulk Metallic Glasses. T. Mei-Bo, Z. De-Qian, P. Ming-Xiang, and W. Wei-Hua. 5, 2004, Chinese Physics Letters, Vol. 21, p. 902.


4. Phase Separation and Crystallization in Cu-Zr Metallic Glasses. N. Mattern, U. Kuhn, A. Concustell, A. Schops, M. D. Baro and J. Eckert. 7, 2007, Materials Transactions, Vol. 48, p. 1639-1643.


5. Experimental and computer simulation determination of the structural changes occurring through the liquid–glass transition in Cu–Zr alloys. M.I. Mendelev, M.J. Kramer , R.T. Ott , D.J. Sordelet , M.F. Besser , A.Kreyssig , A.I. Goldman , V. Wessels , K.K. Sahu, K.F. Kelton, R.W. Hyers, S. Canepari & J.R.Rogers. 29, October 14, 2010, Philosophical Magazine, Vol. 90, p. 3800.


6. Cu-Zr (Copper-Zirconium). H. Okamoto. July 25, 2012, Journal of Phase Equilibria and Diffusion, Vol. 33, p. 417-418.