Crystal Structure Analysis of Irradiated Ni₃Al Using Molecular Dynamics Simulation

Abstract

The structural changes of irradiated Ni₃Al were simulated by a molecular dynamics (MD) method. The irradiation event was modeled as the energy deposition of thermal energy produced by a high-energy ion beam, i.e. effective stopping power gSe. The L1₂ structure was taken as the initial structure and the effects of the irradiation event on the atomic structure were investigated. The relative degree of order, defined as the ratio of the calculated diffraction peaks of the L1₂ structure before and after the irradiation, and the number of site-exchanged atoms were calculated and found to show good correlation with gSe. The strength of the specimen was estimated from potential energy and it was decreased after the irradiation. Results of the uniaxial extension test done in the MD simulation suggest that the off-site atoms and site-exchanged atoms are the major cause of the reduction of specimen strength.

Fig. 5 Time changes of ordered L1₂ lattice structure in Ni₃Al after being irradiated with gSe = 2.0 keV/nm. Green spheres are atoms in the ordered L1₂ lattice structure and red spheres are recovered atoms from the disordered structure. Both green and red atoms are on fcc sites, while grey atoms are in the amorphous structure.

1. Introduction

Intermetallic compounds have attracted a lot of attention recently because they show various desirable characteristic properties such as high-temperature strength,¹⁾ good oxidation resistance,²⁾ hydrogen storage capability,³⁾ superconductivity⁴⁾ and shape memory.⁵⁾ These properties arise from their ordered lattice structure. This means that the properties of intermetallic compounds can be modified and controlled by irradiating the compounds using high-energy beams of electrons and ions.

Among intermetallic compounds, the dual-phase Ni₃Al–Ni₃V system is known to have high-temperature strength and wear resistance and therefore is expected to be applied as a new construction material.^6–8) These excellent characteristics are due to the stability of two-phase microstructures to a high phase transformation temperature. It is an interesting theme to clarify the effects of energetic ion irradiation on Ni₃Al from the viewpoint of modification of material properties in a confined region. Yoshizaki et al.⁹⁾ carried out an energetic ion irradiation of Ni₃Al intermetallic compounds and investigated their crystal phase transformation and hardness modification. They found that the relative order of the crystal, deduced from the X-ray diffraction data, was changed by the irradiation and correlated well with the density of energy deposited through elastic collisions. The hardness of the materials was also changed as a consequence of the structural change caused by the irradiation. However, a detailed analysis only from actual experiments is difficult because the modification process of properties is very rapid and proceeds in a non-equilibrium state. An atomistic computer experiment, in other words a numerical simulation, is an alternative approach to overcome the difficulties of actual experimental research because it can provide detailed information about atomic arrangements in the non-equilibrium state. In a pioneering simulation study of the ion irradiation effects on metallic compounds, Devanathan et al.¹⁰⁾ used molecular dynamics (MD) simulations to examine electron-irradiation-induced amorphization of the ordered intermetallic compound NiZr₂. They introduced chemical disorder and Frenkel pairs in the specimens rather artificially: chemical disorder was created by exchanging randomly selected Ni and Zr atoms, and Frenkel pairs were introduced by removing atoms at random from their sites and introducing them into interstitial positions. Devanathan et al. found that both random atom exchanges and Frenkel-pair introduction can make NiZr₂ amorphous. Nagase et al.¹¹⁾ applied the same method as Devanathan et al. to C16-Zr₂Ni compound and found that the electron-irradiation-induced amorphization was achieved through the accumulation of lattice defects and that thermal relaxation of the lattice defects in the atomic configuration induced the amorphization in intermetallic compounds. These studies clarified the basic mechanism of the amorphization in intermetallic compounds; however, the physical processes that show how the defects are introduced by irradiation are still unclear. We have utilized the MD simulation technique to study ion beam irradiation based on the thermal spike model.^12–16) This technique provides detailed information about the defect formation process by ion beam irradiation. In this paper, we investigated the structural changes of irradiated Ni₃Al by using MD simulation. The effects of irradiation on atomic order and structure were clarified from atomistic structural analysis, and the specimen strength was estimated by the uniaxial extension test simulated by the MD method and its correlation with the structural change was considered.

2. Method of Simulation

2.1 Modeling the irradiation event

The ion beam irradiation on Ni₃Al was simulated by the LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator).¹⁷⁾ LAMMPS is one of the most reliable MD simulator available today and it is in wide use.^18–20) Furthermore, LAMMPS offers graphics processing unit (GPU) acceleration for high-efficiency calculation of many particles and accelerates the processing speed by parallelizing the calculation.^21–23) These features can reduce the simulation time drastically and increase reliability of the MD simulation.

As the initial structure, Ni₃Al single crystal with L1₂ was generated. The lattice constant was 0.35896 nm and the dimensions of the simulation cell were 12 lattice cells in x- and y-axis directions, and 8 lattice cells in the z-axis direction. The size of the specimen was 4.25 nm in x- and y-directions, and 2.89 nm in the z-direction. The atomistic arrangement is shown in Fig. 1(a). For the interaction between Ni and Al atoms, the embedded-atom method (EAM) potentials were used.²⁴⁾ We examined the stability of Ni₃Al single crystal with L1₂ by a preliminary MD simulation using the EAM potential and confirmed that the structure was maintained up to its melting temperature, which is consistent with actual experimental behavior.

Fig. 1

The configuration of the specimen and targeted irradiation area. (a) Ni₃Al in the ordered L1₂ structure where green spheres are Ni and grey are Al atoms. (b) Irradiation area inside a specimen. (c) Irradiation area viewed from ⟨001⟩ direction.

We simulated the ion irradiation event as follows. At first, the Ni₃Al single crystal was thermalized at 298 K. In this simulation, the ion beam irradiation was modeled as the energy deposition just after the irradiation. The present simulation does not consider the elastic collision directly; however, this modelling can provide valuable information to clarify the structural transition of irradiated materials.^12–16,25) The energy deposited to the irradiated system as thermal energy is called effective stopping power (gSe). gSe is expressed as thermal energy deposition per unit length, keV/nm. gSe was set to 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0 and 5.0 keV/nm. This thermal energy was considered to be deposited to the irradiation area, a cylinder with 3 nm radius, as shown in Figs. 1(b) and (c). The kinetic energy just after the irradiation (E_{k_after}) was calculated as the summation of the total kinetic energy of the irradiation area before irradiation (E_{k_before}) and the energy from effective stopping power (gSe). Energy from gSe was calculated by multiplying gSe value by the Ni₃Al specimen size in the ⟨001⟩ direction (l_z = 2.89 nm) as follows,   

\begin{equation} E_{\textit{k_after}} = E_{\textit{k_before}} + (gSe \times l_{z}). \end{equation} (1)

From E_{k_after}, the temperature of the irradiated region was calculated from the equipartition law. The velocities of individual atoms inside the irradiation area were set according to the Boltzmann distribution law. The temperature outside the targeted irradiation area was kept at ambient temperature, 298 K. Then, the MD simulation was done for 10,000 time-steps (10 ps, where 1 time-step was 0.001 ps). During the simulation, the temperature outside the irradiation area was controlled using the Nose-Hoover method which was designed to calculate atomic positions and velocities in an isothermal-isobaric NPT ensemble. The periodic boundary condition was imposed in all the directions, x, y and z, to simulate the irradiation event in the bulk Ni₃Al specimen.

2.2 Structural analysis

Experimental results showed that when the Ni₃Al interatomic compound was irradiated with energetic ion beams, the order of atoms was changed from the ordered L1₂ structure into the disordered A1 structure.⁹⁾ Figure 2 shows Ni₃Al atoms in the ordered L1₂ structure and the disordered A1 structure. In this paper, we examined the changes of order and structure of atoms and strength changes of specimens after irradiation. For the observation of the order and structure changes, two technique were used: the relative degree of order (RDO) of the structure and polyhedral template matching (PTM).

Fig. 2

(Left) Ordered L1₂ structure where green spheres are Ni and grey are Al atoms in Ni₃Al. (Right) Disordered A1 structure where grey atoms can be either Ni or Al atoms.

RDO can be calculated from the intensities of the 110 and 220 peaks in the X-ray diffraction pattern obtained by the LAMMPS. According to the actual experiment by Yoshizaki et al.,⁹⁾ the RDO is defined as the ratio of the peaks before and after irradiation,   

\begin{equation} \mathit{RDO} = \frac{I_{110}/I_{220}}{U_{110}/U_{220}}, \end{equation} (2)

where I_hkl is the intensity of the hkl peak for the specimen after irradiation and U_hkl is the intensity of the hkl peak for the specimen before the irradiation.

The polyhedral template matching (PTM) algorithm is provided by the software Open Visualization Tool (OVITO).²⁶⁾ The PTM algorithm can identify the local alloy type, such as the L1₂ structure, by considering the chemical species of atoms. This algorithm defines alloy type of a lattice atom by comparing the types of its neighbor atoms to its own type. In addition, the PTM algorithm can also assign a structure type property to each atom. This is done by using the convex hull formed by the set of neighbor atoms that describes the local structure around an atom. The convex hull is then represented as a planar graph, and this graph is used to classify the structure. Since this method does not use the concept of bonds between atoms, it is less sensitive to thermal fluctuations.

It is challenging to evaluate specimen hardness by MD simulation. This is because the specimen in the present MD is not a finite structure due to the periodic boundary condition, so the specimen does not have a real surface to perform a test like the Vickers hardness test. Although we can create an artificial surface for the specimen by not applying the periodic boundary condition on one of the surfaces, the result will be influenced by the size effect. Therefore, in this paper, we observed specimen strength changes rather than specimen hardness changes. Specimen strength was estimated using the uniaxial tension technique. In the uniaxial tension technique, the fix-deform command in LAMMPS was used to deform the specimen along the x-axis direction. The specimen was deformed until the strain value reached 0.15. During the deformation, the stress tensor was calculated, and the first maximum stress tensor was used as the index of the specimen strength. The maximum strain of 0.15 for each condition was selected to be within the range of plastic strain to estimate Vickers hardness. There should be a generation process and/or an interaction of dislocations; however, they are complicated processes and we leave them to a subsequent study.

3. Results and Discussion

In this paper, MD simulation was conducted to further understand the detailed structure and strength changes after irradiation. The X-ray diffraction calculation and PTM algorithm provided by OVITO were used to analyze the structure changes. Figure 3 shows the X-ray diffraction patterns for unirradiated and irradiated specimens. We compared the intensities of the 110 and 220 peaks with those of the unirradiated specimen and calculated its RDO for every gSe. The results are shown in Fig. 4. We see clearly that the RDO decreased as the gSe value increased. However, when gSe value was more than 3.0 keV/nm, the RDO converged to a constant value around 0.3. A similar result was also obtained in the actual experiment by Yoshizaki et al. (see Fig. 4(b) in Ref. 9)): the change in RDO decreased with the increase of elastically deposited energy density.

Fig. 3

Calculated diffraction intensity profiles of the irradiated specimens with the characteristic wave length of Cu.

Fig. 4

Relative degree of order as a function of gSe value.

Figure 5 shows an example of time changes of the L1₂ lattice structure in Ni₃Al after the irradiation with gSe = 2.0 keV/nm. Immediately after the irradiation, the atomic structure at the central region of the (001) surface became an amorphous structure (represented by grey circles in Fig. 5). The amorphous structure then recovered gradually to the crystal structure (red circles). However, the order of the Ni and Al atoms had changed from the ordered L1₂ lattice structure into the disordered A1 structure, i.e., the location of Ni and Al atoms changed from their original sites. Even at the end of the simulation, some of the atoms were unable to recover to the lattice sites, i.e., grey circles remained in the structure at 10 ps in Fig. 5. These atoms were separated at the distance larger than half the first neighbor distance between Al and Ni from their original sites in the L1₂ structure and we define these atoms as off-site atoms. In contrast, the atoms represented as red circles at 10 ps were the on-site atoms after the irradiation. However, some atoms were on different sites from their original sites, Ni to an Al site and vice versa. Such atoms are defined as site-exchanged atoms.

Fig. 5

Time changes of ordered L1₂ lattice structure in Ni₃Al after being irradiated with gSe = 2.0 keV/nm. Green spheres are atoms in the ordered L1₂ lattice structure and red spheres are recovered atoms from the disordered structure. Both green and red atoms are on fcc sites, while grey atoms are in the amorphous structure.

Figure 6(a) shows the number of the off-site and site-exchanged atoms at the end of every irradiation as a function of gSe. The specimen irradiated at gSe = 0.5 keV/nm showed no site-exchanged atoms at the end of the simulation. However, off-site atoms were found in this specimen. It should be noted that the site-exchanged atoms were defined only if the atoms were on fcc sites. As the gSe value increased, the number of site-exchanged atoms increased. The site-exchanged atoms were apparently unchanged from gSe = 3.5 keV/nm and above. The number of the site-exchanged atoms as a function of gSe was inversely correlated with the RDO results shown in Fig. 4. Meanwhile, it seems that the off-site atoms do not follow a simple function of gSe. We can consider that the random arrangement of the off-site atoms disturbs the recovery process to the crystal structure regardless of the magnitude of gSe when larger than a critical value between 0.5 and 1.0 keV/nm.

Fig. 6

(a) Changes of the number of the off-site (blue histogram) and site-exchanged atoms (red diamonds) as a function of effective stopping power gSe. (b) Images of the atomic structures viewed from ⟨001⟩ at the end of simulation. The value of gSe is in ascending order from upper left (0.5 keV/nm) to lower right (5.0 keV/nm). Green spheres are the atoms in the ordered L1₂ lattice and red spheres are recovered atoms from the disordered structure. Both green and red atoms are on fcc sites, while grey atoms are in the amorphous structure.

Figure 6(b) shows the atomic structures of the specimens viewed from the ⟨001⟩ direction after irradiation. The area not exposed to irradiation would have fewer site-exchanged atoms. For gSe = 3.0 keV/nm and above, the size of the disordered area was the same. This shows that the area far from the irradiation area was not disturbed even when the gSe value increased. From our observation, the irradiation-free area is robust to the pressure from the irradiation area that has high kinetic energy. This can be seen in Fig. 5 where this area blocks the amorphous structure area from extending into the entire specimen.

Figure 7 shows the maximum stress tensor during the uniaxial tensile test as a function of gSe. The initial atomic positions for the uniaxial tension are the structures shown in Fig. 6(b). The maximum stress tensor was damped as gSe increased on average, and the additional oscillation was imposed upon this simple damping behavior. The damping trend correlated to the change of the number of the site-exchanged atoms and the imposed oscillation correlated to the change of the number of the off-site atoms (compare Fig. 6(a) and Fig. 7). It should be noted that the relationship between the oscillation of the value of stress tensor and the change of the number of the off-site atoms showed a negative correlation: the value of the stress tensor was low when the number of the off-site atoms was large and vice versa. The above observation leads to the consideration that the maximum stress tensor during the uniaxial tensile test as a function of gSe can be interpreted as a superposition of the effects of the site-exchanged atoms and off-site atoms. However, each effect on the stress is not clear and it is difficult to separate them qualitatively because of an equivocal tendency of the off-site atoms and of ambiguity of each contribution to the maximum stress tensor. The accumulation of off-site atoms could encourage slip and dislocation creation. This could be one of the causes of reduced specimen strength. When the number of site-exchanged atoms increases, the number of strong bonds between Ni and Al decreases. This also causes the reduction of specimen strength.

Fig. 7

Maximum stress tensor as a function of effective stopping power.

In the actual experiment, the Vickers hardness test was done to investigate the hardness of Ni₃Al after irradiation.⁹⁾ The experimental results showed that Vickers hardness does not correlate with the elastically deposited energy density. On the other hand, the simulation by the uniaxial tensile method showed that the maximum stress tensor is likely to be well correlated with the gSe value. However, both experiment and simulation agree in that the irradiation can reduce the hardness or strength of the specimen. Furthermore, the simulation results also suggested that the off-site atoms and site-exchanged atoms created by the irradiation could be the major cause of the reduction of specimen strength. The original sites of off-site atoms become point defects if they are not occupied by other atoms. The point defects will accumulate and grow to voids which degrade the strength of Ni₃Al. We point out that irradiation-induced void formation has been observed in-situ using high-resolution transmission electron microscopy.²⁷⁾

4. Conclusion

The ion beam irradiation effects on Ni₃Al atomic order and structure, and strength changes were investigated by the MD simulation. The irradiation was considered as the thermal energy deposited during the irradiation and it is called effective stopping power (gSe). Nine gSe values were examined for Ni₃Al specimens. The results showed that the RDO and the number of site-exchanged atoms correlated well with gSe value. However, the off-site atoms did not correlate with gSe value. The specimen strength decreased after the irradiation. Results of the uniaxial tensile test performed by the MD simulation suggested that the off-site atoms and site-exchanged atoms were the major cause of the decreased specimen strength. It is an interesting theme to apply our simulation method to other metallic compounds such as Ni₃Nb, Ni₃Ta and Ni₃Ti because they have different types of ordered structure and were found to show various structural and hardness changes under ion irradiation.^28,29)

Acknowledgements

This research is financially supported by Grants-in-Aid for Scientific research under Contract Nos. 21360469 and 16K06963, and by the Light Metal Educational Foundation, Inc. under funds for Encouragement and Promotion of Research, Study and Education. Mr. C.C. Woo, former student of Ibaraki University, performed preliminary MD simulation for Ni₃Al single crystal.

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