J. Cent. South Univ. Technol. (2008) 15: 443-448

DOI: 10.1007/s11771-008-0083-y

Microscopic phase-field simulation for coarsening behavior of

L1₂ and DO₂₂ phases of Ni₇₅CrxAl_25-x alloy

LU Yan-li(卢艳丽)¹, CHEN Zheng(陈 铮)^{1, 2}, ZHANG Jing(张 静)¹

(1. School of Materials Science and Engineering, Northwestern Polytechnical University,

Xi’an 710072, China;

2. State Key Laboratory of Solidification Processing, Northwestern Polytechnical University,

Xi’an 710072, China)

Abstract: Based on the microscopic phase-field dynamic model and the microelasticity theory, the coarsening behavior of L1₂ and DO₂₂ phases in Ni₇₅Cr_xAl_25-x alloy was simulated. The results show that the initial irregular shaped, randomly distributed L1₂ and DO₂₂ phases are gradually transformed into cuboidal shape with round corner, regularly aligned along directions [100] and [001], and highly preferential selected microstructure is formed during the later stage of precipitation. The elastic field produced by the lattice mismatch between the coherent precipitates and the matrix has a strong influence on the coarsening kinetics, and there is no linear relationship between the cube of the average size of precipitates and the aging time, which does not agree with the results predicted by the classical Lifshitz-Slyozov-Wagner. The coarsening processes of L1₂ and DO₂₂ phases are retarded in elastically constrained system. In the concurrent system of L1₂ and DO₂₂ phases, there are two types of coarsening modes: the migration of antiphase domain boundaries and the interphase Ostwald ripening.

Key words: Ni₇₅Cr_xAl_25-x alloy; coarsening behavior; microscopic phase-field; elastic strain; coherent precipitates

1 Introduction

The microscopic phase-field kinetic model^[1-4] based on the Ginzburg-Landau kinetic equation makes the non-homogeneous system described at atomic-scale, and the ordering and phase separation are treated simultaneously by the non-equilibrium free energy function contacting with concentration and long-range ordering parameters. This model has the unique advantage of describing the highly nonlinear and non-equilibrium process during alloy precipitation.

Classical coarsening theory assumes that under an infinitesimal volume fraction of the second phase particles coarsening in a strain energy-free system, the coarsening process is driven only by the reduction of interfacial-energy, and the shape and spatial distribution of precipitates are not influenced by the coarsening process. But in the actual metallic alloy, there is elastic constrain more or less because of different lattice parameters between coherent precipitate phases and matrix phases. So there exist elastic strain fields around precipitates, and their superposition results in long-range elastic interactions, which directly affects the morphology and coarsening behavior of the precipitates. Coarsening under the influence of coherent strain energy is significantly more complicated than the strain energy-free coarsening, mainly because of the anisotropic elastic properties and the long-range elastic interaction. The anisotropic elastic properties result in nonspherical precipitates, and the long-range elastic interaction causes strong spatial correlation between the precipitates and makes their alignment along certain crystallographic directions. These influences lead to the fact that the classical coarsening theory shows distinct defect in explaining the coarsening behavior of precipitated system including the elastic strain^[5-7].

The coarsening behavior of ternary nickel bases alloy was studied based on the microscopic phase-field kinetic model in this work. The morphology evolution and the coarsening behavior of L1₂ and DO₂₂ phases in the Ni₇₅Cr_xAl_25-x alloy were simulated by incorporating microelasticity theory.

2 Theoretical model

2.1 Microscopic phase-field kinetic model of ternary system

The microscopic phase-field kinetic model describes atomic structure morphology by single-site occupation probability. Let P_A(r, t), P_B(r, t) and P_C(r, t) represent the probabilities of finding atoms A, B or C at a given site r and at a given time t, since P_A(r, t)+P_B(r, t)+ P_C(r, t)=1, only two equations are independent at each lattice site, and there will be two independent kinetic equations at each lattice site for atoms A and B. So the microscopic phase-field kinetic model of ternary system is described as^[8]

      (1)

where  L_αβ(r-r′) 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; ξ(r, t) is the thermal noise term, which is assumed to be Gaussian-distributed with the average value of zero and uncorrelated with space and time, and obeys the fluctuation-dissipation theory; F is the total free energy including the elastic strain energy contribution. Based on the mean-field approximation, F is given by

                              (2)

where  V_αβ(r-r′) is the interaction energy between α and β at lattice sites r and r′, including short-range chemical interaction energy V_αβ(r-r′)_ch and long-range strain- induced elastic interaction energy V_αβ(r-r′)_el:

V_αβ(r-r′)=V_αβ(r-r′)_ch+V_αβ(r-r′)_el                  (3)

2.2 Microelastic theory

In the microelastic theory, the strain energy of solid solution is given as a sum of two physically distinct terms^[9]: 1) the configuration-independent term describing the self-energy and image force-induced energy; 2) the configuration-dependent term associated with concentration inhomogeneity. The first term is not affected by spatial redistribution of solute atoms and therefore it can be ignored. The second term, however, gives a substantially nonlocal strain energy change associated with spatial distribution of solute atoms, and it affects the morphology of the precipitation phase.

In a real space, the configuration-dependent strain energy associated with an arbitrary atomic distribution n(r) is

             (4)

The Fourier transformation of Eqn.(4) yields

                    (5)

where  N is the total lattice number. The prime “ ′ ” in Eqn.(5) implies that the point k=0 is excluded; V(k)_el is the Fourier transformation of the density function of elastic energy V(r)_el. The long-wave approximation for V(k)_el can be described as

                    (6)

where  n(n=k/k) is a unit vector in direction k; n_x and n_y are components of the unit vector k along axes x and y, respectively; ε₀ (ε₀=da(c)/(a₀dc)) is the concentration coefficient of crystal lattice expansion caused by the atomic size difference, a(c) is the crystal lattice parameter of solute, a₀ is the crystal lattice parameter of a solid solution, and c is the atomic fraction of solute atoms; H(n) is the strain energy parameter which characterizes the elastic properties and the crystal lattice mismatch, and

                (7)

where  δ(δ=C₁₁-C₁₂-2C₄₄) is the elastic anisotropy constant and C_ij is the elastic constant of the studied system. In this work, C₁₁=210.5 MPa, C₁₂=149.0 MPa, C₄₄=98.1 MPa were chosen based on the value of Ni-based alloy at 950 K^[10].

3 Simulation results and analysis

For Ni-Cr-Al alloy, JIA et al^[11] studied the laws of atom occupation and substitution, and they suggested that there may be the quasibinary section Ni₃Al-Ni₃Cr and Ni₃Al-Cr₃Al in solution lobe in the ternary Ni-Cr-Al phase diagram. Ni₇₅Cr_14.5Al_10.5 alloy is within the γ+γ′ two-phase field in the ternary isothermal section phase diagram at 950 K^[12]. The structures of precipitates are identified by different arrangement states of different atoms on the plane of projection.

3.1 Evolution of precipitates morphology

Ni₇₅Cr_14.5Al_10.5 alloy was aged at 950 K. The occupational probability of atom is represented by a gray scheme on which the black indicates Ni atom, the white indicates Cr atom and the gray indicates Al atom. Thus, if the occupation probability of Cr equals 1.0, then that site is assigned to the white, and so on. Therefore, the DO₂₂ phase appears to be white, the L1₂ phase appears to be gray and a black background is formed. The simulation was performed in a square lattice consisting of 128×128 unit cells, periodic boundary conditions were applied along both dimensions, and the time step was 0.000 1.

The atomic morphology evolution of precipitates for Ni₇₅Cr_14.5Al_10.5 alloy is presented in Fig.1. At this composition, the phase transformation will occur only if the thermal noise term is given at the initial state. Many smaller L1₂ order phases are formed in the disordered solution at 1 800 time steps as shown in Fig.l(a), which distribute randomly and present irregular shape. The phase boundary between order phase and disorder phase is diffuse, and the transition region is about several lattices wide. As time proceeding, the new precipitates begin to be formed near the phase boundary of L1₂ phase, as shown in Fig.1(b), and their atomic arrange is identical to the 2-D projection of Ni₃Cr order phase (DO₂₂ structure). So the precipitate phase is regarded as Ni₃Cr. The nuclei of DO₂₂ phases are mainly formed at the edge and corner of L1₂ phases. At this time, the corresponding nucleation potential barrier is descended more rapidly. With the growth of L1₂ and DO₂₂ phases, their shapes begin to change from the initial irregular shape to cuboidal shape with round corner and their arrangement becomes more regularly. In the end, the cuboidal precipitates link together to form rods, distributed regularly along directions [100] and [001], as shown in Figs.1(c)-(e). At 6.0×10⁵ time steps, the highly preferential selected microstructure is formed in the matrix, as shown in Fig.1(f).

Fig.1 Temporal evolution of occupation probabilities of solute atoms for Ni₇₅Cr_14.5Al_10.5 alloy: (a) t=1.8×10³; (b) t=5.0×10⁴; (c) t=   1.0×10⁵; (d) t=2.0×10⁵; (e) t=4.5×10⁵; (f) t=6.0×10⁵

From the atomic morphology evolution, it is found that two types of coarsening modes simultaneously exist in the coarsening process: 1) the migration of antiphase domain boundaries (APBs); and 2) the interphase Ostwald ripening. The former case occurs among different L1₂ order domains or DO₂₂ order domains separated by the APBs. The diffusion distance when one APB is migrated is about the width of one APB, so the larger domain grows and the smaller domain vanishes. The growth of order domain is also the process of decrease of APBs, and its driving force is the reduction of the APBs energy. The latter occurs between the L1₂ order phase and DO₂₂ phase separated by the phase boundary. Smaller DO₂₂ phase is similarly eaten by the larger L1₂ phase, as shown in Figs.1(b)-(e). Particle A grows instantly at the early stage of precipitation, but gradually changes smaller, shrinks and dissolves due to the continuous growth and coarsening of surrounding L1₂ phase. This behavior is regarded as the interphase Ostwald ripening.

Fig.2 shows the variation of average size of DO₂₂ phase along directions [100] and [001] with time. It can be seen that the size along both directions increases with increasing time step, and the average size along direction [100] increases more rapidly than that along direction [001], which shows that the growth velocity of DO₂₂ phase along direction [100] is greater than that along directions [001], and DO₂₂ phase grows along direction [100] and extends along direction [001] simultanously. Therefore, DO₂₂ phase presents the elongated cuboidal shape in the later stage of precipitation.

Fig.2 Change of average size of DO₂₂ phase along directions [100] and [001] with time step

In the precipitation process of Ni-based alloy, because of different lattice parameters between coherent precipitate phases and matrix phases, there exist elastic strain fields around precipitates, and their superposition results in long-range elastic interactions. So the phase transformation will be controlled by both the elastic strain energy and the interface energy. Being different from interface energy, elastic strain energy is not only proportional to the volume friction of precipitate phase, but also relates to the morphology of precipitate phase. During the initial precipitation, particles prefer to becoming similarly equiaxed or irregular shape in order to reduce interface energy. The contribution of anisotropic elastic energy becomes obvious as the precipitation phase grows up, gradually dominates the precipitation process, and then the precipitate phases develop to quadrate shape in order to reduce the strain energy.

For Ni-based alloy, elastic anisotropic factor is negative^[13], and the minimum strain energy is achieved along direction <001>. So the precipitates arrange along directions [100] and [001] in the later stage when elastic strain energy dominates the precipitation process.

3.2 Growth and coarsening of ordered phases

In classical Liftshitz-Slyozov-Wagner (LSW) theory and the modified LSW theory, the effect of the elastic stress on the coarsening process is neglected, equally only short-range chemical interaction exists in the system. However, during actual phase transformation of crystal solid solution, the difference of lattice parameters between the precipitates and matrix leads to the rearrangement of the crystal structure, resulting in elastic strain. The overlap of the elastic field of the individual precipitates induces elastic interactions between the precipitates. The elastic field is relied on the volume and configuration of precipitates. Therefore, the reduction of the bulk free energy is the driving force of phase transformation. In this case, the coarsening process does not totally obey the Ostwald ripening^[14].

Variation in the cube of the average radius () with time for L1₂ and DO₂₂ phases is presented in Fig.3. Because the precipitate is nonspherical, the average particle size was taken as the standard of identifying the size of precipitates^[15]. From Fig.3, it can be seen that the cube of the average size of the particles is not a linear function of the aging time, which is required for the LSW theory. So the classical LSW theory is unfitted to this system.

Fig.3 Variation in cube of average radius with time for L1₂ and DO₂₂ phases

The LSW theory predicts that during the coarsening process, the growth law of precipitates follows:∝t^m, where  is the average radius, t is the aging time (number of time step), and m is the coarsening rate time index and is equal to 1/3 in the LSW theory. In an elastic-contained system, a finite lattice mismatch introduces significant internal stress in the system and has remarkable effect on the coarsening kinetics of the particles^[16].

The mean radius of particles of L1₂ and DO₂₂ phases as a function of aging time t in a logarithm representation is shown in Fig.4. The slope of the line represents coarsening rate time index. The coarsening rate time index of L1₂ phase is calculated as m=0.183, and that of the DO₂₂ phase is calculated as m=0.25. So the coarsening rate time indexes of L1₂ and DO₂₂ phases are both smaller than the predicted values by LSW theory, namely the average radius of precipitates in elastic- constrained system is smaller than that of elastic stressfree system at the same time step. Therefore, the coarsening process in the elastic-constrained system is retarded, which is in agreement with the experiment observed in nickel based alloy by QIU^[17]. This retarded coarsening phenomenon is the result of elastic energy or elastic interaction, and it is a typical example of elastic effect.

Fig.4 Mean radius of particles of L1₂ and DO₂₂ phases as function of aging time

The evolution of volume fraction of DO₂₂ and L1₂ phases in Ni₇₅Cr_xAl_25-x alloy with time is shown in Fig.5. When x=14.5, the L1₂ phase is formed first, and its volume fraction reaches the maximum value. At this time, the volume fraction of DO₂₂ phase begins to increase, both reach the equilibrium values in the end. When x=19.0, the DO₂₂ phase is formed first, the changes of volume fraction of L1₂ and DO₂₂ phases are just inverse to those of x=14.5. The final microstructures of Ni₇₅Cr_xAl_25-x alloys with x=19.0, DO₂₂ phase precipitated early and x=17.0, L1₂ phase precipitated early at 950 K are shown in Fig.6. From Fig.6, it can be seen that the two final microstructures are the same and similar to the morphology in Fig.1(f), only the corresponding quantities of both precipitates are different. Seen from the final equilibrium value, the volume fraction of the first precipitated phase is also greater than that of the later precipitated phase. However, the medium concentration alloy with x=17.0 presents different changes, the first precipitate is L1₂ phase, and the later precipitate is DO₂₂ phase. But the volume fraction of DO₂₂ phase exceeds that of L1₂ phase, which is also observed from the simulated morphology, as shown in Fig.6(b). Although the amounts of precipitate phases are different, their final structural morphologies are the same, and present cuboidal shape and align along directions [100] and [001] regularly.

Fig.5 Evolution of volume fraction of DO₂₂ phase (a) and L1₂ phase (b) in Ni₇₅Cr_xAl_25-x alloy

Fig.6 Final microstructures of Ni₇₅Cr_xAl_25-x alloys at 950 K:  (a) x=19.0, DO₂₂ phase precipitated early; (b) x=17.0, L1₂ phase precipitated early

4 Conclusions

1) For Ni₇₅Cr_xAl_25-x alloy, the coarsening processes of L1₂ and DO₂₂ phases are retarded. There are two types of coarsening modes: the migration of APBs and the interphase Ostwald ripening.

2) The elastic field produced by the lattice mismatch between the coherent precipitates and the matrix has a strong influence on the coarsening kinetics. There is no linear relationship between the cube of the average precipitates size and the aging time, which is entirely different from the results predicted by the classical Lifshitz-Slyozov-Wagner theory.

3) The precipitation sequences and the amounts of L1₂ and DO₂₂ phases are not the same in the alloys with different concentrations, but their final morphologies are the same, and present cuboidal shape and align regularly along directions [100] and [001].

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Foundation item: Project(50671084) supported by the National Natural Science Foundation of China; Project(20070420218) supported by China Postdoctoral Science Foundation

Received date: 2007-10-21; Accepted date: 2007-12-25

Corresponding author: LU Yan-li, PhD; Tel: +86-29-88494460; E-mail: luyanli@nwpu.edu.cn

(Edited by CHEN Wei-ping)