Reversion Process in FCC Ising System By H

Ⅰ

Transactions of the Japan Institute of Metals, Vol. 29, No. 3 (1988), pp. 198 to 207

Reversion

Process

By H. Okuda*

in FCC

Ising

System

and K. Osamura**

Reversion phenomena in the FCC Ising system was investigated using the kinetic Ising model. The results were compared with the experimental results obtained for the metastable G.P. zones in Al-Zn binary alloys. The present simulations have made it clear that the kinetic Ising model can qualitatively well reproduce the structure change during reversion in the Al-Zn metastable system. Comparison with the parameters obtained from direct observation in the real space and calculated ones in k-space has also defined that the reversion process in the Ising system is essentially the same as the one in the binary alloys, and two-phase

analysis

and its interpretation

applied

to Al-Zn

system is appropriate.

(Received July 5, 1987)

Keywords: computer simulation, kinetic Ising model, Monte Carlo methods, reversion, structure function, Guinier-Preston zones

.

Al-Zn binary alloy suggests that the latter process follows the former(7). We are going to show that the time evolution of the structure

Introduction

Recently, the kinetic Ising model has been intensively used to inquire into the nonequilibrium phenomena in a condensed matter, especially the kinetics of phase transition. For example, the phase decomposition of a binary system in square(1) or cubic lattice(2), order-disorder transition in triangle or cubic lattice (3) and so on(4). Computer simulations using the kinetic Ising model and the Monte Carlo method have the advantage in that we can observe the system in the real space and at any time during the process. In the present study, the reversion phenomena in the FCC Ising system was investigated as the system with some extent of temperature dependent mutual miscibility. The phase diagrams for the Ising system(5) and an Al-Zn binary alloy(6) are shown in Fig. 1. For such a kind of systems, two elementary processes can be considered in the reversion process. One is the process where the concentration of the precipitates decreases, and the other is the one that shrink to dissolve. Experimental *

**

Graduate

Student

the precipitates results for the

, Kyoto University, Sakyo-ku, Kyoto 606. Present address: Fujitsu Labolatories, Atsugi, Morinosato-Wakamiya 10-1, Atsugi 243-01, Japan. Department of Metallurgy , Faculty of Engineering, Kyoto University, Sakyo-ku, Kyoto 606, Japan.

parameters is qualitatively the same between the Ising system and the Al-Zn alloy, and interpretations of the parameters calculated in kspace were confirmed by the direct observation in the real space. Ⅱ .

Simulation

Procedure

The model used in the present study was the kinetic Ising model on the FCC lattice. Dimensions of the system were 15, 20, 30 unit cells for each side, and periodic boundary condition was used to avoid the finite size effect. Each simulation was averaged over eight independent runs, so that the effect of initial condition and statistical fluctuation were well eliminated. The temperature is measured by the unit of critical temperature Tc, and the length is measured by the unit cells of the FCC lattice. MCS (Monte Carlo Steps per spin) is used for the time unit. Solvent and solute atom species are denoted by A and B atom, respectively, and assigned to be s(r)=-1 and s(r)=+1 at each lattice site, r. The concentrations of the solute atom (spiece B), CB, examined here were 7.5%, 10% and 20%. The aging and reversion process was achieved by the spin-exchange mechanism according to the transition probability(8):

Ⅲ

Reversion

Process

in FCC

Ising

System

199

Fig. 1 Phase diagram for three-dimensional Ising systems (a) and the Al-Zn binary alloy system (b). Circles denote the reversion temperatures examined and the boxes denote the pre-aging temparature.

Wi,j=τ-1exp(-δH/KT) /(1+exp(-δH/KT)) where the

i and

j denote

exchange,

energy defined

difference by(9)

the

where related regular

state

before

respectively.

δH

between

δH=δ

solution,Ω(9),

the

and

after

denotes

the

states

which

ΣSiSj

J is the exchange to the interaction and

. (1)

is

(2)

integral parameter the

sum

Results

A set of 'snapshots' of the (111) cross-section during reversion at 0.8 Tc obtained for CB=20% is shown in Fig. 3. As the reversion proceeded, it was seen that the precipitates became less concentrated and dissolve into the matrix, and the number of small clusters such

which is of the is

taken

over all the nearest sites. The system was pre-aged for about 103MCS at 0.5 Tc, and then reverted at the temperatures between 0.6 and 1.5 Tc. From the atomic configuration obtained at each time, the structure function S(k, t) and the structure parameters such as averaged radius R, concentration Cp of the precipitates, and SRO were calculated, and the time evolution of these parameters was discussed. In order to reduce the statistical error, the parameters were averaged for at least eight independent runs. The simulation was performed on FACOM M382/380 system at Kyoto University and pseudo-random number was generated by the shuffle method registered on the library SSLII(10).The general flow chart of the simulation is shown in Fig. 2.

Fig.2

General

flow chart

of the simulation.

200

H.

Fig. 3

Okuda

and

Snapshots of the (111) cross section during reversion at 0.8 Tc.

as mono-, di- or trimers increases in the matrix during the process. This change was also seen in the (100) and (110) cross section. From the atomic configulation obtained at each time, structure function S(k, t) was calculated by the spherized averaging of Fourier transform of the density-density correlation function G(r, t)(1)(2), Si(k,

K. Osamura

t)=∫G(r,t)exp(-iR・r)dr

structure function decreased in the first stage, and after about 300MCS of reversion, it began to increase (b). The former stage corresponded to the reversion process, and the latter to the coarsening process at 0.7 Tc. It was also found

(3)

where G(r, t)is defined by G(r,t)=

(4)

where < > means the ensemble average and s denotes the average of s(r). S(k, t) corresponds to the small-angle scattering intensity from the polycrystalline sample, I(k). Structure function S(k, t) calculated for the reversion at 1.5 Tc is shown in Fig. 4. While the reversion temperature is above the phase boundary in this case, it is seen that the structure function decreases monotonously to vanish. The locus of the maximum of structure function, km, was almost constant in the early stage, and then shifted to the smaller k, which suggests that the average interparticle distance increases. Figure 5 shows an example of the change of the structure function during reversion within the phase boundary, i.e. partial reversion at 0.7 Tc. As shown in Fig. 5(a), the

Fig.4 above

Change

of the structure

the phase

boundary.

function

during

reversion

Reversion

Process

in FCC

Ising

System

201

that the decrease of the structure function in the first stage as mentioned above was more remarkable as the reversion temperature became higher. The time evolution of the structure function showed qualitatively good agreement with the experimental result of SAXS intensity measured for the reversion process of Al-Zn binary alloy(7)(10),which is shown in Fig. 6 for the reversion above the miscibility gap, and Fig. 7 for below the gap. In the SAXS measurements, the change of the zone radius during reversion is estimated from the Guinier approximation of the scattering intensity at time t, I(k, t)=I0 exp (-(kRG)2/5) or from the autocorrelation relation function obtained transform of the intensity,

(5)

part of the corby the Fourier

(6) The average radius for the Ising system can also be calculated by replacing I(k, t) by the structure function, S(k, t). These parameters obtained from the structure function enable us to compare directly with the experimental results obtained by SAXS measurements. An example of the correlation function calculated

Fig. 5 Change of the structure function during reversion below the phase boundary. (a) The first stage corresponding to reversion. (b) The second stage corresponding to coarsening.

Fig. 6 Change of the small-angle scattering intensity I(k) for AI-Zn alloys reverted above the miscibility gap of G.P. zone. The scattering intensity decreases to vanish, corresponding to Fig. 4.

202

H.

Okuda

and

K.

Osamura

Fig. 8 Correlation function at 0.7 Tc for 230 MCS.

calculated

for the reversion

Fig. 7 Change of the small-angle scattering intensity I(k) for Al-Zn alloys reverted below the miscibility gap of G.P. zone. The scattering intensity decreased firstly (a) and then increased gradually (b), which corresponds to Fig. 5(a) and (b) respectively.

for the reversion at 0.7 Tc is shown in Fig. 8. Changes of the radius during reversion at 1.5 Tc and 0.7 Tc were calculated for CB=20% from the structure function as shown in Fig. 9(a) and (b). For the reversion above the phase boundary (a), the radius remained almost constant in the early stage, and exceeding about 80MCS, it began to increase rapidly. In the case of the reversion below the phase boundary as shown in (b), both Guinier radius RG and the radius from the correlation function(11), R*, increased gradually in the first stage where the structure function decreased, and after exceeding about 350MCS, the radius began to increase rapidly. The onset of this increase of the radius was almost coincident with the time when the structure function began to increase and corresponded to the coarsening process. As shown above, the present simulation proved that the radius of the precipitates apparently never decrease during reversion. This behavior is very similar to the ones observed in the reversion above and below the zone solvus for Al-Zn binary alloys (7)(12). For the reversion process below the phase boundary, the Warren-Cowley SRO can be used to distinguish the reversion process and the coarsening process. In the present simula-

Fig. 9 Change of the average radius of precipitate during reversion above the phase boundary (a) and below it (b).

tion, the atomic configuration was known in real space, so that the W-C SRO was directly calculated using the definition (13), αB(i,j,k)=1-P(i,j,k)(B￨A)/P(B).

Here,

the indices

(i, j, k) denotes

(7)

the neigh-

Reversion

Process

in FCC

Ising

Fig.11 tron Al-15

Fig. 10 Warren-Cowley SRO 0.7 Tc as a function of time.

for CB=20%

reverted

at

bor site defined by the indices (i, j, k) of FCC lattice, and p(B|A) denotes the probability of finding B atom in the neighbor sites of A atoms defined by the indices. The change of W-C SRO for CB=20% reverted at 0.7 Tc is displayed in Fig. 10. The distance r is defined from the indices by r=√i2+j2+k2

.

(8)

It is seen that the order parameters decreased with time in the first stage, and after about 380MCS of reversion, they began to increase. The increase of the order parameters may be interpreted in a way that the solute atoms again began to coagulate, i.e., the coarsening of the precipitates occurred after 350MCS. The onset of the coarsening obtained from the short range order parameter for the nearest-neighbor is shown by arrows in Fig. 8(b). Under the assumption of the two phase analysis, it was concluded experimentally that the reversion process is divided into two stages for Al-Zn binary alloys(7)(12).The early stage was characterized by the nearly constant radius and the decrease in the concentration of the precipitates and the later stage by the decrease in the volume fraction and the number of precipitates. Figure 11 shows an example of the experimental result for an Al-15 at% Zn alloy reverted at 423K. The relative change of the volume fraction Vf and the average electron density difference between the precipitates and phase

the

matrix model.

Δρwere The

calculated average

using

electron-density

two-

System

Relative density at%Zn

203

change difference reverted

of

integrated

Δρ, and

volume

intensity fraction

Q,

elecVf

for

at 423K.

difference, which is proportionl to the difference in the solute concentration between the precipitates and matrix, decreased rapidly within 50s or so, and the volume fraction Vf decreased after about 100s. The coarsening process with constant volume fraction and electron density difference was seen after 200s. In the kinetic Ising model, the concentration of the precipitates, Cp, and the volume of precipitates Vp were calculated directly from the real space without any assumption. In order to calculate the concentration and volume of the precipitates in the discrete lattice, a criterion that defines 'inside the cluster' should be defined. In the present simulation, a lattice site is defined to be 'inside the cluster' if (1) The lattice site is occupied by a solute (B) atom and connected to the site which is determined to be inside the cluster by the nearestneighbor site, or (2) The lattice site is occupied by the solvent (A) atom and surrounded by no less than nc solute atoms, at least one of which is judged to be inside the cluster. As the critical nearest-neighbor number nc, decreased, the volume of the precipitate was estimated to be slightly larger, and then the concentration slightly smaller. This nc dependence was found to occur because the judgement at the precipitate-matrix interface is sensitive to the critical nearest-neighbour number. But the time-dependent behavior of the concentration was the same for nc=6, 7, 8 and 9. In the following discussion, the critical nearestneighbor number nc was chosen to be 8, i.e., two third of the total nearest-neighbor. The time evolution of the concentration during

Ⅳ

204

H. Okuda

Fig. 12 The concentration of the precipitates 10% as a function of time.

and

for CB=

reversion at 0.6 and 0.7 Tc for CB=10% are shown in Fig. 12. In order to estimate the concentration more accurately, the calculation was performed only for the clusters whose size were larger than 200 atoms. The concentration decreased monotonously in the early stage, and reached a constant value at about 22MCS for 0.7 Tc and 26MCS for 0.6 Tc. The concentration was found to converge to the lower value as the reversion temperature became higher. After the concentration converged to a constant, the reduction of the volume fraction of precipitates is expected to occur until the coarsening process at the reversion temperature began. The onset of the coarsening process estimated from the SRO were about 560MCS for 0.6 Tc and 1880MCS for 0.7 Tc, which is shown by the arrows in the figure. The onset time of the later stage of reversion and

Table

1 Onset time of the later stage of reversion sion below the phase boundary.

K. Osamura

coarsening process are listed in Table 1. The estimation concerning the reduction of the volume fraction has some ambiguity for the systems containing many particles, because the total number of the solute atom is conserved and no reasonable criterion exists in order to determine the cut-off size of the precipitates. The time evolution of the precipitated volume during reversion will be discussed later for single spherical particle with the change of its concentration. The concentration change during reversion for CB=20% at higher reversion temperatures was not calculated accurately for the later stage, because the average concentration exceeded the site-percolation threshold (Pc =0.198)(14). .

From structure

Discussion

the simulation results, change in the parameters during reversion has been

presented for the systems which had undergone phase decomposition at lower temperature, and the time evolution of the structure parameters was found to agree well with the experimental results for Al-Zn binary alloys. In order to make clear the reversion behavior for the individual precipitate, a rather artificial initial condition was introduced, i.e., the reversion process of single spherical particle was examined. In this case, the effect of interparticle diffusion of solute atoms was counted only through the periodic boundary condition. The

and the coarsening

process

obtained

for rever-

Reversion

Process

in FCC

Ising

System

205

Fig. 13 Snapshot of single particle during reversion at 0.9 Tc for (a) 50, (b) 330, (c) 1030 and (d) 2520MCS.

simulation size

was

was

performed

30×30×30

in the

in the unit

system cell,

and

whose as the

initial condition, a spherical particle with Cp= 100% and the initial radius of 4.5, 6.0 and 9.0 was used. The change of the system with initial radius R0=9.0 at 0.9 Tc was seen as a set of snapshots of (111) cross section in Fig. 13. In the figure, the circles denote the solute atoms and the solid line denotes the boundary of the particle determined by the program according to the rule mentioned already It is seen that in the early stage, the size of the particle was almost constant, and the concentration of the

particle was gradually decreasing as the solvent atom (A) was penetrating into the particle. And after the concentration of the precipitate has decreased, it began to shrink. The relative change of the concentration and the volume of the particle during reversion at 0.8 Tc is shown in Fig. 14. The concentration of the particle decreased in the early stage while its volume remained almost constant. After the concentration of the zone has almost converged to a constant, the volume began to decrease. It is seen that this sequence was the same for all the initial radii examined here. In the later stage

206

H.

Okuda

and

K.

Osamura

98.2%, and that for the clusters between 50 and 200 was 97.9%. This tendency of the sizedependent particle concentration is reasonable in consideration of the surface energy effect on the local equilibrium concentration. Ⅴ .

Fig. 14 The time evolution of concentration cp and the volume of particles Vp during reversion at 0.8 Tc.

where

the volume

of the particle

decreases,

particle with smaller initial radius completely dissolved into the matrix and disappeared. In contrast, the particle with the largest radius, i.e. R0=9.0, decreased its volume for a while, and exceeding 4000MCS, the volume of the particle remained constant, because volume fraction of the particle lied inside the phase boundary at 0.8 Tc and the partial reversion is expected for this particle. It has been made clear from the above discussion that for an individual particle, the reversion was clearly divided into two stages regardless of the situation whether the reversion was inside or outside the phase boundary at the reversion temperature. The early stage is characterized by the decreasing concentration and the nearly constant volume, and the later stage by the nearly constant concentration and the decreasing volume. In the reversion process of more realistic system with size distribution, these processes overlap because the smaller particle has changed its concentration earlier and then dissolve earlier. The concentration change shown in Fig. 12 was interpreted to be the concentration average of these particles which were to some extent in a different stage of reversion. It is seen that the converged value in Fig. 14 became slightly lower as the original particle radius was larger. This tendency was also observed in the reversion process in such a multiparticle system as discussed in Fig. 12. The converged value at 0.6 Tc was 97.7% in this case, while the average concentration for the clusters whose sizes were between 20 and 50 was

Conclusion

The reversion process in the FCC Ising lattice was investigated by means of a spinexchange mechanism and the Monte Carlo method. The results obtained in both k- and real-spaces was compared with the experimental results for Al-Zn binary alloys. The results obtained are as follows: (1) Changes of the structure function and other structure parameters were coincident with those obtained experimentally by the insitu SRSAXS measurement for Al-Zn binary alloys. (2) From the direct calculation in real space, it was made clear without the assumption of two-phase model that the reversion process was devided into two stages, the early one with decreasing concentration and the later one with decreasing volume fraction. Subsequently, coarsening at the reversion temperature was observed. Acknowledgment The authors acknowledge Mr. H. Kita of Kyoto University for valuable suggestions concerning the programming technique. EFERENCES R (1) M. Rao:Phys. Rev.,B13(1976),4328. (2) M. Kalos,J. Lebowitz,O. Penroseand A. Sur: J. tat. Phys., 18 (1978), 39; O. Penrose, J. L. S Lebowitz,J. Marro, M. Kalosand A. Sur: ibid., 19 (1978),243;P. Franzl,J. Lebowitz,J. Marroand M. Kalos:ActaMetall.,31 (1983),1849;O. Penrose,J. Lebowitz,J. Marro, M. Kalos and J. Tobochnik: ibid., 34 (1984),399. (3) M. Phaniand J. Lebowitz:Phys.Rev.Lett.,4 (1980), 366. (4) J. Marroand R. Toral:Physica,122A(1983),563. (5) J. Essamand M. Fisher:J. Chem.Phys., 38 (1963), 802. (6) V. Geroldand W. Merz:Scr.Metall.,1 (1967),33. (7) K. Osamura, H. Okuda, H. Hashizumeand Y. Amemiya:ActaMetall.,33 (1985),2199;H. Okuda, K.Osamura,H. Hashizumeand Y.Amemiya:submitted to ActaMetall.

Reversion

Process

(8) K. Kawasaki: Phys. Rev. 145 (1965), 224; Phase Transition and Critical Phenomena, ed. by C. Domb and M. S. Green, Academic Press, New York, Vol. 2. (9) K. Binder, M. H. Kalos, J. L. Lebowitz and J. Marro: Adv. Colloid Interf. Sci., 10 (1979), 173. (10) Fujitsu Scientific Subroutine Library II. (11) K. Osamura and Y. Murakami: J. Japan Inst. Metals, 43 (1979), 537.

in FCC Ising System

207

(12) K. Osamura, H. Okuda, H. Hashizume and Y. Amemiya: KEK Activity report 1985/1986, 317; K. Gerstenberg and V. Gerold: Cryst. and Tech., 20 (1985), 79. (13) B. E. Warren: X-ray Diffraction, Addison-Wesley, (1969). (14) D. Stauffer: Introduction to Percolation Theory, Taylor & Francis, (1985), p. 17.