Research on Microstructural Evolution Behavior of Ni-Based Single-Crystal Alloy with Re Based on Non-Linear Ultrasonic Lamb Wave and Molecular Dynamics Method

Abstract

Interface dislocation networks have a great influence on the mechanical properties of the new Ni-based single-crystal alloy (NSC) containing Re, but it is difficult to find out the structural evolution behaviors at the micro-level. Thus, molecular dynamics (MD) simulation is used to analyze the atomic potential energy change and dislocation evolution mechanism, and non-linear characteristic parameters are used to analyze the microstructure evolution of NSC. First, a new model of Ni-Al-Re that is closer to the real properties of the material is established using the MD method according to the optimal volume ratio of matrix phase to precipitate phase. Then, the MD models of NSC with different contents of Re are calculated and analyzed under compressive and tensile loads. The results show that with an increase in Re atoms, the atomic potential energy at the interface dislocation networks is reduced; thus, the stability of the system is enhanced, and the hindrance of the interface dislocation networks to the dislocation movement of the matrix phase is strengthened. At the same time, the number of HCP structures and OISs formed by the destruction of the intact FCC structures also decreases. In the non-linear ultrasonic experiment, with the increase in Re atoms, the non-linear enhancement of the microstructure of the NSC leads to an increase in the corresponding non-linear characteristic parameters. Accordingly, the microstructural evolution behaviors of the phase interface of the new NSC can be effectively explored using the combination of MD simulation and non-linear ultrasonic experimentation. The results of this study lay a foundation for the subsequent research of the microscopic defects of NSCs by using ultrasonic phased-array technology.

1. Introduction

1.1. Research Background

The higher the temperature of the high-temperature and high-speed airflow before entering the turbine, the greater the thrust of the aero-engine [1]. However, the higher the temperature, the stronger the impact of the airflow on the aero-engine turbine blades [2]. Therefore, it is highly necessary to improve the temperature and pressure bearing capacities of the aero-engine turbine blades. A Ni-based single-crystal alloy (NSC) is the core material for realizing high aero-engine thrust, which is mainly used in the manufacture of aero-engine turbine blades [3,4]. At present, the demand for new fighters capable of supercruise without reloading is very strong, but the temperature and pressure bearing capacities of the aero-engine turbine blades made of traditional NSCs re no longer sufficient for use.

In recent years, the application of NSCs containing Re has improved the temperature and pressure bearing capacities of aero-engine turbine blades and extended the creep and fatigue life. The tensile properties of NSCs are improved after the addition of the Re element [5], and the Re atom can delay the shear of the matrix phase and the diffusion of elements [6]. In addition, the Re atom can prolong the Kear–Wilsdorf locking time at the late creep stage [7]. In the study of the creep life of new NSCs containing 4.5 wt.% Re, the creep life of an NSC can be improved by adding the Re element [8]. For the creep behavior of NSCs containing Re under different heat treatment processes, the results show that at 760 °C and 800 MPa tensile stress, an NSC exhibits better creep resistance [9].

The above research on the new NSC containing Re is mainly focused on the characterization of the mechanical behavior or creep life, but the microstructural evolution behaviors of phase interface are seldom reported. However, it is easy to form micro-cracks at the phase interface and extend the service of the NSC. Therefore, it is necessary to study the microstructural evolution behaviors of phase interface of the new NSC containing Re. Moreover, the recent research on NSCs has primarily focused on surface roughness analysis [10], surface crack analysis [11], hydrogen embrittlement studies [12], fracture behavior analysis [13], and the observation of the microstructural evolution of an NSC through electron microscope [14]. Due to the high cost of an NSC turbine blade, it is not suitable for destructive testing during service. However, in the above studies of NSC performance, although a variety of experimental techniques and methods are used, the adopted technologies do not overcome the challenge of rapid and non-destructive testing. For instance, the use of both SEM and TEM requires prior sample preparation. And these methods are not only destructive but can only analyze the specific section or local area of the material and cannot achieve the overall analysis of the material. The RBS/C technology is suitable for the analysis of the surface coating structure of the NSC, but it cannot deeply examine the internal structure of the entire NSC turbine blade. In contrast, the non-linear ultrasonic Lamb wave technique can achieve fast and non-destructive detection. More importantly, due to its short wavelength, a non-linear ultrasonic Lamb wave is extremely sensitive to micro-defects and can propagate within the entire turbine blade material. Therefore, the whole material can be analyzed and evaluated using a non-linear ultrasonic Lamb wave.

1.2. Research Status of Propagation Characteristics of Lamb Wave in NSC

An NSC is mainly used to manufacture turbine blades that belong to the thin plate structure, and Lamb wave is more suitable to propagate in the thin plate structure [15]. Based on this, the propagation of a Lamb wave in an NSC is greatly affected by changes in the microstructure. The Re content in the new NSC is different, which will change the microstructure. Therefore, it is necessary to investigate the effect of microstructure changes on Lamb wave propagation of new NSCs containing Re. In the non-linear ultrasonic experiment, the Model RAM-5000-SNAP system (Advanced Radiation Measurement & Analysis Ltd., Woodinville, WA, USA) is adopted, and an organic glass tilt sensor is used to send and receive the signal. In addition, according to the propagation characteristics of a Lamb wave in an NSC, the incidence angle and receiving angle are both $19.5°$, the excitation frequency is 2.05 MHz, and the distance between the transmitting end and the receiving end is the same (2.5 cm) in the four tests. In the analysis of the results, the ratio of the displacement amplitude of the second harmonic $A_{2f}$ to the square of the displacement amplitude of the fundamental frequency ${A_{1f}}^2$ is used as the non-linear characteristic parameter [15]:

$${\displaystyle \frac{A_{2f}}{{A_{1f}}^2}}={\displaystyle \frac{a{k}^2(48\sigma R^3{E_1}^3\mathsf{\Omega }{L}^4\Lambda +5{\mu }^3{b}^2{E}_2)}{80E_1{\mu }^3{b}^2}}$$

(1)

where a is the Joseph-Louis Lagrange coordinates, k is the wavenumber, $\sigma$ is the internal stress of the Ni-based single-crystal alloy sample, R is the tangential decomposition factor, E₁ is the second-order elastic constant of NSC, $\mathsf{\Omega }$ is the transition coefficient from shear strain to longitudinal strain, L is half the length of the dislocation ring, $\Lambda$ is the dislocation density inside the material, $\mu$ is the shear modulus of the material, B is the Berman vector, and E₂ is the third-order elastic constant. In the above formula, it can be seen that the non-linear characteristic parameter is proportional to the dislocation density and the fourth power of the dislocation loop length. When the Re content increases, the mismatch of the phase interface changes, and the corresponding dislocation distribution state also changes, which will eventually affect the non-linear characteristic parameter. The microstructural evolution behavior of an NSC containing Re is analyzed using the molecular dynamics (MD) method in the subsequent section.

1.3. Research Status of MD Simulation

At present, MD simulation is widely used to study the microstructural evolution behavior of material. MD simulation can be used to analyze the micro-wear behaviors of metal materials [16], the micro-mechanisms of nano-fluids [17], the micro-mechanism of the early phase of titanium oxidation in Ti/CuO materials [18], ab initio simulation of molten salt [19], micro-defects caused by radiation [20], and periodic boundary conditions for the microstructure evolution behavior of material [21]. As for the microstructural evolution behaviors of NSCs, recent studies are mostly focused on analyzing the evolution mechanism of cracks in NSCs under fatigue damage [22], the cyclic fatigue fracture behavior of an NSC [23], the evolution mechanism of micro-cracks in an NSC at 500 °C [24] and the evaluation of the surface preparation quality of NSCs [25]. The above studies used MD simulation, but most of them are based on the Ni-Al model, which is mainly used for the first-and second-generation NSC, and is out of step with the application of the new NSC containing Re. This is because of the fact that the mechanical properties of NSCs are improved by adding the Re element, especially the creep life. Therefore, it is necessary to explore the MD of the Ni-Al-Re model.

MD simulation provides a new technological approach to analyze the materials as a whole in the nano-scale range, which can be used to study the microstructural evolution behaviors of NSCs under complex loading conditions and calculate the dislocation density and dislocation loop length. As for the theoretical relationship between the non-linear characteristic parameters and microstructural evolution behaviors of NSCs, some research results have been accumulated [15]. Therefore, the research on microstructural evolution behaviors of the phase interface of an NSC containing Re is based on the MD simulation and non-linear ultrasonic experiment.

2. Research Methods

Because of the complexity of the MD simulation, the software MATLAB (R2016 version, MathWorks, Natick, MA, USA), LAMMPS [26], and OVITO [27] are used during the simulation. MATLAB is utilized to build the new NSC model, and the old NSC model can be modeled directly in LAMMPS. The calculation of the new and old models can be performed in LAMMPS. OVITO can be used to analyze the number, type, and loop length of dislocations directly, and the new and old models, as well as the calculation results, can be displayed by OVITO.

During the process of modeling, the lattice parameters of the γ phase are different from those of the γ′ phase ($a_{\gamma }$ = 0.352 nm; $a_{{\gamma }^{\prime }}$ = 0.3567 nm); thus, there are lattice mismatches at the interface between the two phases. Given a mismatch degree of δ, we obtain [28]:

$$\delta ={\displaystyle \frac{2(a_{{\gamma }^{\prime }}-a_{\gamma })}{(a_{{\gamma }^{\prime }}+a_{\gamma })}}$$

(2)

And the relationship between $a_{\gamma }$ and $a_{{\gamma }^{\prime }}$ is as follows [28]:

$$n{a}_{{\gamma }^{\prime }}=(n+1)a_{\gamma }$$

(3)

From the above equation, n = 75. Accordingly, in order to fully explore the microstructural evolution behaviors of dislocation networks at the phase interface, the MD models of the new NSC with different contents of Re (0 wt.%, 2 wt.%, 4 wt.%, and 6 wt.%) are established in this study.

During the MD simulations under tensile load, the periodic boundary constraint condition is used, the model is uniaxial tensile along 001 crystallographic direction, the simulation system is set to an N (number of atoms)/P (pressure)/T (temperature) ensemble, the ambient temperature is set at 900 K, the relaxation time is set at 45 ps, the calculation steps is set to 38,000, and the strain rate is set at 3.5 × 10⁹ s⁻¹. During the MD simulations under compressive load, the same simulation environment and operational parameters as mentioned above are used. In the non-linear ultrasonic experiment, the Model RAM-5000-SNAP system is used. The excitation frequency of the Lamb wave is 2.05 MHz, and the incidence angle and reception angle are both 19.5°. In addition, the elemental contents of the NSC samples are shown in

Table 1. Elemental contents (wt.%) of NSC samples.

Samples	Re	Ni	Ti	Cr	Co	Al	Mo	W	Ta
Sample 1	0	62.49	1.82	9.73	6.14	6.21	2.78	7.14	3.69
Sample 2	2	60.49	1.82	9.73	6.14	6.21	2.78	7.14	3.69
Sample 3	4	58.49	1.82	9.73	6.14	6.21	2.78	7.14	3.69
Sample 4	6	56.49	1.82	9.73	6.14	6.21	2.78	7.14	3.69

.

3. Results and Discussion

The MD model of a traditional NSC is a sandwich model, as shown in Figure 1 (old model): the upper atom is in γ phase (Ni), and the lower atom is in γ′ phase (Ni₃Al). Figure 1a shows the whole three-dimensional model, and the γ phase (Ni) is magnified. In order to observe the inside of the model, the three-dimensional model is cut into blocks, and the γ′ phase (Ni₃Al) is magnified, as shown in Figure 1b. The MD model of the new NSC containing Re consists of a γ phase and γ′ phase, while the Re atoms (yellow atoms) are randomly distributed in the γ phase, as shown in Figure 2. Because the γ′ phase cannot be observed, the whole three-dimensional model (see Figure 2a) must be sliced (see Figure 2b). It can be seen in Figure 2b that the γ′ phase exists inside the model; thus, this model is closer to the real properties of the material [28]. When the volume ratio of γ phase to γ′ phase reaches or approaches 3:7, the material has good mechanical properties.

The 001-crystallographic-direction NSC is representative and characteristic [29,30], possessing extremely high tensile strength and excellent creep resistance. It can maintain its strength and stability at temperatures close to 80% of its melting point, which further enhanced the corrosion and oxidation resistance [31]. Therefore, the 001-crystallographic-direction NSC is used for the modeling analysis in this study. The square interface dislocation networks have a strong inhibitory effect on dislocation movement, based on which the MD model, with square interface dislocation networks, of the new NSC containing Re are established (see Figure 3). It can be seen that the blue lines are dislocation loops, which are relatively complete, thus forming an obvious square interface dislocation network.

The newly constructed MD model is shown in Figure 4. Because the γ′ phase exists inside the model, only the γ phase can be seen in Figure 4. There are no Re atoms, and the γ phase contains only Ni atoms in Figure 4a, and in Figure 4b–d, the yellow Re atoms are randomly distributed in the γ phase. The number of atoms in the MD model of a traditional NSC is relatively small [15,25], while the number of atoms in the MD model of the new NSC established in this study exceeds 2.4 million (the number of atoms is calculated in terms of the volume in which the model is built). Because the periodic arrangement of atoms in the γ′ phase cannot be observed in Figure 4, a Ni₃Al atomic cluster with a periodic arrangement structure is shown in Figure 5a. In the sandwich model, the γ phase is Ni, while in the new MD model, Re atoms are randomly substituted for Ni atoms in the γ phase, as shown in Figure 5b.

In order to clearly reveal the microstructural evolution behaviors of dislocation networks at the phase interface of an NSC containing Re, it is necessary to focus on the analysis of the phase interface. Four different phase interfaces are shown in Figure 6. There are no Re atoms in Figure 6a, and the Ni atoms are periodically arranged. Figure 6b–d show the phase interfaces of the MD models of NSCs containing 2 wt.%, 4 wt.%, and 6 wt.% Re, respectively. It can be seen that the NSC containing Re has a great influence on the phase interface.

To further investigate the microstructural evolution behaviors of dislocation networks at the phase interface of NSCs containing Re, the MD simulations are carried out on the four models (NSC containing 0 wt.%, 2 wt.%, 4 wt.%, and 6 wt.% Re) under the tensile load and compressive load, respectively. Then, the Model RAM-5000-SNAP system is used to test the acoustic characteristics of the four NSC samples, and the microstructural evolution behaviors of the materials are analyzed by the changes in the non-linear characteristic parameters.

Figure 7a shows the atomic distribution of the MD model of the NSC without Re under tensile load; it can be seen that the atoms at the top of the model move violently, and there is obvious tensile deformation. In addition, the atomic distribution structure of the model is a typical face-centered cube structure, and the structural state of the model is changed to that of 4093 HCP (Hexagonal Close-Packed) structures and 89,320 other irregular structures (OISs) appear in the model system. Figure 7b shows the atomic distribution of the MD model of the NSC containing 2 wt.% Re under tensile load; it can be seen that the atomic motion activity at the top of the model is relatively weak. There are 3091 HCP structures and 71,422 OISs in the model system, indicating that the stability of the model is enhanced, and the resistance to tensile deformation is strengthened after Re atoms are integrated into the model. Figure 7c shows the atomic distribution of the MD model of the NSC containing 4 wt.% Re under tensile load; there are 2696 HCP structures and 69,842 OISs in this model system, and the stability of the model is further enhanced when the Re content continues to increase. Figure 7d shows the atomic distribution of the MD model of the NSC containing 6 wt.% Re under tensile load; there are 2195 HCP structures and 67,643 OISs in this model system. According to the simulation results, it can be concluded that the stability of the model can be strengthened when the Re content is suitably increased.

The stability of the model is affected by the integration of Re atoms, and it is necessary to reveal the effect mechanism from the perspective of energy change. Because the temperature is kept constant (900 K), and the average atomic kinetic energy of the model system remains unchanged, the energy of the model system is only changed by the average atomic potential energy. After calculation and analysis, the average atomic potential energies of the four model systems are −11,283,900 ev, −11,356,300 ev, −11,425,100 ev, and −11,493,600 ev, respectively (see Figure 8). It can be seen, from the initial potential energies of the four model systems, that the PE decreases with the increase in the Re content, which is beneficial to the stability of the model. However, under the initial action of tensile load, the average atomic spacing between the models increases, and the PE increases. After Re atoms are incorporated into the model system, the increase in potential energy is reduced, which further indicates that Re atoms can reduce the PE, thus enhancing the stability of the model.

The γ/γ′ interface of the NSC has the ability to inhibit dislocation movement [11]; thus, it is necessary to analyze the dislocation display in the model system. The dislocation displays of the four models under tensile load are shown in Figure 9. The total lengths of all types of dislocation loops in Figure 9a–d are 6782 Å, 6466 Å, 6423 Å, and 5391 Å, respectively. Because the volumes of the four models are the same, the dislocation densities of the four models decrease gradually under the tensile load. According to the change in the total length of the dislocation loops, the formation of new dislocations is relatively reduced after Re atoms are integrated into the model system, indicating that Re atoms have the ability to hinder the dislocation movement. Furthermore, with the increase in Re atoms, the dislocation movement is further hindered, and the stability of the model is improved.

The above analysis results only reveal the change mechanism of the total length of the dislocation loops, while the process analysis of dislocation movement at the phase interface has not been performed. Therefore, it is necessary to analyze the dynamic process of dislocation movement in each model and analyze the variation characteristics of the atomic structure state of the model. The MD simulation results of the dislocation loop length of NSCs containing 0 wt.%, 2 wt.%, 4wt.%, and 6 wt.% Re under tensile load are shown in Figure 10, Figure 11, Figure 12 and Figure 13, respectively. There are six small images in Figure 10, Figure 11, Figure 12 and Figure 13, and the calculation steps of these six small images are 200, 5000, 10,000, 15,000, 20,000, and 25,000. In addition, statistical analyses are conducted on the HCP structures and OISs during the MD simulation, and the detailed results are presented in

Table 2. The results of MD simulation.

Calculation Steps	0 wt.% Re		2 wt.% Re		4 wt.% Re		6 wt.% Re
	HCP	OIS	HCP	OIS	HCP	OIS	HCP	OIS
200	95	40,446	89	40,003	73	39,921	0	39,222
5000	156	41,274	151	40,869	114	40,454	32	39,952
10,000	279	41,721	164	40,994	118	40,525	78	40,227
15,000	321	41,951	221	41,399	173	40,876	96	40,680
20,000	594	42,996	312	41,983	309	41,375	243	41,091
25,000	969	44,307	580	43,226	538	42,227	458	41,287

. It can be seen in Figure 10, Figure 11, Figure 12 and Figure 13 and Table 2 that with the increase in tensile damage, the total length of the dislocation loops increases, and the number of HCP structures and OISs also show an increasing trend. When the Re content increases, the dislocation loop length decreases, and the number of HCP structures and OISs also decreases. This indicates that the inhibition of interface dislocation networks on the dislocation movement is enhanced [32]; thus, the tensile deformation resistance of the material is improved.

The simulation results in Figure 10, Figure 11, Figure 12 and Figure 13 only reveal the process change mechanism of dislocation movement at the phase interface of NSCs under tensile load; the microstructural evolution behaviors of the phase interface under compressive load are not clear. Therefore, it is necessary to analyze the dynamic process of dislocation movement in each model and analyze the variation characteristics of the atomic structure state of the model under compressive load. The MD simulation results of the dislocation loop length of NSCs containing 0 wt.%, 2 wt.%, 4wt.%, and 6 wt.% Re under compressive loading are shown in Figure 14, Figure 15, Figure 16 and Figure 17, respectively. There are also six small images in Figure 14, Figure 15, Figure 16 and Figure 17, and the calculation steps of these six small images are the same as those for Figure 10, Figure 11, Figure 12 and Figure 13. The results of the statistical analysis on HCP structures and OISs are presented in

Table 3. The results of MD simulation under compressive load.

Calculation Steps	0 wt.% Re		2 wt.% Re		4 wt.% Re		6 wt.% Re
	HCP	OIS	HCP	OIS	HCP	OIS	HCP	OIS
200	1683	63,027	1102	61,909	1080	57,631	817	54,442
5000	1715	63,425	1216	62,435	1207	59,275	1143	56,690
10,000	3205	65,899	2205	62,739	2106	61,877	2001	60,088
15,000	3620	69,421	2306	64,425	2232	62,496	2102	61,126
20,000	8318	70,125	6410	66,368	5056	63,798	5004	62,207
25,000	9781	73,118	9229	69,239	7386	67,802	7182	66,903

. It can be seen in Figure 14, Figure 15, Figure 16 and Figure 17 and Table 3 that with the increase in compressive damage, the total length of the dislocation loops increases [25], and the number of HCP structures and OISs also show an increasing trend. When the Re element is integrated into the model, the variation trend of the dislocation loop length, HCP structures, and OISs is consistent with the results of the MD simulation under tensile load. According to the above simulation results, it can be concluded that it needs more energy to promote the motion of Re atoms, and the integration of Re atoms reduces the PE, which contributes to the stability of the model. With the increase in Re atoms, the inhibition of interface dislocation networks on the dislocation movement is further enhanced under the condition of tension damage or compression damage, which is beneficial to improve the plastic deformation resistance of the material [32].

The microstructure evolution states of NSCs at different stages have specific non-linear characteristic parameters [15], which are proportional to the density of the interface dislocation networks. To fully explore the impact of the Re element on the density of the interface dislocation networks, it is necessary to conduct a non-linear ultrasonic experiment on the prepared qualified NSCs under non-damage conditions. The Lamb wave incident and stimulation signals of the new NSCs containing 2 wt.%, 4 wt.%, and 6 wt.% Re are shown in Figure 18a–d, and the corresponding non-linear characteristic parameters are 0.6671 e⁻⁴/mm, 0.7238 e⁻⁴/mm, 0.7773 e⁻⁴/mm, and 0.9912 e⁻⁴/mm (see Figure 19), respectively. It can be seen that with an increase in the Re content, the density of the interface dislocation networks of the NSC increases [32], which enhances the non-linear sound contact effect, resulting in the increasing trend of the corresponding non-linear characteristic parameter. This research result will provide technical reference for further exploring the micro-structural evolution behaviors of the phase interface of the new NSC.

4. Conclusions

In order to fully reveal the microstructural evolution behaviors of the phase interface of the new NSC containing Re, MD simulation is used to analyze the four models (NSCs containing 0 wt.%, 2 wt.%, 4 wt.%, and 6 wt.% Re) under tensile and compressive loads, respectively. And then the non-linear characteristic parameters of the four NSC samples are tested using the non-linear ultrasonic experiment. The main results are as follows:

(1) A new Ni-Al-Re MD model of the NSC is established based on the optimal volume ratio (3:7) of γ phase to γ′ phase, which is closer to the real properties of the material in comparison with the traditional Ni-Al sandwich model.

(2) The integration of Re atoms in the MD model of the NSC reduces the PE, thus improving the stability of the model.

(3) Under tension load or compressive load, the number and total length of the new dislocation loops gradually decrease with the increase in Re atoms. In addition, the number of HCP structures and OISs formed by the destruction of the intact FCC structures also decreases.

(4) With the increase in the Re content, the inhibition of interface dislocation networks on the dislocation movement is further enhanced, which is beneficial to improve the plastic deformation resistance of NSC.

(5) In the non-linear ultrasonic experiment, the density of interface dislocation networks of the NSC increases with the increase in the Re content, which enhances the non-linear acoustic contact effect and leads to an increase in the corresponding non-linear characteristic parameters. This conclusion provides a technical reference for the further study of the micro-defects of NSCs using ultrasonic techniques.