Trans. Nonferrous Met. Soc. China 26(2016) 1073-1078

Quantitative calculation on atomic site occupation during precipitation of Ni₃(Al_1-xFe_x) by microscopic phase-field study

Hai-hong LIAO, Min-jie LIANG, Pei-kang BAI

School of Materials Science and Engineering, North University of China, Taiyuan 030051, China

Received 8 December 2015; accepted 26 February 2016

Abstract: Microscopic phase-field method was used to simulate the site occupation of a series of alloys with a stoichiometric composition of Ni₇₅Al_25-xFe_x (x=0, 5-10) aged at 1273 K. With the change of Fe content, quantitative calculations were made on each atomic site occupation probability (SOP) in L1₂-Ni₃ (Al_1-xFe_x), so as to find out the dynamic response law. The result of the study shows that, with the increase of Fe content, the Fe atom preferentially occupies the B sites (corner sites of FCC) with its SOP value being increased gradually, and the SOP of the Al atom on the B sites is greatly decreased. Meanwhile, Al_Ni and Fe_Ni anti-sites form in the precipitation of L1₂ phase. Moreover, with the increase of Fe content, the formation of Al_Ni and Fe_Ni anti-sites becomes much easier. In addition, it has been found that the instantaneous dynamic evolution of the atomic SOP is completed at the early stage of the growth of L1₂ phases.

Key words: intermetallics; Ni₇₅Al_25-xFe_x alloy; atomic site occupation; precipitation; microscopic phase-field

1 Introduction

Atomic site occupation behavior in ordered intermetallic compounds is of great practical interest in connection with their high-temperature applications and mechanical properties [1,2]. In particular, Ni₃Al intermetallic compound with L1₂-type structure was extensively focused. Various studies have shown that the addition of certain alloying elements can effectively improve its mechanical properties and the effect of the improvement depends on the site occupation of ternary alloying element in Ni₃Al [3-5]. Thus, the investigation of the atomic site occupation behavior in Ni₃Al intermetallic compound is extremely useful in controlling and optimizing alloy properties.

Ni₃Al has L1₂ type face-centered cubic (FCC) structure, in which Ni and Al atoms occupy face-centers (A sites) and corners sites (B sites) of the FCC unit cell, respectively. However, the site occupation behavior of the different alloying elements in Ni₃Al exhibits great diversities [4-8]. Fe is one of the most important alloying elements for Ni₃Al. It can not only improve the ductility and yield strength of alloy, but also drastically affect the electrical and magnetic properties [1,4,6]. Several ways have recently been used to study the site preference of Fe in Ni₃Al, such as atom probe field-ion microscopy, X-ray diffraction measurement, Mossbauer spectroscopy, the statistical thermodynamic calculation and the first-principles calculations [1,3-9]; however, there are still some controversies about site preference of Fe and the factors affecting site occupation in Ni₃Al-Fe alloy. For example, IVANOVSKIA et al [4] believed that Fe atoms have preference for Ni sites in nickel deficient Ni₃Al alloys. NICHOLLS and RAWIINGS [10] have indicated a weak composition-dependent preference for the Al sublattices. SHINDO et al [11] revealed no discernible site preference in Ni₃Al-Fe system. WU et al [12] showed that site distribution of Fe is a function of the concentration, as well as the matrix composition and temperature. ALMAZOUZI et al [8] indicated that Fe occupies mostly Al sites, regardless of Fe concentration and the matrix composition. By using the Wagner-Schottky model in combination with first-principles calculations, JIANG and GLEESON [13] found that the site preference of Fe strongly depends on the temperature.

The previous investigations were mainly focused on the site preference of Fe. Very limited experimental data about the effects of Fe on atomic site occupation in L1₂-Ni₃(Al_1-xFe_x) can be found in the literature and the temporal evolution of atomic site occupation probability (SOP) and anti-site behavior are also not well revealed.

The purpose of the present study is to predict the effect of Fe addition on the atomic SOP during the precipitation process of the L1₂-Ni₃(Al_1-xFe_x) with respect to stoichiometric Ni₇₅Al_25-xFe_x alloy aged at 1273 K. By using the microscopic phase-field model, the temporal evolution of atomic SOP and the anti-sites behavior will be revealed in detail with the varying of Fe content.

2 Microscopic phase-field kinetics model

In recent years, phase-field model has been used extensively by the materials scientists. The temporal morphology and the dynamics processes can also be well presented [14-16]. Microscopic phase-field kinetics model was firstly proposed by KHACHATURYAN [17], and is a deterministic phase-field method based on Onsager and Ginzburg-Landau theory. It relates composition to long range order parameter (LRO) by nonequilibrium free energy of atom single-site occupation probability function, and can be used to solve the kinetic equations of ordering and diffusion. Nonhomogeneous system can be simulated with microscopic phase-field kinetics model at atomic-scale, with which, the precipitation caused by atom jumpiness at crystal lattice site can be described [17,18].

Microscopic phase-field dynamic model is used to describe the atomic configurations and phase morphologies by single-site occupation probability functions P(r,t), which is the probability that a given lattice site r is occupied by an atom at time t. In this model, the atomic jump is supposed as a direct exchange instead of the vacancy mechanism, and the lattice defects, such as the dislocation, vacancy, are neglected. The change rates of these probabilities are linearly proportional to the thermodynamic driving force [17,18].

     (1)

where F is the total free energy function of P(r',t), and L(r-r′) is the symmetry matrix of the microscopic kinetics related to the probability of an elementary diffusion jump from r to r' per unit of time.

Microscopic diffusion equation of the ternary system was developed by CHEN et al [18,19]. In this model, P_A(r,t), P_B(r,t) and P_C(r,t) represent the probabilities of locating an A, B or C atom, respectively. Since P_A(r,t)+P_B(r,t)+xP_C(r,t)=1, only two equations are independent at each lattice site. So, the microscopic diffusion equation of ternary system can be obtained. In order to describe the nucleation, a random noise item was added to the right-hand side of the equation, and then the microscopic Langevin equations become

      (2)

where L_αβ(r－r′) is a constant related to the exchange probabilities of a pair of atoms, α and β(= A, B or C), at lattice site r and r' per unit time, ζ(r,t) is the noise term which is taken to be Gaussian-distributed and its correlation properties meet the requirements of the fluctuation dissipation theory, k_B is the Boltzmann constant, and F is the total free energy of the system, based on the mean-field approximation [14]:

                         (3)

where V_aβ(r－r') is the interaction energy between a and β at lattice sites r and r'.

Substituting Eq. (3) into Eq. (2), Fourier transforming both sides of the kinetic Eq. (2) gives

     (4)

where k is the lattice site vector in the reciprocal space,  are Fourier transforms of corresponding functions in the real space.

In this simulation, Eq. (4) is solved by Euler method, and then two-dimensional (2-D) projection of a three-dimensional (3-D) system along the [001] direction is used. Finally, the temporal evolution information of atomic site occupation probability is acquired as a function of the aging time.

3 Results and discussion

3.1 Temporal evolution of atomic site occupation probability

Figure 1 shows the temporal evolution of atomic SOP on A and B sites in L1₂ phase with the increase of Fe content. At the content of 0 Fe (x_Fe), before 2000 time steps, the alloy system is disordered; this stage is the incubation period of precipitation, thus the SOP of Al atoms on A and B sites are identical on all sites. The incubation period is followed by an ordering process, i.e., the SOP of Al atoms on B sites starts to increase quickly, and reaches the maximum at the 5500 time steps (Fig. 1(a)). On the contrary, their SOP on A sites decreases quickly to the minimum (0.007) and then keeps unchanged (Fig. 1(c)), which indicates Al atoms almost fully occupy B sites. However, with increasing Fe content, the SOP of Al atoms on B sites obviously decreases and the atomic ordering is obviously postponed. In the meanwhile, it is further found that a very small amount of Al atoms begin to occupy Ni sites, indicating that the Al_Ni anti-site behavior occurs. Also, Al_Ni anti-site behavior becomes easier and easier with the addition of Fe. Furthermore, it is found that Fe atoms prefer to occupy B sites (Fig. 1(b)), and a few of Fe atoms occupy A sites, exhibiting the anti-site behavior (Fig. 1(d)), and their SOP gradually rises not only on B sites but also on A sites. The results of the atomic SOP in L1₂-Ni₃(Al_1-xFe_x) with the increase of Fe content at the final equilibrium state, are listed in Table 1.

From Table 1, it can be seen that for the stoichiometric Ni₇₅Al_25-xFe_x alloy, Fe and Al atoms occupied B sites in common, whereas the dependence of the anti-site behavior of Al atoms on composition is relatively weak. The anti-site behavior of Fe atoms becomes stronger and stronger with increasing Fe content.

Fig. 1 Temporal evolution of atomic SOP on B sites

Table 1 At equilibrium state, variations laws of atomic site occupation probability in L1₂-Ni₃(Al_1-xFe_x) with increasing Fe content

3.2 Evolution of volume fraction of L1₂-Ni₃(Al_1-xFe_x)

Figure 2 displays the variation of the volume fraction of L1₂-Ni₃(Al_1-xFe_x) as a function of time. For x_Fe=0.05% alloy, due to the addition of noise item in high temperature, the incubation period of the precipitation shows a disordering fluctuation before 5000 time steps. Then, the volume fraction of L1₂ phases increases quickly and reaches the maximum at the 15000 time steps, which corresponds to a quick growth stage of L1₂ phases. Within 15000 to 20000 time steps, the volume fraction increases slowly and gradually tends to be equilibrium, which is the coarsening stage of L1₂ phases. From Figs. 1(b) and (d) and Fig. 2, it is noticed that during the incubation period of the precipitation, the SOP of Fe atoms on both A and B sites is also abnormal which is consistent with the state of the volume fraction. From 5000 to 10000 time step, the SOP of Fe atoms on B sites gradually increases, and decreases on A sites. At the later stage of the growth of L1₂ phases (10000-15000 time steps), the SOP of Fe atoms on both A and B sites approaches to the equilibrium, and then keep unchanged. As a result, it shows that the site occupation of Fe atoms is already accomplished before the growth of L1₂ phase finishes. Furthermore, with increasing Fe content, the time of the SOP of Fe atoms arriving at the equilibrium is much earlier than that of the volume fraction of L1₂ phase, which indicates that the evolution of atomic SOP is already completed at the much earlier stage of the growth of L1₂ phases, and there is almost no change at the later stage of the growth and coarsening stage of L1₂ phases.

Fig. 2 Variation of volume fraction with different Fe contents

3.3 Variation of long-range order (LRO) parameter and concentration

Figure 3 shows the variation of LRO parameter and concentration of Al atoms in L1₂ phase with x_Fe=0 and 0.10% alloys. From Fig. 3(a), it can be seen that at 3000 time steps, the LRO parameter shows the larger fluctuation and continuous increase, and then reaches the equilibrium value (about 1) at 4800 time steps. Then, the curves of LRO parameter keep almost unchanged in their height although the width is broadened, which corresponds to the coarsening stage of L1₂ phases. Whereas, at the initial stage of aging, the value of concentration only exhibits smaller fluctuation and slower decrease, and finally reaches the equilibrium value (Fig. 3(b)). From the variation of LRO parameter and concentration curves, it can be indicated that the precipitation mechanism of L1₂ phases has the characteristic of spinodal decomposition.

However, for x_Fe=0.10% alloy, the fluctuation of LRO parameter is postponed (Fig. 3(c)), and the equilibrium values of LRO parameter of both Al and Fe atoms in L1₂ phases are significantly decreased, as shown in Fig. 4, which indicates that ordering process of L1₂ phases gradually slackens and their ordering degree obviously descends. Meanwhile, the corresponding concentration curves also present larger fluctuations in the center (Fig. 3(d)). From above analysis, it can be concluded that the formation of L1₂ phases shows more characteristics of non-classical nucleation and growth with increasing Fe content. The addition of Fe obviously changes the precipitation mechanism of L1₂ phases and makes their ordering degree gradually to be decreased.

Some experiments on Ni₃Al-Fe at 1273 K by SHINDO et al [11] indicated that Fe showed preference on Al sublattices, and the fraction of Fe occupied Al sublattices strongly depended on alloy composition. JIANG and GLEESON [13] also showed that Fe preferred to occupy Al sublattices in Ni-riched Ni₃Al at 1273 K. Further, the atomic anti-site behavior was found in binary Ni₃Al alloy [4,20], which resulted in the formation of Al anti-site defects. From the view of formation enthalpy, JIANG and GLEESON [13] and FU and PAINTER [20] also explained the formation of three structure types of anti-site defects, including Al_Ni and X_Ni anti-sites. In this work, our simulation results are consistent with the experimental and theoretic results very well.

Fig. 3 LRO and concentration profiles

Fig. 4 Variation of equilibrium value of LRO parameter of Al atoms (a) and Fe atoms (b) in L1₂ phases with increasing Fe content

4 Conclusions

1) In stoichiometric Ni₇₅Al_25-xFe_x alloy, Fe atoms prefer to occupy B sites, so the formed L1₂ phases are a complex Ni₃(Al_1-xFe_x). With the increase of Fe content, the SOP of Fe atoms on B sites increases gradually, while the case for Al atoms is opposite.

2) Anti-site behaviors including Al_Ni and Fe_Ni appear in precipitation process of L1₂ phases. Meanwhile, with increasing Fe content, Al_Ni and Fe_Ni anti-sites all tend to increase, whereas the dependence of Al_Ni anti-sites on composition is relatively weak.

3) Evolution of the atomic SOP is already completed at the early stage of the growth of L1₂ phases, while the SOP almost keeps unchanged at the later stage of the growth and the coarsening of L1₂ phases.

4) As Fe content increases, the precipitation mechanism of L1₂-Ni₃(Al_1-xFe_x) changes from spinodal decomposition to non-classical nucleation and growth. The lower ordering degree of L1₂ phases appears with the continuous increase of Fe content.

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定量计算Ni₃(Al_1-xFe_x)沉淀中原子占位的微观相场研究

廖海洪，梁敏洁，白培康

中北大学 材料科学与工程学院，太原 030051

摘  要：采用微观相场方法，模拟研究了化学计量比为Ni₇₅Al_25-xFe_x(x=0, 5~10)系列合金在时效温度为1273 K时的原子占位情况。通过该方法，定量计算了L1₂-Ni₃(Al_1-xFe_x)沉淀相中各原子的占位几率(SOP)，并获得了其随Fe含量变化的动态响应规律。研究结果表明：随着Fe浓度的增加，Fe原子优先占据B格点位置(FCC结构的角位)，且其原子占位几率数值逐渐增高；Al原子在B格点位置的占位几率则明显降低。同时，在L1₂相沉淀过程中出现了Al_Ni 和Fe_Ni反位现象；且随着Fe含量的升高，Al_Ni 和Fe_Ni原子反位的形成将会变得更加容易。原子占位几率的瞬时动态演化早在L1₂相长大的初期阶段就已完成。

关键词：金属间化合物；Ni₇₅Al_25-xFe_x 合金；原子占位；沉淀；微观相场

(Edited by Yun-bin HE)

Foundation item: Project (2013011014-1) supported by the Natural Science Funds of Shanxi Province, China; Project (2009021028) supported by Science and Technique Foundation for Young Scholars of Shanxi Province, China

Corresponding author: Hai-hong LIAO; Tel: +86-351-3557392; E-mail: 499257098@qq.com

DOI: 10.1016/S1003-6326(16)64204-8