Trans. Nonferrous Met. Soc. China 25(2015) 2361-2366

Simulation of pre-precipitation in Ni₇₅Al₁₄Mo₁₁ alloy by microscopic phase-field model

Jing-jing LIANG, Rui-qin LI, Yao-Hong ZHAO

School of Mechanical and Power Engineering, North University of China, Taiyuan 030051, China

Received 8 January 2015; accepted 10 June 2015

Abstract: The early precipitation process of Ni₇₅Al₁₄Mo₁₁ alloy was simulated by microscopic phase-field model at different temperatures. The microstructure of the alloy, the precipitation time of L1₀ structure and occupation probability of the three kinds of atoms were investigated. It is indicated that the non-stoichiometric L1₀ (I/II) phases are found in the precipitation process. With the temperature increasing, the appearance time of L1₀ is brought forward. The L1₀ (II) structure always precipitates earlier than the L1₀ (I) structure. Compared with lower temperature, higher temperature brings the formation time of L1₀ phase forward and makes L1₀ phase have a higher order degree. But lower temperature shortens the process time of the L1₀ phase to the L1₂ phase. Al and Mo atoms tend to occupy γ site, Ni atom tends to occupy α and β sites. At the same temperature, Al atom has stronger occupation ability than Mo atom in the same site. Ni, Al and Mo collectively form the composited L1₂ structure.

Key words: Ni₇₅Al₁₄Mo₁₁ alloy; microscopic phase-field model; precipitation incubation period; L1₀(I/II) phase

1 Introduction

γ′ phase which has the L1₂ phase is the main strengthening phase of Ni-based alloys, the existence of which endows the Ni-based alloys with excellent properties at high temperatures. In Ni₇₅Al₁₄Mo₁₁ alloy, the L1₂ structure is a highly symmetric face-centered cubic lattice, the Ni atoms occupy the eight corner points in the face-centered cubic lattice and the Al atoms occupy the face-centered points [1,2]. Phase-field simulation has been widely used to investigate the phenomenon in material science. For instance, CHEN et al [3] used phase-field modeling to study α phase transformation in Ti-Al-V alloy, and found that the results agreed well with the DICTRA simulations. YAMANAKA et al [4] simulated the microstructural formation and deformation behavior of ferrite-pearlite. They found that it could predict the formation and morphological change of α phase in the Fe-C alloy during the γ→α transformation. KOYAMA [5] utilized the phase-field modeling to demonstrate the microstructure changes in magnetic materials, such as Ni₂MnGa ferromagnetic shape memory alloy, Fe-Pt nanogranular thin film, Co-Sm-Cu rare-earth magnet, and Fe-Cr-Co spinodal magnet [5]. Phase-field model is also used in nucleation and grain growth [6-9], bimodal particle size distribution [10], rafting [11], precipitation [12] and coarsening [13]. Some researches have been done [14-20] in Ni-Al-X (X=metallic element) alloys. Few works about Ni-Al-Mo alloy have been performed so far [12].

In the pre-precipitation process, atoms diffusion appears in the L1₂ phase. Therefore, a variety of transient phases could appear in this process. The main purpose of this work is to find the transitional phase in the early stage of γ′ precipitation and the relationship between temperature and transitional phase.

2 Theoretical model

In the phase-field model, all phases or domains in the matrix are characterized by field variables, compositions and order parameters. These field variables are continuous across the interface regions. Microscopic phase-field model based on the diffusion equations which are the discrete lattice forms of the Cahn-Hilliard equation is firstly proposed by KHACHATURYAN [21] and developed by PODURI and CHEN [22,23] for the binary and ternary real alloy systems. Equations of ternary alloy systems are

    (1)

where L_αβ(r-r′)(α, β=A, B or C) is a constant which expresses the exchange probabilities between a pair of atoms, α and β, at lattice sites r and r′ per unit time, F is the free energy, T is the temperature, k_B is the Boltzamann constant, ξ(r, t) is the thermal nose which is assumed to be Gaussian-distributed with average value of zero, t is the aging time, P_A(r, t), P_B(r, t) and P_C(r, t) stand for atom (A, B, C) occupation probabilities at a given lattice site r and a given time t. In ternary alloy systems, P_A(r, t)+P_B(r, t)+P_C(r, t)=1, and the free energy F can be approximately expressed by the mean-field theory as

                          (2)

where V_αβ(r-r′) expresses the effective exchange interaction energy between α and β (α, β=A, B, C), which contains chemical interaction energy V_αβ(r-r′)_ch and elastic energy V_αβ(r-r′)_el [24]:

        (3)

Using four-neighbor atoms interaction energy is more precise in describing V_αβ(r-r′) than two-neighbor atoms interaction energy for three ternary alloys.

In the reciprocal space, V_αβ(r-r′) is expressed as

                (4)

Substituting Eqs. (2), (3) and (4) into Eq. (1), a ternary alloy kinetic Eq. (5) in reciprocal space is given:

(5)

where

V_AB(k), V_BC(k), and V_AC(k) are the Fourier transformations of the corresponding functions in the real space.

3 Simulation results

3.1 Transformation process of L1₀ to L1₂

Figure 1 indicates the ordered crystal structures of L1₀phase, L1₂ phase and their projections along [001] direction. The white ball indicates the Al atom and the black ball indicates the Ni atom. The L1₀ phase has two projection styles, the L1₀ (II) phase is received by rotating the L1₀ (I) structure by 90° along the tetrad-axis in [010] orientation.

Figures 2 and 3 show the microstructure evolution processes at 873 K and 1073 K, respectively. The white grid point expresses the occupation probability of Al atom, the lighter the white color is, the greater the probability is.

Fig. 1 2D structure projections along [001] of projection images of 3D order structure and its typical image in atomic evolution figure

Fig. 2 Microstructure evolution of Ni₇₅Al₁₄Mo₁₁ at 873 K

Fig. 3 Microstructure evolution of Ni₇₅Al₁₄Mo₁₁ at 1073 K

From Figs. 2 and 3, we can observe that the L1₀ (II) phase (Fig. 2(b) and Fig. 3(b), the enlargements in the white circle are shown Fig. 2(h) and Fig. 3(h)) precipitates earlier (t=8700) from the disordered matrix than the L1₀(I) structure (Fig. 2(c) and Fig. 3(c), the enlargements in the white circles are shown in Fig. 2(i) and Fig. 3(i), t=9500) at 873 K or 1073 K. Ahead of this step, in the disordered matrix, the atoms cluster firstly at t=5600, then the short-range ordered phase appears, along with aging proceeding (t=8700-16100), the L1₀ and L1₂ phases appear. At last, a majority of L1₀ structure transforms into L1₂ phase and a portion of it disappears.

Figures 2 and 3 are at the same steps, but the white grid points in Fig. 2 are lighter than those in Fig. 3. This shows that the phase in Fig. 2 has higher ordering extent at 1073 K.

3.2 Effect of temperature on precipitation of L1₀ phase

In order to investigate the effect of temperature on the L1₀ precipitation, five temperatures were chosen to observe the effect on the aging of Ni₇₅Al₁₄Mo₁₁ alloy. From Fig. 4, we can observe that as the temperature increases, the precipitation reveals regularity. The structure firstly precipitated is the L1₀ (II), and the increase of the temperature does not change the precipitation sequence of the L1₀ (II) and L1₀ (I).

Fig. 4 Effect of temperature on pre-precipitation of Ni₇₅Al₁₄Mo₁₁ alloy

Figure 5 shows the microstructure evolution of the Ni₇₅Al₁₄Mo₁₁ alloy. At this time, the L1₀ (II) and L1₀ (I) reach the maximal volume fraction. Obviously, we can find that the volume fractions of two different projection styles are almost the same under different temperatures. Therefore, the effect of temperature on the L1₀ structure’s volume fractions of two styles is weak in Ni₇₅Al₁₄Mo₁₁ alloy.

3.3 Occupation probability evolution of Ni, Al and Mo atoms at pre-precipitation stage

From Fig. 6, we can have a further realization about the transformation process of L1₀ to L1₂. The α, β and γ sites are labeled in Fig. 1. The rise or decline part of the curve indicates the transformation process. At α site, Al and Mo atoms firstly undergo balance stage, then the curve starts to descend, but they are chaotic at 873, 923, 1023 and 1073 K. The tardiest transformation occurs at 973 K compared to the other temperatures. At β site, Al atom goes through a transient rising stage, this rise of curve signs the formation of L1₀, at a higher temperature, the curve has a distinct rising trend and the time of rise is brought forward (this agrees with Fig. 4), which indicates that at the same time step, the L1₀ phase has a better order degree. From Figs. 1(a) and (b), at this time, the occupation probability of Ni atom at the same site should decline accordingly, and it is demonstrated in Fig. 6(h). Along with aging, the curve declines, and the L1₀ phase starts to transform to the L1₂ phase. Compared with lower temperature, the same transformation at a higher temperature comes up later.

Fig. 5 Microstructure evolution of Ni₇₅Al₁₄Mo₁₁ alloy at different temperatures

Fig. 6 Atom occupation probability curves of Al (a, b, c), Mo (d, e, f) and Ni (g, h, i) atoms at α (a, d, g), β (b, e, h) and γ (c, f, i) sites

The Mo atom has the similar tendency with the Al atom. The Al atom has a higher number than the Mo atom at the same time step and the same temperature. They tend to occupy the γ site.

4 Conclusions

1) By utilizing the microscopic phase-field model, the aging of Ni₇₅Al₁₄Mo₁₁ alloy was simulated at 873, 923, 973, 1023 and 1073 K, and it is found that the L1₀ phase precipitates with L1₀ (I) and L1₀ (II) two projection styles.

2) The temperature does not change the precipitation sequence of L1₀ (I) and L1₀ (II). The L1₀ (II) structure always precipitates earlier than the L1₀ (I) structure. Temperature has little effect on the volume fractions of L1₀ (I) and L1₀ (II). Compared with lower temperature, higher temperature brings the formation time of L1₀ phase forward and makes L1₀ phase have a higher order degree. But lower temperature shortens the process time of the L1₀ phase to the L1₂ phase.

3) Al and Mo atoms tend to occupy the γ site, Ni atom tends to occupy α and β sites. At the same temperature, Al atom has stronger occupation ability than Mo atom in the same site. Ni, Al and Mo collectively form the composited L1₂ phase.

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Ni₇₅Al₁₄Mo₁₁ 合金早期沉淀过程的微观相场法模拟

梁晶晶， 李瑞琴，赵耀红

中北大学 机械与动力工程学院， 太原 030051

摘  要：采用微观相场动力学模型研究不同温度下Ni₇₅Al₁₄Mo₁₁合金的早期沉淀过程, 研究合金的微观结构、L1₀相的析出时间以及3种原子的占位概率。结果表明：沉淀过程中析出L1₀非化学计量比有序相，L1₀相有I型和II型2种结构，随着温度的增加，L1₀相析出的时间提前。在沉淀的过程，II型L1₀结构的析出时间比I型L1₀结构的析出时间早。温度升高缩短了L1₀相的形成时间，使L1₀相有序度更高；温度越低，L1₀相向L1₂相的转变时间越短。Al原子和Mo原子占据 γ 位， Ni原子占据α位和β位，在同样的温度和格点下，Al原子的占位几率大于Mo原子的占位几率。Ni、Al和Mo 3种原子构成复合L1₂相。

关键词：Ni₇₅Al₁₄Mo₁₁合金；微观相场模型；沉淀孕育期；L1₀ (I/II) 相

(Edited by Yun-bin HE)

Foundation item: Project (51275486) supported by the National Natural Science Foundation of China

Corresponding author: Rui-qin LI; Tel: +86-13835113159; E-mail: liruiqin@nuc.edu.cn

DOI: 10.1016/S1003-6326(15)63851-1