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The effect of second neighbour repulsion on fcc binary alloy phase diagrams: A Monte Carlo study

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PRAMANA __ joumal of physics

© Printed in India

July 1993 pp. 41-49

The effect of second neighbour repulsion on fcc binary alloy phase diagrams: A Monte Carlo study

RITA KHANNA and T R WELBERRY

Research School of Chemistry, The Australian National University, P.O. Box 4, Canberra, ACT 0200, Australia

MS received 19 November 1992; revised 21 April 1993

Abstract. We report the results of the Monte Carlo simulation of the phase diagram of fcc binary alloys using a 3-D Ising model with nearest and next-nearest neighbour repulsive interactions. Calculations are carried out for a ratio of second- to first-neighbour interaction energies of 0-4. The resulting phase diagram contains three superstructures A2Bz, A2B and AsB, each separated by a disordered fcc phase. There was no evidence for the formation of an A3B phase. These results are in qualitative agreement with CVM results.

Keywords. Monte Carlo simulation; phase diagram; binary alloys.

PACS No. 81.30 I. Introduction

A problem of great importance in the thermodynamic properties of alloys is the informa- tion regarding the range and the stability of various possible phases. Nearest neighbour Ising models have long been used to approximate binary alloys and the calculated phase diagrams have found qualitative agreement with the experimentally observed ones. Despite such unquestionable success, it is, however, clear that such models cannot hope to explain the tremendous variety of phase diagrams observed in nature.

The model can be made more realistic and general by adding second and even further neighbour and/or multibody interaction [1, 2]. The ground state problem in the case of fcc binary alloy with nearest and next-nearest neighbour (nn and nnn) pair interactions has been completely solved. Kanamori and Kakehashi [3] have derived a list of ground state structures up to fourth neighbour interactions. However the results of this simple energy model show that a large number of observed fcc superstructures can be accounted for by nn and nnn interactions. Many of the structures predicted by the fourth neighbour interaction have not been experimentally observed. In this paper we focus our attention on ordering in a fcc binary alloy using nn and nnn repulsive interactions.

The model adopted here consists of a binary system with atomic species A and B occupying a rigid fcc lattice. In the Ising spin language, a site i occupied by an A atom is represented by an up spin Si = 1, and a site occupied by a B atom by a down spin Si = - 1. The configurational energy is determined, within the framework of 3D Ising model, by the sum of effective pair interactions between the first and second nearest neighbours, denoted, respectively, by J1 and J2 [4, 5]. The Hamiltonian is

41

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Rita Khanna and T R Welberry

equivalent to an Ising magnet in a magnetic field H

H=SIESiSj+S2ESiSj-HES,.

nn nnn

(1)

where the sum nn extends over all nearest neighbour pairs and nnn over all next-nearest neighbour pairs. For antiferromagnetic ordering (J1 > 0), three distinct types of families of ordered structures are observed depending on the magnitude of the ratio O~ = J2/J1 and are labelled as: (1) < 1 0 0 ) family for ct < 0, (2) <11/20> family for 0 < ct < 0"5, and (3) <1/21/21/2> family for • > 0"5 [6,7]. In a recent study of cation ordering in oxide systems [8], we have looked at the ordering in the regime (¢t > 0.5) and found that ct has a strong influence on the range and stability of various observed phases. In this paper, we will solely concentrate on the ordered structures in the region defined by (0 < ~ < 0"5). The ground state structures expected in this parameter range are to be found at the stoicbiometries: A 2 B2, A 2 B, Aa B and A5 B (see figure I).

Using the cluster variation method (CVM), Sanchez and deFontaine [4, 6] have obtained a prototype phase diagram for (0 < 0t < 0"5). The phase diagram reported for ct = 0"25 was only partially complete as the low symmetry structure AsB was not

b" ~,. {o. 1.o1

I

a - ( i . o . o ) /

A2B2

C - (0.0.2~

I I , ~ T

L . . o . o , / - -

A2B

b ' - (o. t.ol

a = ( # , ~ . 0 ) /

- - II

c = ( ~ ' o

' I I

, 6 1 I

b = (0.0, I) b

c = {÷,÷.17

o) l

L___I _

/ 0 ~ -

A3B AsB

Figure 1. Schematic unit cells for the structures of the <11/20) family: (a) A2B2, (b) A2 B, (c) A3B, (d) AsB.

42 P r a m a n a - J. Phys., Vol. 41, No. 1, July 1993

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Phase diagram of fcc binary alloys

included in the calculations. CVM results, in general, agree well with Monte Carlo results near stiochiometric compositions, but they have been found to deviate considerably in regions away from the stiochiometric compositions. Even though CVM calculations have been known to produce many of the topological features of the phase diagrams, it has been argued that a CVM phase diagram is a product more of the approximation used to produce it than the true phase diagram of the model Hamiltonian [9]. At the present time, one of the most reliable phase diagrams for binary alloys (Cu-Au, ~ < 0) has been obtained by Binder [10] using Monte Carlo simulation. A complete Monte Carlo phase diagram for 0 < ~ < 0"5 is not available, though Binder et al [11] have obtained transition temperatures at a few isolated points for ct = 0"25. In this paper, we carry out a complete phase diagram calculation using Monte Carlo methods for ~t = 0.4.

2. The M o n t e Carlo m e t h o d

We consider a system of N = 4L 3 spins on an fcc lattice with periodic boundary conditions. L was measured in units of the lattice constant, a, and has to be chosen such that the periodic boundary conditions are consistent with the expected ordered structures without the need of introducing antiphase boundaries in the structure.

From some of the expected orderings shown in figure 1, A2 B ordering will require L to be a multiple of 3, while A3B ordering will require L to be an even number.

We chose L to be a multiple of 6(12, 18,24 etc), so that it is consistent with all the expected structures. The structures A2 B2 and As B are also consistent with this choice.

The simulations were performed using single spin-flip Glauber dynamics in the grand canonical ensemble, with the concentration, CB, varying as a function of temperature and magnetic field. Starting from an initial configuration, the system was allowed to evolve according to the Metropolis algorithm [12]. The data were collected for typically five to ten thousand Monte Carlo steps per site (MCSS).

Labelling the four sublattices of the unit cell of the fcc lattice from 1 to 4, we introduce the sublattice magnetizations m~,

m v = ( 1 / N ) E ( S i ) , v = 1 to 4 (2)

where ( ) is computed by taking time averages over different Monte Carlo runs. The total magnetization M is given by

M = m t + m 2 + m 3 +m4

and is related to the average concentration c e of B atoms by

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c~ = (1 - M)/2. (4)

Order parameter distribution functions were used to locate the phase boundary and to determine the order of the transition 1-13]. P(~)d~b is defined as the probability the order parameter will take on a value in the range [q~, tk + 6q~]. Instead of using specific order parameters for each superstructure, we have carried out the distribution function analysis using only the magnetization order parameter M. From the peak positions of P(M), one can unambiguously identify the composition of various stable phases present in the system. There is no need for a priori assumptions regarding the Pramana - J. Phys., Voi. 41, No. 1, July 1993 43

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Rita Khanna and T R Welberry

presence of different phases and no phases can be left out of the phase diagram as in the case of CVM calculations for ~ = 0"25. In this work, a peak position in P(M) could be located with an accuracy of 0.002. The lattice size dependence of the order parameter was used to identify the order of the transition.

3. Simulation results

The phase diagram obtained from the analysis of distribution function data is shown in figure 2. As only pair interactions have been used in this simulation and the interaction parameters have been assumed to be independent of the concentration, this phase diagram is symmetric about ca = 0.5. The phase diagram is dominated by three superstructure phases separated by a disorded fcc phase (labelled here as F).

The three superstructures observed are: A 2 B2, A2 B and A s B and are labelled as such in the phase diagram. There is no evidence whatsoever for the A3B phase observed in some of the CVM simulations.

The A2B 2 phase extends from cs = 0.5 to 0.425 at low temperatures. The region of stability of this phase narrows with the increasing temperature and transforms into a disordered phase at k a T/J1 ~ 1.05 (c a = 0"5). The transition from the A2B 2 phase to the disordered F phase is clearly of first order and is also marked by a well defined co-existence region between the phase boundaries. Distribution function plots of magnetization as a function of magnetic field and lattice size are shown in figure 3.

The closest gap between the two peaks in the transition region identifies the limits of the co-existence region and can be computed with a fair degree of accuracy.

The next stable superstructure A2 B, observed in the concentration range ca = 0.28 to 0"38, is surrounded on both sides by the disordered F phase and' is stable below kB T/J~ ~ 1.0. The distribution function plots for F to A2B transition and A2 B to F transition are shown in figures 4 and 5 respectively. Both these transitions are of first order and have well defined co-existence regions. According to CVM simulations, we should observe the next superstructure around ca --- 0.25 with a formula of A 3 B.

We however could not see any evidence of a superstructure in this composition range.

1.4 J 1 '=1.0; J2:0.4 1.2

F

0.8

0.6

AsB

0.4 ' '

u.O 0.1 0.2

kBT/Jl 1.0

A2B A2B2

I I

0 . 3 0 . 4 0 . 5

C#

Figure 2. Temperature-composition phase diagram for the parameter set:

J1 = 1"0; J2 = 0.4. Various observed ordered structures are indicated as such in the phase diagram.

44 P r a m a n a - J. Phys., Vol. 41, No. !, July 1993

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Phase diagram of fcc binary alloys

1500

1000

A

v

13.

500

u . 0 8 0 (a)

2 . 5 2 5

~ t - - - 2 . 5 7 5 . . . • . . . . 2 . 7 0

A2B2 ----> F

, I t l I

r, 7 t

\ /i,,

J_),

0 . 1 0 0 . 1 2 0 . 1 4 0 . 1 6 M

1000

A2B2 --~ F (b)

1 2

800 .... o - - la

600

o., 400

200

0 . 0 9 0 0.11 0.1:3 0 . 1 5 M

Figure 3. Plots of the distribution function P(M) vs M for J2 =0-4 and k B T/J 1 = 0.7 for the A2B 2 to F transition. (a) Plots of P(M) for three different values of the field H/J~. The magnitudes of H/J 1 are indicated against the plot symbol. The lattice size for these simulations was L = 12. (b) Plots of P(M) for two different lattice sizes across the transition. The lattice size L is indicated against the plot symbol.

The peaks in the distribution function data were very similar to those observed for a disordered phase (figure 6).

The third superstructure observed was A s B in the concentration range Ca = 0"14 to 0"21. The overall shape of the phase boundaries for this phase is very similar to those for the A2 B phase. The distribution function plots for F to As B transition and A 5 B to F transition are shown in figures 7 and 8 respectively. Both of these transitions are also of the first order. N o new superstructure was observed in the concentration range cB = 0.0 to 0.14.

P r a m a n a - J. Phys., Vol. 41, No. 1, July 1993 45

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R i t a K h a n n a a n d T R W e l b e r r y

v

13.

2000

0 u . 2 2 1000

F ---) A2B

-..-.o---- 4.0 ... * ... 4.25 ... a .... 4.3 ----o.---. 4.5

/ l

0 . 2 6 0 . 3 0 0 . 3 4 M

Figure 4. Plots of the distribution function P(M) vs M for 12 = 0 . 4 and k B T / J 1 = 0-7 for the F to A2B transition for four different values of the field H / J 1 . The magnitudes of H / J 1 are indicated against the plot symbol. The lattice size for these simulations was L = 12.

2000

A2B ---) F

7.25

. . . . t - - - 7 . 5

... • .... 7.55

~ ; ~ 7.6

13.

1000

. / ~ " , t "

o

0 . 3 4 0 . 3 8 0 . 4 2

M

Figure 5. Plots of the distribution function P ( M ) vs M for J : = 0.4 and k , T / J t = 0"7 for the A2B to F transition for four different values of the field H / J l . The magtitudes of H / J ~ are indicated against the plot symbol. The lattice size for these simulations was L = 12.

4. D i s c u s s i o n

Sanchez and de Fontaine [4,6] have computed temperature-composition phase diagrams for fcc binary alloys for ~ = 0"25, 0"35 and 0.45 using the cluster variation method. Apart from the As B phase which was left out in the ~ = 0.25 simulation, the three other observed superstructures in all three simulations were at A2 B2, A2 B and A 3 B compositions. O u r M o n t e Carlo results for ~ = 0"4 are consistent with C V M results in this regard. The Monte Carlo phase boundaries a p p e a r to be quite similar 46 Pramana - J. Phys., Vol. 41, No. 1, July 1993

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Phase diagram of fcc binary alloys

v

2000

1000

u.46

0J

0.48

. . . U . . . .

I I

0.50 0.52

8 . 5 8 . 7 5 9 . 0 9 . 2 5

0.54 M

Figure 6. Plots of the distribution function P(M) vs M for J2 = 0.4 and ks T/J 1 =0.7 in the composition region where A3B phase should have been observed. The magnitudes of H/Jl are indicated against the plot symbol. The lattice size for these simulations was L = 12.

2000

1 2 . 5 . . . * ... 1 2 . 7 7 5 . . . q= . . . . 1 3 . 0

AsB ---) F

1000

~ ~ .,, : :

0 . . . . . .

u . 6 4 0 . 6 8 0.72

M

Figure 7. Plots of the distribution function P(M) vs M for J2 = 0"4 and

k s T/J~ =0.7 for the F to AsB transition for three different values of the field H/Jt. The magnitudes of H/J~ are indicated against the plot symbol. The lattice size for these simulations was L = 12.

to the CVM phase diagram for ~ = 0-45 with the following differences. In the CVM phase diagram, the A2 B2 to F phase boundaries (co-existence region) appears to be the broadest around ks T/da "-" 0-8 to 0.9 region and narrows down considerably with the decreasing temperature. In addition, this phase extends below cs = 0.4 at low temperatures and finally joins the A2B phase. The M o n t e Carlo results, which were computed above ks T/J 1 =0"5, due to the increased scatter in data at low temperatures, do not show this narrowing of co-existence region and also indicate a smaller region of stability for the A2 B2 phase. The transition temperature at cs = 0.5 P r a m a n a - J. Phys., Vol. 41, N o . 1, J u l y 1 9 9 3 4 7

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Rita Khanna and T R H/elberry

v 13.

2000

F --) AsB

1 0 . 0 0 ... ' • . . . . 1 0 . 1 2 5 . . . • . . . . 1 0 . 2

1 0 0 0

A

~ ~.

k b"

0.56 0 . 5 8 0.60 0.62 0.64

M

Figure 8. Plots of the distribution function P(M) vs M for J2 =0-4 and kB T/J1 =0'7 for the AsB to F transition for three different values of the field H/J1. The magnitudes of H/JI are indicated against the plot symbol. The lattice size for these simulations was L = 12.

appears to be identical in both simulations. The boundaries for A 2 B and As B phases are similar in both cases apart from the fact that the region of stability of these phases is over-estimated in the CVM phase diagram. Also, the transition temperature for the As B phase is lower in our Monte Carlo results. In addition, the CVM results indicate the presence of an A 3 B phase at very low temperatures.

As the second neighbour repulsion (~) decreases, the CVM phase diagrams tend to become fairly complicated. Main feature of these phase diagrams appears to be the gradual disappearance of the A3B phase as ~ approaches 0"5. Canonical ensemble Monte Carlo simulations of Binder et al [11"1 at ~ = 0.25 indicated the presence of an AaB phase at cB=0"25 ( k n T / J ~ = l ' l l 6 ) . Our simulations at ~ = 0 . 4 in the temperature range k B T/J 1 > 0.5 show a complete absence of the A3B phase. A more complete investigation of the A 3 B phase as a function of ~ by Monte Carlo methods is in progress. The other three superstructures A2B2, A2B and AsB are, however, stable in the entire parameter range (0 < ~ < 0"5).

We have considered here a very simple model, which cannot be expected to represent faithfully any real alloy system. We find that the behaviour of stoichiometric alloys is rather insensitive to the parameters of the model. Observing the ordering behaviour in the non-stoichiometric region would be a much more sensitive tool for checking whether a model correctly represents a real system. Many body interactions, concentration dependence of interaction strengths and contributions from more distant neighbours are expected to make simulation results more accurate. In addition, the Ising model in its present form cannot describe lattice-dynamical phase diagrams as one is looking only at the atomic ordering process and lattice relaxations/distortions are not allowed. There is still need for a lot of future work in this field.

References

[1] D P Landau and K Binder, Phys. Rev. BI7, 2328 (1978)

[2-1 R Kikuchi, J M Sanchez, D de Fontaine and H Yamauchi, Acta Metall. 28, 651 (1980) 48 Pramana - J . Phys., Vol. 41, No. 1, July 1993

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Phase diagram of fcc binary alloys

[3] J Kanamori and Y Kakehashi, J. Phys. (Paris) 38 Suppl. C7, 274 (1978) I-4] J M Sanchez and D de Fontaine, Phys. Rev. B21, 216 (1980)

1"5] M Allan and J W Cahn, Acta Metall. 20, 423 (1972)

[6] J M Sanchez and D de Fontaine, Phys. Rev. B25, 1759 (1982) [7] M S Richards and J W Cahn, Acta Metall. 19, 1263 (1971)

[8] R Khanna, T R Welberry and R L Withers, J. Phys. C5, 4251 (1993) I-9] K Binder and D P Landau, Phys. gev. B30, 1477 (1984)

[10] K Binder, Z. Physik 1345, 61 (1981)

[11] K Binder, J L Lebowitz, M K Phani and M H Kalos, Acta Metall. 29, 1655 (1981) 1-12] K Binder, in Monte Carlo methods in statistical physics (Springer, Berlin, 1979) vol. 1 1"13] O G Mouritsen, in Computer studies of phase transitions and critical phenomenon (Springer

Verlag, New York, 1984) p. 18

Pramana - J. Phys., Vol. 41, No. 1, July 1993 49

References

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