007) 974–978www.elsevier.com/locate/matlet

Materials Letters 61 (2

Microscopic phase-field simulation of atomic migration characteristicsin Ni75AlxV25−x alloys

Yong Sheng Li ⁎, Zheng Chen, Yan Li Lu, Yong Xin Wang, Qing Bo Lai

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

Received 25 March 2006; accepted 14 June 2006Available online 3 July 2006

Abstract

Atomic migration characteristics of phase transformation in Ni75AlxV25−x alloys were investigated with the microscopic phase-field simulation.The results show that the preferential growth direction of D022 is the [100] direction, L12 phase is substituted by D022 along this direction, theD022 results in a plates shape and is parallel to L12 at the final. The atomic substitution behavior in the [100] direction also exhibits the sitepreference, which results in the changeless of Ni atom plans of L12 and D022, thus the substitution modes have the least atom jump quantities, theminimizing energy of atomic jump is accordant with the particles alignment along the elastic soft directions. It is also found that the atomicsubstitution modes of L12→D022 in the early stage are similar to that of D022→L12.© 2006 Elsevier B.V. All rights reserved.

Keywords: Atomic migration; Site preference; Intermetallic alloys and compounds; Computer simulation

1. Introduction

The atomic diffusion in ordered solid solutions, such as theA3B-type intermetallic phases, has attracted many researchinterests [1,2,3], for the substitution-type ordered structuresgive an understandability model for the selection of atom jumpsites. The L12 as first order precipitates of face-centered sub-stitutional solid solutions has been broadly studied [4,5], andthe transformations of the two kinds of similar structures L12and D022 present the site and direction preference. The phasetransition dynamics basically depends on the long-range atom-ic diffusion, so the atomic jump and site occupation modebased on atomic-scale simulations will give helpful under-standing about the diffusion process. Due to the elasticinteractions between the precipitates, the selective orientationof the particles is obvious [6,7]. These phenomena can beexplained by the atom site preference that results in the energyminimizing of the system during the coarsening stage. Two

⁎ Corresponding author. Tel.: +86 29 8847 4095; fax: +86 29 8846 0502.E-mail address: ysli@mail.nwpu.edu.cn (Y.S. Li).

0167-577X/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.matlet.2006.06.038

ordered phases L12 (Ni3Al,γ′) and D022 (Ni3V,θ) precipitate inNi75AlxV25−x alloys by the eutectoid transformation [8,9], andthe precipitation order of the two phases is different as theconcentration changes, so the transformation between L12 andD022 can be investigated simultaneously for the early and laterstage phase separation.

In this paper, we focus on the direction selective and atomicmigration characteristics of D022→L12 transformation. Theconfiguration of the Ni75AlxV25−x alloy including elastic in-teractions was analyzed, and the atomic substitution modes ofthe D022↔L12 transformation were clarified by the microscop-ic phase-field simulation.

2. Model and numerical methods

2.1. Microscopic phase-field model

The microscopic phase-field dynamic model was founded byKhachaturyan [10], which was developed from the time-depen-dent Ginzburg–Landau equations. In this model, the atomicconfigurations and morphologies are described by a single-siteoccupation probability function, x(r,t), which is the probabilitythat a given lattice site r is occupied by an atom at time t. So the

975Y.S. Li et al. / Materials Letters 61 (2007) 974–978

atomic jump is supposed as a direct exchange mechanism. Thechange rates of these probabilities are linearly proportional tothe thermodynamic driving force

∂xðr; tÞ∂t

¼Xr V

Lðr−r VÞ yFyxðr V; tÞ ; ð1Þ

where F is the free energy function of x(r′,t), L(r− r′) is thesymmetry matrix of microscopic kinetic related to the prob-ability of an elementary diffusion jump from site r to r′ per unitof time.

In the ternary system, the occupation probability functionsPA(r,t)+PB(r,t)+PC(r,t)=1.0, so there are only two equationsthat are independent at each lattice site. In order to describe thenucleation, we add a random thermal noise to the right-handside of the equation, the microscopic Langevin equations [11]are given by

dPAðr; tÞdt

¼ 1kBT

Xr V

LAAðr−r VÞ ∂F∂PAðr V; tÞ þ LABðr−r VÞ ∂F

∂PBðr V; tÞ� �

þ nðr; tÞdPBðr; tÞ

dt¼ 1

kBT

Xr V

LBAðr−r VÞ ∂F∂PAðr V; tÞ þ LBBðr−r VÞ ∂F

∂PBðr V; tÞ� �

þ nðr; tÞ;

8>>><>>>:

ð2Þwhere Lαβ(r− r′) is a constant related to the exchangeprobabilities of a pair of atoms, α and β, at lattice site r and r′per unit time, α, β=A, B or C. kB is the Boltzmann constant.ξ(r,t) is assumed to be Gaussian-distributed with the averagevalue of zero, which is uncorrelated with space and time, and it

Fig. 1. Microstructure evolution of alloy with x=8.0 at 1230 K. (a) t⁎=2; (b) t⁎=6; (particle D in (c) is the polymer of A and B.

obeys the so-called fluctuation dissipation theory. F is the totalfree energy of the system,

F ¼ −12

Xr

Xr V½VABðr−r VÞPAðrÞPBðr VÞ

þ VBCðr−r VÞPBðrÞPCðr VÞ þ VACðr−r VÞPAðrÞPCðr VÞ�þ kBT

Xr

½PAðrÞlnðPAðrÞÞ þ PBðrÞlnðPBðrÞÞþ PCðrÞlnðPCðrÞÞ�; ð3Þ

where Vαβ(r− r′)=Vαβ(r− r′)at+Vαβ(r− r′)el is the interactionenergy between α and β at lattice site r and r′, including short-range chemical interaction Vαβ(r− r′)at and long-range strain-induced interaction Vαβ(r− r′)el.

2.2. Microelasticity field

According to the linear elasticity theory of multi-phasecoherent solid developed for the homogeneous modulus case,the strain energy generated by an arbitrary concentration orstructure heterogeneity can be presented as a function of theconcentration or long-range order parameter fields. If the strain ispredominantly caused by the concentration heterogeneity, whichis the case for the Ni-based superalloys, and if the Vergard's law(the lattice parameter is linearly dependent on composition) isfulfilled, the configuration-dependent part of the strain energy canbe expressed in an extremely simple form of pairwise interaction

c) t⁎=10.6; (d) t⁎=18. Particles A, B, and C in (b) are D022 before aggregation,

Fig. 2. Average aspect ratio of D022 particles as a function of time.

976 Y.S. Li et al. / Materials Letters 61 (2007) 974–978

[12]. The long-range strain-induced interaction associatedwith anarbitrary atomic distribution p(r) is given by

Eel ¼ 12

Xrr V

Vabðr−r VÞelPaðrÞPbðr VÞ: ð4Þ

For the decomposition process is determined by thedevelopment of a packet of concentration waves with wavevectors close to zero, the long-wave approximation for theFourier transformation of Vαβ(r− r′)el can be used

VabðkÞelcMðnÞ ¼ Mele20ðn2xn2y−hn2xn2yiÞ; ð5Þ

n=k /k is the unit vector in the reciprocal space, and (nx, ny) arethe Cartesian coordinates of a unit vector n, ε0=da(c) / (a0dc) isthe concentration coefficient of the crystal lattice parameter, a(c) is the lattice parameter of a solid solution with concentrationc, a0 is the lattice parameter of pure solvent, and ⟨nx

2ny2⟩ is the

average of nx2ny

2.

Mel ¼ −4ðC11 þ 2C12Þ2

C11ðC11 þ C12 þ 2C44Þ f; ð6Þ

where ζ=C11−C12−2C44 characterizes the elastic anisotropyof the system, Ci,j are elastic constants of Ni-based solid

Fig. 3. Schematic diagram of the atom site at the interphase boundary formed by [100] di(002)L12 planematching.B1, B2 are vanadium atoms, B1 andB2 lie on the corner sites ofatoms; C1 and C2 are nickel atoms.

solution, which are C11=201.1, C12=149.5, C44=87.5 GPa at1250 K [13].

The elastic strain energy is given by a dimensionless pa-rameter ξ=Melε0

2 /Δf, where ξ=1200 is an input date, Δf is thelocal chemical free energy. The parameter ε0 can be estimatedfrom the lattice misfit and the content diversity. For the misfitstrain of γ′ /γ is 0.0056 [14] and θ′ /θ is 0.007 [15], with thehomogeneous modulus approximation, we use the averagevalue 0.0063. The content diversity is 0.09 for alloy withx=8.0, so inputting ε0=0.07.

3. Results and discussion

3.1. Growth direction preference and configuration

The simulation is performed with 256×256 mesh points, a per-iodical boundary condition is imposed along both dimensions, the timestep Δt=0.0002, and the thermal fluctuations are removed after nuc-leation, then the system chooses the dynamic path automatically.

For the different precipitation driving forces of L12 and D022 atdifferent regions below the eutectoid temperature, there are two kinds oftransformation pathways: fcc→D022→L12+D022 at low aluminumconcentration regions, and fcc→L12→L12+D022 at high aluminumconcentration regions. Fig. 1 shows the morphology evolution of thealloy with x=8.0 at 1230 K. The simulated pictures are two-dimensionalprojection of the fcc lattice along the [001] direction, which is depictedwith different color schemes. If the occupation probability of aluminum is1.0, then that site is green, if the occupation probability of vanadium is1.0, then that site is red, therefore the L12 phase appears to be green, theD022 phase appears to be red and all the nickel sites in both phases appearto be blue. It can be seen from Fig. 1(a) that L12 precipitates firstly, thenD022 precipitates along the interphase boundaries of L12.

In the later stage precipitation, the particles align along certaincrystallographic directions due to the long-range elastic interactionsbetween the precipitates. For the tetragonal structure D022 has threedifferent variants, we prescribe the two directions [100] and [001] inthe projection picture, as shown in Fig. 3(a). The [100] direction ofD022 is dominant during the growth and coarsening, so D022 results inplates shape at the final, as shown in Fig. 1(d), the L12 also presents asimilar shape, which aligns along the <001> directions, and the L12and D022 are the interval distribution in space. The simulated mor-phology is consistent with the experimental results [16].

rections of D022 and L12. (a) (001) Planes of two phasesmatching, (b) (001)D022 andD022 and face-center sites of (002)D022 plane respectively;A1 andA2 are aluminum

Fig. 4. Average atom occupation probability of the transformation of D022→L12 (a) and L12→D022 (b).

977Y.S. Li et al. / Materials Letters 61 (2007) 974–978

Now we study the migration direction preference of D022. Asshown in Fig. 1(b), the D022 particles A, B, and C are separated by L12,as the coarsening progresses, the particles A and B merge into a largeparticle D along the [100] direction. But the particles D and C form theinterphase boundaries, as shown in Fig. 1(c). This process is dominatedby the D022 substituting L12 along the [100] direction. Fig. 2 shows theparticle average aspect ratio (the ratio between the longer edge length tothe shorter one) of D022 as a function of time, and it can be seen that theaverage aspect ratio increases as coarsening progresses, which demon-strates that one direction of D022 is preferential.

The L12 exhibits the same orientation selective characteristic at thecoarsening stage, since L12 has only one orientation variant, it growsalong the <100> directions. However, the D022 does not almost growalong the [001] direction and is perpendicular to the L12, and thereforms the orientation relationship: {001}L12 // (001)D022, as shown inFig. 1(d) labeled with arrows. So the D022 and L12 particles are parallelto each other at spatial alignment.

3.2. Atomic migration characteristic

Nowwe discuss the atomic substitutionmodes during the D022→L12transformation. In the simulation, the vacancy effect was ignored, wefocus on the initial and final state of the atom sites. Fig. 3 shows theprojection schematic diagram of the atom sites of the interphaseboundaries formed by the [100]D022 direction and L12. As shown inFig. 3(a), the (001) planes of the two phases match at the interphaseboundaries, which are denoted as the first matching style in this paper. A1and A2 are aluminum atoms that lie on the corner sites of L12, which areequal in crystallography. B1 and B2 are vanadium atoms, which corre-spond to the corner of D022 and face-center sites of (002)D022 plane,respectively, and C1 and C2 are nickel atoms. As the two phase structuresare transforming, the A1 atom substitutes B1 atom, A2 atom substitutesC1 atom, and the C1 atom substitutes B2 atom, then the same processesgo on. It can be seen from the atomic migration processes that the nickelatom plans of D022 need not change; atomic substitution only happens atthe planes including vanadium atom, and this kind of substitution modehas the least atommigration quantities, so the jumppotential is the lowest.

If the matching atom plans are (002)L12 and (001)D022 at the interphaseboundaries, which is denoted as the second matching style, as shown inFig. 3(b), all of the planes happen to exchange atoms for the two phasestransformation, so the atom jump potential is bigger than that of the firstmatching style. It progresses difficultly during the coarsening process, asshown in Fig.1(c)–(d) labeled with circle, when the other D022 particleshave aggregated or formed the interphase boundaries through the firstmatching style, and the two D022 particles just begin to aggregate.

Fig. 4(a) shows the average atom occupation probability (AOP) ofL12 and D022 during the two phases transformation with the firstmatching style. The three regions I, II and III plotted out by the dash-dotted line delegate L12 phase, transition region and D022 phaserespectively. It can be seen that the D022→L12 transformation is theincrease of L12 average AOP, and the decrease of D022 average AOP.The two phases average AOP presents alternative transition state at theinterphase boundaries regions. The transformation of L12→D022 isjust reverse to that of D022→L12 for the average AOP of two phases,as shown in Fig 4(b). The atomic substitution processes of L12→D022are B1 atom substitutes A1 atom, B2 atom substitutes C2 atom, and C2atom substitutes A2 atom. The average AOP of the two phases is verysimilar to the composition profiles derived from the 3D atom probereconstruction [17].

4. Conclusions

The atomic migration characteristics during the D022↔L12transformation in Ni75AlxV25−x alloys were investigated bymicroscopic phase-field simulation. The particles growthdirection and atomic substitution exhibit preferential character-istic. When (100) planes of the two phases match at theinterphase boundaries formed by [100]D022 direction and L12,the atom jump quantities are the least during the two phasestransformation, thus the [100]D022 direction is dominant, and theD022 results in the plates shape. This reveals that the atom jumppotential minimizing is accordant with the particles alignmentalong the elastic soft directions. The atomic substitution pro-cesses of D022→L12 are as follows: the Al atoms substitute Vand Ni atoms of the corner sites of D022, and the Ni atomssubstitute V atoms of (002)D022 face-center sites, the Ni atomplanes of L12 and D022 are changeless. It is also found that theatomic substitution modes of L12→D022 in the early stageprecipitation are similar to that of D022→L12.

Acknowledgements

This work is financially supported by the National NaturalScience Foundation of China (No.50071046), Natural ScienceFoundation of Shaanxi Province (2003E103), and theDoctorate Foundation of Northwestern Polytechnical Univer-sity (CX200507).

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