Microscopic Phase-Field Simulation for Correlations of Ni3Al Precipitates in Ni-Al Alloys

Rare Metal Materials and Engineering Volume 38, Issue 11, November 2009 Online English edition of the Chinese language journal

Cite this article as: Rare Metal Materials and Engineering, 2009, 38(11): 1890-1893.

ARTICLE

Microscopic Phase-Field Simulation for Correlations of Ni3Al Precipitates in Ni-Al Alloys Lu Yanli,

Chen Zheng,

Zhong Hanwen,

Zhang Jing

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

Abstract: The effect of Al concentration on the correlations of Ni3Al precipitates (γ′) in Ni-Al alloys was studied based on the microscopic phase-field kinetic model and the microelasticity theory. The simulation results show that γ′ precipitates are arranged along the [001] and [010] directions regarding alloys with all concentration, but there exists different correlations. For the alloy with lower Al concentration, γ′ precipitates are independent each other; for the alloy with medium Al concentration, there are mainly four types correlations among the γ′ precipitates: L-shaped pattern, doublet pattern, triplet pattern and quartet pattern; for the alloy with higher Al concentration, the correlation of γ′ precipitates is mainly doublet pattern. Key words: microscopic phase-field; elastic strain; morphological evolution; precipitation process; Ni-Al alloy

Ni3X intermetallics are the main precipitates in nickel-based alloys, which have received great interests of many researchers because of their own attractive properties as high temperature structural and chemical materials[1-3]. The volume fraction, morphology, size and spatial distribution of γ′ precipitates strongly affect the mechanical properties of the alloys. Therefore, the control of the γ′ precipitate morphology is one of the most important topics for developing new types of precipitation-strengthened alloys. When the precipitate morphology is examined, the individual shape, distribution of precipitated particles and their correlation should be considered. Based on the Ginzburg-Landau dynamic equation, Khachaturyan established the microscopic phase field model, which interpreted qualitatively phase transformation by atomic jumping between crystal lattices. The atomic ordering, the structure of antiphase boundary and the atomic clustering during the precipitation have been simulated based on the microscopic phase field method. This model has been successfully used in the study of alloy precipitation process [4-7], but the above studies do not consider the effect of elastic strain energy. For many alloys, because of the different lattice parameters between coherent precipitates and matrix phases, elastic strain energy may exist around precipitates, which can strongly af-

fect precipitation process. The purpose of this paper is to study the effect of Al concentration on the correlation of γ′ precipitates in Ni-Al alloys using the microscopic phase-field simulation; the effect of the elastic strain energy is considered sufficiently.

1

Theoretical Modeling

1.1 Microscopic phase-field model The microscopic phase-field model was developed by Khachaturyan[8]. In this model, the atomic configurations and morphologies are described by a single-site occupation probability function P(r, t), which represents the probability of finding an atom at a given lattice site r and at a given time t.

dP ( r , t ) c0 (1 − c0 ) δF = ∑' L(r − r ′ ) δ P (r , t ) + ξ (r , t ) (1) dt k BT r Where L(r–r′) is the exchange probability between a pair of atoms from site r to r′ per unit of time. F is the total free energy of the system. c0 is the atomic fraction of solute atoms. T is temperature. ξ(r, t) is the thermal noise term, which is assumed to be Gaussian-distributed with the average value of zero, is uncorrelated with space and time, and obeys the so-called fluctuation dissipation theory: <ξ(r, t)>=0

Received date: November, 24, 2008 Foundation item: Supported by the National Natural Science Foundation of China(50671084); China Postdoctoral Science Foundation (20070420218); Natural Science Foundation of Shaanxi Province Corresponding author: Lu Yanli, Post-Doctor, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, P. R. China, Tel: 0086-29-88494460, E-mail: [email protected] Copyright © 2009, Northwest Institute for Nonferrous Metal Research. Published by Elsevier BV. All rights reserved.

1890

Lu Yanli et al. / Rare Metal Materials and Engineering, 2009, 38(11): 1890-1893

<ξ(r, t) ξ(r′, t′)>= –2kBTL(r–r′)δ(t–t′)δ(r–r′) Where <…>denotes an averaging, <ξ(r, t)> is the average value of the noise over space and time, <ξ(r, t) ξ(r′, t′)> is the correlation and δ is the Kronecker delta function. In the mean-field approximation, the free energy in equation (1) is given by: F＝

1 2

∑ ∑ W ( r − r ′) P ( r ) P ( r ′) + r′

r

(2)

k B T ∑ [ P ( r ) ln P ( r ) + (1 − P ( r )) ln(1 − P ( r ))] r

Where W(r–r′)=W(r–r′)ch+W(r–r′)el is the interaction energy between two atoms at lattice sites r and r′, W(r–r′)ch is short-range chemical interaction and W(r–r′)el is long-range strain-induced interaction.

1.2

Microelasticity theory

In the phase-field model, the contribution of elastic strain energy to the total free energy can be described as a function of field variable. For Ni-based superalloys, if we assume that the strain is predominantly caused by the concentration heterogeneity, and if the lattice parameter is linearly dependent on composition, thus, the morphology-dependent part of the strain energy can be expressed in an extremely simple form of pairwise interaction[9]. The long-range strain-induced interaction associated with an arbitrary atomic distribution P(r) is given by

E el =

1 2

∑ W ( r − r ′) rr ′

el

P ( r ) P ( r ′)

(3)

4 (C 11 + 2 C 12 ) δ C 1 1 (C 1 1 + C 1 2 + 2 C 4 4 ) 2

el

= −

a

b

c

d

e

f

(4)

Where n=k/k is a unit vector in the k direction. nx, ny are components of the unit vector k along the x and y axes parallel to the [001] and [010] directions in the reciprocal space. ε0=da(c)/(a0dc) is the concentration coefficient of crystal lattice expansion caused by the difference of atomic size. a(c) is the crystal lattice parameters of solute, while a 0 is the crystal lattice parameters of a solid solution. c is the atomic fraction of solute atoms. H

nected each other; with the time proceeding, the shape of γ′ precipitates gradually changes to the column shape and the arrangement becomes more and more regular; at the later stage, the γ′ precipitates regularly distribute along the [100] and [010] directions, as shown in Fig.2f.

For the decomposition process is determined by the development of a packet of concentration waves with waves vectors close to zero, the long-wave approximation for the Fourier transformation of W(r–r′)el can be described as: V ( k )el ≈ H ( n ) = H el n x2 n y2ε 02

boundary conditions are applied along both dimensions, and the time step is 0.0001. The occupational probability of Al atom is represented by a gray scheme on which black indicates 1 and white represents 0. Fig.1 shows the atomic morphology evolution of Ni-9.0at% Al alloy. At the beginning, the smaller γ′ precipitates are distributed randomly and present spherical or irregular shape, which are independent each other. As aging proceeds, the γ′ precipitates gradually grow, whose shape and arrangement become more and more regular; at the later stage of aging, the γ′ precipitates present a tendency to be aligned along the elastically soft directions, e.g.<100> directions, and some of them present column shape, while others present irregular shape. For Ni-based alloy, the elastic anisotropic factor is negative and the minimum strain energy is achieved along the <001>directions, so the γ′ precipitates arrange along the [100] and [010] directions at the later stage when elastic strain energy dominates precipitation process. Fig.2 shows the atomic morphology evolution of Ni-14.0at% Al alloy. At the initial stage of precipitation process, a large number of γ′ particles are precipitated from the disordered matrix, which are randomly distributed and con-

(5)

Where δ=C11–C12–2C44 is the elastic anisotropy constant and Cij are the elastic constants of the studied system, which are C11=206.8 GPa, C12=148.5 GPa, and C44=94.4 GPa at 1050 K[10].

[001] [010]

2 Simulation Results and Analysis The Ni-9.0at%Al, Ni-14.0at% Al and Ni-16.7at%Al alloys are aged at 1050 K. The 2-D simulation is performed in a square lattice consisting of 512×512 unit cells, periodic

Fig.1 Morphological evolution of γ′ phase for Ni-9.0at%Al alloy: (a) t=20 000, (b) t=120 000, (c) t=200 000, (d) t=300 000, (e) t=400 000, and (f) t=500 000

1891

Lu Yanli et al. / Rare Metal Materials and Engineering, 2009, 38(11): 1890-1893

a

b

a

b

c

d

c

d

f

e

f

e N M

Q

P

Fig.2 Morphological evolution of γ′ phase for Ni-14.0at% Al alloy:

Fig.3 Morphological evolution o f γ′ phase for Ni-16.7at% Al alloy:

(a) t=20 000, (b) t=120 000, (c) t=200 000, (d) t=300 000, (e)

(a) t=20 000, (b) t=120 000, (c) t=200 000, (d) t=300 000, (e)

t=400 000, and (f) t=500 000

t=400 000, and (f) t=500 000

tates in Ni-16.7at% alloy is larger than that of Ni-14.0at% Al alloy; the correlation of precipitates is mainly the doublet pattern, and we did not find other three correlations as shown in Ni-14.0at% Al alloy. Fig.4 illustrates the variation of volume fraction of different Al concentration alloys with time. The volume fractions for different Al concentration alloys rise with the increase of time, the rising trend change from the rapider to the slower. Whereas, the time steps that the volume fraction needs to begin to change and get the maximum value are different. The 1 Volume Fraction

Different from the Ni-9.0at.%Al alloy, the γ′ precipitates of Ni-14.0at% Al alloy are no longer independent each other, but present four correlations. There are four types antiphase domains of L12 ordered γ′ phase. During the process of growth and coarsening, if the particles belong to the same domain, they simply coalesce to form larger particles; if they belong to different domains, they don’t coalesce, but form the different morphological patterns depending on the correlative arrangements and the sizes of the particles. In the case that a pair of adjacent particles belongs to the same domain and the remaining two particles are a part of another domain, doublet morphology is formed, corresponding to P zone of Fig.2f. When three particles belong to a single domain and one particle belongs to another domain, an L-shaped pattern is formed, as shown in Q zone of Fig.2f. When all the four particles belong to different domains, an identical quartet is formed as shown in M zone of Fig.2f. In other situations, triplet pattern is similar to that shown in N zone of Fig.2f. Because the elastic strain energy dominates the precipitation process at the later stage, the particles tend to form the different correlation in order to relax the elastic strain energy. Fig.3 illustrates the atomic morphology evolution of γ′ precipitates in Ni-16.7at% Al alloy. The variation regularity of the morphology and the arrangement of γ′ precipitates are similar to that of Ni-14.0at% Al alloy. However, the size of γ′ precipi-

Ni-9.0at% Al alloy Ni-14.0at% Al alloy Ni-16.7at% Al alloy

0.8 0.6 0.4 0.2 0 0

2

4

6

Time Step/×10

8

10

4

Fig.4 Variation of volume fraction of different Al concentration alloy

1892

Lu Yanli et al. / Rare Metal Materials and Engineering, 2009, 38(11): 1890-1893

P1

P2

300 nm Q1

M

M

300 nm

Q2

N2

N1

300 nm

300 nm

Fig.5 Comparison of simulation results (P1-N1) with the corresponding experiment results (P2-N2)[12]

difference of volume fraction variation is arisen by the difference of early stage precipitation mechanism, caused by the difference of Al concentration, which was studied in the Ref[11]. This shows why the precipitate correlations are different in the alloys with different Al concentrations from another aspect.

3 Comparison with Experiment Observations Fig.5 (P1-N1) is the enlarged pictures corresponding to the P-N zone of Fig.2f. Fig.5 (P2-N2) is the dark-field of transmitssion electron micrograph of Ni-14.0at Al% alloy aged for 15 min by Y. Y. Qiu[12]. Making the experiment observations projected along the [100] direction, we can find that the simulation results are in good agreement with the experiment observations, which prove that the simulation results are true.

4 Conclusions The precipitates’ correlations are different for the alloys with different Al concentrations. For the alloy with lower Al concentration, the γ′ precipitates are independent each other; for the alloy with medium Al concentration, there are four types correlations among the γ′ precipitates: L-shaped pattern, doublet pattern, triplet pattern and quartet pattern; for the alloy

with higher Al concentration, the precipitates’ correlation is mainly doublet pattern. For all the alloys, γ′ precipitates gradually change from the non-directional distribution to alignment along the [001] and [010] directions.

References 1 Adorno, A. T. J. Alloy Compd., 2006, 414: 55 2 Numakura, H.; Nishi, K. Mater Sci. Eng. A., 2006, 442: 59 3 Li-Shing, Hsu. J. Alloy Compd., 2006, 413: 11 4 Simmons, J. P. Mater Sci. Eng. A., 2004, 365: 136 5 Poduri, R.; L, Q. Chen. Acta Mater, 1997, 45(1): 245 6 Chu, Zhong. Rare Metal Materials and Engineering, 2006, 35(2): 200 7 Lu, Yanli. Rare Metal Materials and Engineering, 2005, 33(8): 1205 8 Khachaturyan, A. G. Theory of Structural Transformation in Solids. New York: Wiley, 1983: 129 9 Wang, Y. Acta Metal Mater, 1995, 43: 1837 10 Prikhodko, S. V. Scripta Mater, 1998, 38: 67 11 Lu, Yanli. Progress in Natural Science, 2005, 5(12): 1104 12 Qiu, Y. Y. J. Alloy Compd., 1998, 270: 145

1893