Phase-field simulation of phase separation in

Ni₇₅Al_xV_25-x alloy with elastic stress

LI Yong-sheng(李永胜), CHEN Zheng(陈 铮), LU Yan-li(卢艳丽), WANG Yong­-xin(王永欣)

 State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China

Received 28 July 2006; accepted 15 September 2006

Abstract: The phase separation in Ni₇₅Al_xV_25-x alloys incorporated with the elastic stress was investigated using the microscopic phase-field model. The final morphology of γ′ and θ is similar in spatial alignment, but the volume fraction of γ′ phase increases and that of θ decreases as the Al concentration increases. For the small elastic interactions of early-stage phase separation, the coarsening of γ′ and θ can be approximated by a linear growth law as predicated by Lifshitz and Slyozov and Wangner (LSW) theory. As the elastic interactions increase at late-stage coarsening, the growth rate decreases, and the growth presents quick increase at early-stage and slows down at late-stage.

Key words: Ni₇₅Al_xV_25-x; phase separation; elastic stress; microscopic phase-field

1 Introduction

It is known that the intermetallic precipitation phases play an important role in enhancing the mechanical properties of alloy, such as the distribution and size of the Ni₃X style precipitates in the Ni-based superalloys [1,2]. Generally, the lattice styles of produced phase and matrix are different, thus the elastic stress is induced due to the lattice mismatch between the precipitates and matrix. The elastic interactions are the key factor that controls the domain structure and its temporal evolution. Furthermore, the kinetics of growth and coarsening are altered by the elastic interactions [3-4]. While the effect of elastic interactions on the phase separation dynamics is not yet clarified, and the studies are mostly focused on the binary system. So it is important to predict the kinetics of phase separation[5-6]. The phase-field model has been used extensively by the materials researchers in recent years[7-9]. The temporal morphology can be obtained using the phase-field simulation, and the dynamic processes can also be presented. The phase-field model is adopted in this paper, and the dynamic behavior of two phases systems including elastic interactions are investigated.

The structural change of happens in the γ(fcc)→γ′(L1₂)+θ(D0₂₂) eutectoid reaction in pseudobinary Ni₃(Al, V) alloys. Both the L1₂ and D0₂₂ are A₃B-type ordered structures, and the D0₂₂ structure can be characterized by the periodic array of the antiphase boundaries introduced in the L1₂ structure. The elastic stress is induced during structure change due to the different lattice styles of precipitates and matrix. In the simulation, the elastic moduli difference between different phases is neglected, which is the homogeneous elastic modulus approximation proposed by KHACHATURYAN[10].

We focused on the phase separation dynamics of Ni₇₅Al_xV_25-x alloys in this paper. The growth and coarsening dynamics of precipitates incorporated with the elastic interactions were discussed qualitatively. The morphology and volume fraction were also described.

2 simulation methods

2.1 Microscopic phase-field model

The microscopic phase-field dynamic model originated from the Onsager-type diffusion equations proposed by KHACHATURYAN[10]. In the model, the atomic configurations and morphologies of a ternary alloy are described by single-site occupation probability functions,,, which represent the probability of finding an A, B or C atom at a given lattice siteand at a given time t. For ternary system, ，only two equations are independent at each lattice site. In order to describe the nucleation, a random noise item is added to the right-hand side of the equation. Thus the microscopic Langevin equations are given by

      (1)

where is a constant related to the exchange probabilities of a pair of atoms, α and β, at lattice site r and r′ per unit time, α, β =A, B or C. k_B is the Boltzmann constant; is assumed to be Gaussian-distribution with an average value of zero, which is uncorrelated with space and time, and it obeys the so-called fluctuation dissipation theory; F is the total free energy of the system,

                            (2)

whereis the interaction energy between α and β at the lattice sites r and r′, including short-range chemical interaction  and long-range  strain-induced interact- ion .

2.2 Microscopic elastic field

In the linear elasticity theory of multi-phase coherent solid developed for the homogeneous modulus case, the strain energy generated by an arbitrary concentration or structure heterogeneity can be presented as a function of the concentration or long range order parameter fields. Therefore, it can be easily incorporated in the field dynamic equations. If we assume that the strain is predominantly caused by the concentration heterogeneity, which is the case for the Ni-based superalloys, and if the Vergard’s law (the lattice parameter is linearly dependent on composition) is fulfilled, the configuration-dependent part of the strain energy can be expressed in an extremely simple form of pairwise interaction[11]. The long-range strain-induced interaction associated with an arbitrary atomic distribution p(r) is given by

               (4)

For the decomposition process is determined by development of a packet of concentration waves with wave vectors close to zero, the long-wave approximation for the Fourier transformation of  can be used:

          (5)

where  is the unit vector in the reciprocal space, and () are the Cartesian coordinates of a unit vector n,  is the concentration coefficient of crystal lattice parameter, a(c) is the lattice parameter of a solid solution with concentration c, a₀ is the lattice parameter of pure solvent, and is the average of .

                 (6)

where characterizes the elastic anisotropy of the system, C_i,j are the elastic constants of Ni-based solid solution, and C₁₁=204.2, C₁₂=147.5, C₄₄=92.6GP at 1 100 K[12].

The elastic strain energy is given by a dimensionless parameter, where =1 200 is an input date,  is the local chemical free energy. The parameter ε₀ can be estimated from the lattice misfit and the concentration diversity, =0.045.

3 Results and discussion

3.1 Configuration

The ordered intermetallic precipitates L1₂D0₂₂ structure transformations in pseudobinary Ni₃(Al,_,V) alloys depend on the eutectoid composition. The D0₂₂→L1₂ structure transformation happens in the low concentration regions (x≤5), and L1₂→D0₂₂ in the high concentration regions (x≥6.5). The simulated pictures are depicted with different grey scheme. If the occupation probability of vanadium is 1.0, then the site is black, so D0₂₂ phases are black. If the occupation probability of aluminium is 1.0, then the site is black, so the L1₂ phases are black, and the color of matrix is gray. The simulation is performed with 128×128 mesh points. A periodical boundary condition is imposed along both dimensions.

Fig.1 shows the microstructure configuration of D0₂₂→L1₂ transformation in Ni₇₅Al_2.7V_22.3 alloy at 1 100 K. The γ′ phase nucleates at the interphase boundaries of θ phase and grows along the interphase boundaries, as shown in Fig.1(b). Due to the different structures of precipitates and the matrix, the elastic stresses increase with the growth of particles. The particles predominant alignment direction is obvious, as shown in Figs.1(c)-(d), the γ′ phase aligns along the <001> directions, and the θ phase aligns along the [100] direction. The final pattern of two phase is characterized by an alternating array of plates shape[13]. For the middle and high concentration regions, such as x=5.5 and 8.1, the final morphology is similar to that of x=2.7, as shown in Fig.2.

The differences of the phase separation for different concentration regions are the change of volume fraction of precipitates. Fig.3 shows the volume fraction as a function of Al concentration for γ′ phase and θ phase. It can be seen that the volume fraction of γ′ phase increases as the Al concentration increases, but that of θ phase decreases. The volume fraction variation of two phase can also be seen from the simulated pictures.

3.2 Growth and coarsening

In this segment, we will discuss the particles growth

and coarsening law at early-stage and late-stage. For the precipitates are nonspherical, the average particle radius (APR) as a valid representation of domain size, is approximated by R=, where S is the domain area. The at a given time is obtained by averaging over all the domains in the system. It is well known that the obeys a scaling form³~t in the absence of any elastic effects, i.e. the growth is consistent with LSW dynamics. However, this theory is valid only in the limit of zero volume fraction. At the early-stage of phase separation, the elastic stress is small and can be neglected. Therefore, the t^1/3 growth law may be valid.

Fig.4 shows the cube of average radius for the early-stage phase separation of γ′ and θ phase for alloy with x=2.7. It can be seen that the particles growth follows a linear power law, which accords with the LSW theory. In this stage, the maximum of the APR of γ′ phase is smaller than that of θ phase. The early-stage particles growth also presents the linear relationship for the high concentration alloy with x=8.1, as shown in Fig.5. But the maximum of the APR of γ′ phase is larger than that of the θ phase (Fig.5(a)), for the θ phase precipitates later than that of the γ′ phase in high concentration regions. GES et al [14] have analyzed the coarsening of γ′ in Ni-based superalloy for long ageing times, and they have found that the coarsening can be approximated by a linear growth for short ageing times. KAWASAKI and ENOMOTO [15] have also found that the t^1/3 law holds in the early stages even in the presence of elastic interactions; but in the later stages, coarsening is accelerated to t^1/2 kinetics if the particles are harder than the matrix. The simulated results in this paper are consistent with the experimental studies.

Fig.1 Microstructure evolution of Ni₇₅Al_2.7V_22.3 alloy at 1 100 K: (a) t*=3.4; (b) t*=14; (c) t*=120; (d) t*=240

Fig.2 Microstructure of alloy with x=5.5(a) and x=8.1(b)

Fig.3 Volume fractions of precipitates as function of Al concentration

As the growth and coarsening progress, the elastic stress increases due to the lattice mismatch between the precipitates and matrix. The phase separation dynamics are changed because now shape changes and coalescence events play an important role. Fig.6 shows the cube of APR of different concentration as a function of time. The growth rate of  γ′ and θ slows down with time going, such as the θ phase APR after t=410⁵ time steps for alloy with x=2.7, as shown in Fig.6(b). The phase separation exhibits the early and late-stage processes. So the late-stage coarsening disobeys the growth law predicted by LSW theory. SAGUI et al[16] and NISHIMORI et al[17] have also found the early and later stages for the coarsening process in the ternary elastic interaction system with simulation study.

Fig.4 Cube of radius as function of time for alloy with x=2.7: (a) γ′ phase; (b) θ phase

Fig.5 Cube of average radius as function of time for alloy with x=8.1: (a) γ′ phase; (b) θ phase

Fig.6 Cube of radius as function of time: (a) γ′ phase; (b) θ phase

  4 Conclusions

1) As the Al concentration increases, the volume fractions of precipitates are alternant, changing from γ′ volume fraction being less than that of θ to being larger  than it.

2) For the small elastic interactions at the early-stage phase separation, the coarsening of γ′ and θ can be approximated by a linear growth as predicated by LSW theory.

3) With increasing elastic interactions at late-stage coarsening, the growth rate decreases, the growth presents quick increase at early-stage and slows down at late-stage.

References

[1] NUNOMRA Y, KANENO Y, TSUDA H, TAKASUGI T. Phase relation and microstructure in multi-phase intermetallic alloys based on Ni₃Al–Ni₃Ti–Ni₃V pseudo-ternary alloy system [J]. Intermetallics, 2004, 12(4): 389-399.

[2] YASHIRO K, KUROSE F, NAKASHIMA Y, KUBO K, TOMITA Y, ZBIB H M. Discrete dislocation dynamics simulation of cutting of c0 precipitate and interfacial dislocation network in Ni-based superalloys[J]. International Journal of Plasticity, 2006, 22: 713-723.

[3] WANG K G, GLICKSMAN M E, RAJAN K. Length scales in phase coarsening: Theory, simulation, and experiment[J]. Comp Mater Sci, 2005, 34(3): 235-253.

[4] CALDERON H A, KOSTORZ G, QU Y Y, DORANTES H J, CRUZ J J, CABANAS-MORENO J G. Coarsening kinetics of coherent precipitates in Ni-Al-Mo and Fe-Ni-Al alloys[J]. Mater Sci Eng A, 1997, 238 (1): 13-22.

[5] THORNTON K, AKAIWA N, VOORHEES P W. Dynamics of late-stage phase separation in crystalline solids[J]. Phys Rev Lett, 2001, 86: 1259-1262.

[6] MURALIDHARAN G, CHEN H. Coarsening kinetics of coherent γ′ precipitates in ternary Ni-based alloys: the Ni–Al–Si system[J]. Sci Technol Adv Mater, 2000, 1: 51-62.

[7] WANG Y U. Computer modeling and simulation of solid-state sintering: A phase field approach[J]. Acta Mater, 2006, 54: 953–961.

[8] YEON D H, CHA P R, KIM J H, GRANT M, YOON J K. A phase field model for phase transformation in an elastically stressed binary alloy[J]. Modelling Simul Mater Sci Eng, 2005, 13: 299-319.

[9] CHOUDHURY S, LI Y L, KRILL III C E, CHEN L Q. Phase-field simulation of polarization switching and domain evolution in ferroelectric polycrystals[J]. Acta Mater, 2005, 53: 5313-5321.

[10] KHACHATURYAN A G. Theory of Structural Transformation in Solids[M]. New York：Wiley, 1983.

[11] WANG Y, CHEN L Q, KHACHATURYAN A G. Particle translational motion and reverse coarsening phenomena in multiparticle systems induced by a long-range elastic interaction[J]. Phys Rev B, 1992, 46: 11194-11197.

[12] PRIOKHODKO S V, CARNES J D, ISAAK D G, ARDELL A J. Elastic constants of a Ni-12.69 at. % Al alloy from 295 to 1300 K [J]. Scr Mater, 1998, 38: 67­­-72.

[13] SUZUKI A, TAKEYAMA M. Formation and morphology of Kurnakov type D0₂₂ compound in disordered face-centered cubic γ-(Ni,Fe) matrix alloys[J]. J Mater Res, 2006, 21: 21-26.

[14] GES A, FORMARO O, PALACIO H. Long term coarsening of precipitates in a Ni-base superalloy[J]. J Mater Sci, 1997, 32: 3687-3691.

[15] KAWASAKI K, ENOMOTO Y. Statistical theory of Ostwald ripening with elastic field interaction[J]. Physica A, 1988, 150(3): 463-498.

[16] SAGUI C, ORLIKOWSKI D, SOMOZA A M, ROLAND C. Three-dimensional simulation of Ostwald ripening with elastic effects[J]. Phys Rev E, 1998, 58(4): 4092-4095.

[17] NISHIMORI H, ONUKI. Pattern formation in phase-separating alloys with cubic symmetry[J]. Phys Rev B, 1990, 42: 980-983.

(Edited by PENG Chao-qun)

Foundation item: Project (50071046) supported by the National Natural Science Foundation of China; Project (2002AA331051)supported by the National Hi-Tech Research and Development Program of China; Project(CX200507) supported by the the Doctorate Foundation of Northwestern Polytechnical University

Corresponding author: LI Yong-sheng, Tel: +86-29-88474095; E-mail: ysli@mail.nwpu.edu.cn