SIMULATION OF THE FCC3 FCC 1 L12 1 DO22

KINETIC REACTION

C. Pareige and D. BlavetteGPM–UMR 6634 CNRS, University of Rouen, 76821 Mont Saint Aignan, France

(Received May 24, 2000)(Accepted in revised form August 24, 2000)

Keywords:Computer simulation; Nickel alloy; Phase transformations; Ordering; kinetics

1. Introduction

The Ni-rich corner of the Ni-Al-V ternary phase diagram contains a three-phase domain where orderedL12 (Ni3Al, g9) andDO22 (Ni3V,u) phases form in the FCC (g phase) solid solution. The decompositionof the solid solution involves both ordering and phase separation processes. Mean field simulations,based on microscopic Onsager-type kinetics equations, were recently carried out by Poduri and Chen(1). Kinetic pathway was studied in a near-eutectoid alloy situated in the pseudo-binaryNi3Al2Ni3Vsection of the phase diagram. For simplicity, they ignored the misfit strain effects. Their simulationsshowed that at small Al content (less than 0.055at.%), precipitation ofDO22 domains precedesL12

phase formation whereas at large Al content theL12 ordered domains appear first followed by thenucleation ofDO22 ordered domains.L12 and DO22 ordering were observed to take place eitherthrough a continuous mechanism or a nucleation and growth mechanism. Their results agree with theexperimental observations of Benderskyet al. (2). In contrast to the work of Poduri and Chen (3), wehave conducted Monte Carlo simulations in the three-phase region of the phase diagram where FCC andbothL12 andDO22 phases coexist. The kinetic pathway was studied in the model system Ni-6.6at.%Al-15.1at.%V at 800°C.

2. Principle of Monte Carlo Simulations

2.1. Atomistic Kinetic Model

The simulated crystal is constructed on a rigid FCC lattice with periodic boundary conditions. A, B andC atoms and one vacancy(v) are distributed on about 1 million of sites. We describe the kinetic in theternary alloy by the following model (4), (5): i) the diffusion proceeds via vacancy jumps towardsneighbour atoms; ii) the vacancy jump is a thermally activated process and its frequency is given by:GXv 5 nXexp(2DEXv/kBT).

The activation energyDEXv is the energy required to extract the jumping species from theirenvironment minus the energy recovered by placing the jumping atom at the saddle-point.DEXv isestimated assuming that the cohesive energy of the alloy is a sum of pair interactionsEXY

i (X,Y 5 A,B,Cor v) between X and Y atoms situated onith neighbours sites:

DEXv 5 EsX 2 O

Yenn~X!

EXYi 2 O

Yenn~n!YÞX

EYv (1)

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The summations are restricted to the first neighbours of the vacancy and up to the fourth neighbours ofthe atom X in this work (nn(v) and nn(X) respectively). The ‘attempt frequency’nX and the saddle pointcontributionEs

X of the atom X depend on the nature of the jumping specie (A, B or C) but not on thealloy configuration.

2.2. Residence Time Algorithm for MC Simulation

Since experiments were performed at 800°C, whereexp(2DE/kbT) ,, 1, a ‘residence-time algorithm’was used rather than the standard Metropolis algorithm (6). To leave a configurationC, the system canundergoZ1 transitions corresponding to theZ1 possible first neighbours vacancy jumps with frequen-cies:G1(C), G2(C), . . .,GZ1(C). After a time t, the probability that the system is still in the configurationC is given by:P(C, t) 5 exp(2t{t(C)) with t(C) 5 { Si51

Z1 Gi(C)} 21. At each MC step, a randomnumber R between 0 and 1 is chosen and the transition j satisfying the inequality:Si51

j21 Gi(C) # Rt(C) ,Si51

j Gi(C) is performed. The physical time attached to this MC step ist(C). One transition is performedat each step even at very low temperature. So it becomes relatively efficient, especially with the vacancyjump mechanism since the number of possible jumps is small.

3. Results

3.1. Microscopic Parameters

The microscopic parameters of the jump frequencies can be related to available thermodynamic orkinetic quantities. Pair interactions between like atoms were fitted to the cohesive energies. Pairinteraction between unlike atoms were deduced from ordering energies (vXY

j 5 EXXi 1 EYY

i 2 2EXYi ).

The ordering energies between Ni and Al (7) and between Ni and V (this work) were chosen to fit thebinary Ni-Al and Ni-V phase diagrams respectively.vNiV

i values are close to those obtained by Caudronet al. (8) and by Poduri and Chen (9).vAlV

i (this work) were chosen so as to fit the experimental Al andV solubility limits in g, g9 andu phases in the ternary system at 800°C (Figure 1).

Pair interaction between vacancy and atom are deduced from formation energies of vacancies in puremetals. Kinetic parameters (attempt frequency and saddle point energies) were deduced from thediffusion coefficients (self diffusion for Ni and impurities diffusion for Al and V). The correlationfactors of the diffusion coefficients were supposed to be equal to 1. Some of the kinetic parameters wereadjusted in order to reduce vacancy trapping. The values of the parameters are listed in Table 1. Moredetails about parametrisation are available in a previous paper (10).

Figure 1. Isothermal section at 800°C of Ni-V-Al phases diagram. a) experimental [8] and b) simulated.

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3.2. Kinetic

Kinetic of decomposition has been simulated in the Ni-6.6at.%.Al-15.1at.%V at 800°C. The initialconfiguration is random. Kinetic was followed by means of three parameters.

1) the integrated intensity of the superlattice reflections:I 5 Skev S(k)/SkeV S(k) whereS(k) is thestructure factor (Fourier transform of the pair correlation function of atomic occupancies),v is thevolume defined asuk2ksua0,0.1 (ks is the superstructure vector anda0 the lattice parameter) andVis the volume which contains all the k vectors of the reciprocal space of the sample. The integratedintensity is calculated at {001} for Ni-Al (Ni-Al and these k-points are representative of theL12

phase—there is a small amount of Al in theDO22 phase) and at {0112} for Ni-V (Ni-V and these

k-points are representative of theDO22 phase—because the V content in theL12 phase is notnegligible (6at.%), the {001} points for Ni-V cannot be taken as representative of theDO22 phase).

2) the integrated intensity in the small k region such aska0,0.3. It gives information about theconcentration field as Small Angle Scattering (SAS).

3) the (001) simulated diffraction patterns for Ni-Al and Ni-V.

Figure 2 shows the time evolution of the integrated intensity in the small k region and around the{001} points for Ni-Al and the {011

2} points for Ni-V. Three different stages are revealed:

Stage 1.No SAS is observed (the decrease of the integrated intensity in the small k is due to thedecrease of the background scattering intensity. The total scattering intensity remaining constant, theincrease of the superlattice reflections induces a decrease of the background). There is a slight increasein the {001} and {011

2} intensities. Simulated diffraction patterns of Ni-Al exhibit intensity reinforce-

ments around the {001} and around {0112} for Ni-V (Figure 3): there is only short range ordering.

Stage 2.SAS of Al is observed: compositional changes in Al have taken place (Figure 2). 3Delemental maps of Al exhibit small Al-enriched and ordered zones (Figure 4a). The size of these zonesdoes not exceed 5 unit cells. Short range ordering develops: mainly {001} because of the presence ofAl-enriched zones (Figure 3).

Stage 3.SAS of V is observed and long range order now develops in the solute rich regions (Figure3 and Figure 4).

TABLE 1Bond Energies, Saddle-Point Contributions and Attempt Frequencies Used

in this Study

ENiNi1 5 20.74 eV EAlAl

1 5 20.557 eV EVV1 5 20.885 eV

ENiNii 5 0 for all i . 1 EAlAl

i 5 0 for all i . 1 EVVi 5 0 for all i . 1

ENiAl1 5 20.7283 eV ENiV

1 5 20.913 eV EAlV1 5 20.821 eV

ENiAl2 5 0.032 eV ENiV

2 5 0.025 eV EAlV2 5 0.05 eV

ENiAli 5 0 for all i . 2 ENiV

3 5 0.005 eV EAlV3 5 20.05 eV

ENiV4 5 20.0136 eV EAlV

4 5 20.015 eVENiV

i 5 0 for all i . 4 EAlVi 5 0 for all i . 4

ENiv 5 20.221 eV EAlv 5 20.223 eV EVv 5 20.273 eVEs

Ni 5 29.75 eV EsAl 5 29.36 eV Es

V 5 210.54 eVnNi 5 1.13 1015 s21 nAl 5 7 3 1014 s21 nV 5 7 3 1014 s21

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4. Discussion and Conclusion

Monte Carlo simulation of the kinetic in a three-phase alloy Ni-6.6at.%.Al-15.1at.%V at 800°C hasshown that precipitation of theL12 and theDO22 phases from the FCC solid solution takes place in threestages. In the first stage, only {001} and {011

2} short range ordering is observed. Some small

Al-enriched and {001} ordered zones appear in the second stage. TheseL12 nuclei, the size of whichdoes not exceed 5 unit cells, do not have the equilibrium composition of theL12 phase. Compositionalchanges of V take place in the last stage. The composition of theL12 zones tends toward the equilibriumcomposition and theDO22 phase appears. {001} and {011

2} long range order develop in the solute rich

regions. The equilibriumL12 andDO22 phases develop. These results are in agreement with simulations

Figure 2. Time evolution of the integrated intensities (a) in small k region and (b) around the {001} k-points for Ni-Al and the{011

2} k-points for Ni-V.

Figure 3. Simulated diffraction patterns for Ni-V in V1) stage 1, V2) stage 2, V3) stage 3 and for Ni-Al in Al1) stage 1, Al2)stage 2, Al3) stage 3.

SIMULATION OF KINETIC REACTION246 Vol. 44, No. 2

of Poduri and Chen (11). In contrast to the mean field simulations without Langevin noise of Poduri andChen, we were able to simulate the early stage of short range ordering of the solid solution.

Acknowledgment

This work was supported by the CRIHAN (Centre de Ressources Informatiques de HAute-Normandie).

References

1. R. Poduri and L-Q. Chen, Acta Mater. 46, 1719 (1998).2. L. A. Bendersky, F. S. Biancaniello, and M. E. Williams, in Solid-Solid Phase Transformations, ed. W. C. Johnson, J. M.

Howe, D. E. Laughlin, and W. A. Soffa, p. 899, TMS, Warrendale, PA (1994).3. P. Bellon and G. Martin, Mater. Sci. Forum. 209, 155 (1994).4. C. Pareige, F. Soisson, G. Martin, and D. Blavette, Acta Mater. 47, 1889 (1999).5. M. Metropolis, A. W. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953).6. T. A. Abinandanan, F. Haider, and G. Martin, in Solids-Solid Phase Transformations, ed. W. C. Johnson, J. M. Howe, D. E.

Laughlin, and W. A. Soffa, p. 443, TMS, Warrendale, PA (1994).7. R. Caudron, M. Sarfati, M. Barrachin, A. Finel, F. Ducastelle, and F. Solal, J. Phys. I. France. 2, 1145 (1992).8. F. H. Hayes, P. Rogl, and E. Schmid, in Ternary alloys, ed. G. Petzow and G. Effenberg, p. 8, VCH, Weinheim (1993).

Figure 4. a) 3D elemental map of Al in stage 2.V 5 7.7 3 5.3 3 0.7nm3. Al-enriched andL12 ordered zones are exhibited. 3Delemental map of b) Al and c) V in stage 3.V 5 10.63 6.8 3 1.2nm3. At this stage long range ordering develops in the soluterich regions.

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