Thermodynamic Analysis of the Mg RE Zn (RE = Y, La) Ternary hcp Phase Using the Cluster Variation Method

Thermodynamic Analysis of the Mg

­

RE

­

Zn (RE

=

Y, La) Ternary hcp Phase

Using the Cluster Variation Method

Satoshi Iikubo

1

, Shuji Hamamoto

2

and Hiroshi Ohtani

3

1_{Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology, Kitakyushu 808-0196, Japan}

2_{Graduate School, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan}

3_{Department of Materials Science and Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan}

In this study, thermodynamic properties of the Mg­RE­Zn (RE=Y, La) ternary hcp phase atfinite temperature have been investigated by

means offirst-principles calculations combined with the cluster variation method (CVM). Free energy calculation, including the configurational

entropy, shows that the Mg­Y­Zn ternary hcp phase has a tendency to phase separation. Conversely, the Mg­La­Zn ternary system does not

exhibit such behavior even around room temperature. Furthermore, the calculated spinodal region extends to a broader composition range and

the maximal spinodal temperature reaches above 1000 K for the Mg­Y­Zn system. Conversely, the spinodal region for the Mg­La­Zn system is

a limited narrow region near the Mg-rich side, and the maximal spinodal temperature is 300 K. Formation enthalpies calculated on the basis of

recent information from structure analyses do not show a definite difference in these two ternary systems. Therefore, we propose that the

dominant factors in the formation of a novel long period stacking ordered structure include spinodal decomposition as well as structure

transformation from 2H to other structures having periodic stacking faults. [doi:10.2320/matertrans.MI201222]

(Received November 29, 2012; Accepted January 23, 2013; Published April 25, 2013)

Keywords: first-principles calculations, cluster variation method, long period stacking ordered (LPSO), magnesium alloy, spinodal

decomposition

1. Introduction

The discovery of the long period stacking ordered (LPSO) structure in Mg-based alloys has attracted enormous interest

in the field of material science.1­10) _{Four types of LPSO}

structure®10H, 14H, 18R and 24R®have been found and

the mechanical properties of these structures have been extensively investigated. A LPSO structure is characterized by the periodical arrangement of a stacking fault (SF) introduced on the hcp lattice. In addition, rare earth and transition metal atoms added to alloys form a chemically ordered structure in the SF layers. These processes result in a simultaneous modulation in composition as well as atomic structure. This novel structure is widely acknowledged to be a prerequisite for an excellent mechanical properties of the Mg-based alloys, such as a yield strength of 610 MPa and an

elongation of 5% at room temperature.2) _{To improve the}

qualities of alloys, it is important to investigate the detailed process of the formation mechanism of the LPSO structure, particularly focusing on the origin of the compositional and structural modulation.

The atomic arrangement in LPSO structures has been

identified by transmission electron microscopy.9,10) _Before

the system reaches the final structure, some intermediate

states possibly exist. For example, in an early stage of the microstructure development, rare earth atoms and transition metal atoms in the alloys are randomly distributed on the hcp-Mg lattices. On the basis of our thermodynamic analysis

of the Mg­Y­Zn ternary system,7) we showed that a solid

solution of the hcp has a strong tendency to phase separation between the Mg-rich phase and the Y- and Zn-enriched phases. This could yield a compositional modulation observed in the LPSO structures. For a structural modulation, on the other hand, we found that long periodic structures become stable because of the vibrational entropy in a high

temperature region.8) _{Using first-principles phonon}

calcula-tions, we propose “disordered”long periodic structures as a

preliminary step in the formation of the LPSO structures.

With these findings as a background, the present study

aims to conduct a more detailed investigation of the thermodynamic properties of the hcp phase in the entire composition and all temperature ranges. Two alloy systems

are investigated, i.e., Mg­Y­Zn and Mg­La­Zn systems.

The former has a typical LPSO structure, while the latter does not. Free energies of the hcp phase at various temperatures

were computed by means of first-principles calculations

combined with the cluster variation method (CVM).

2. Calculation Procedures

In this study, the CVM11,12) has been used to calculate

configurational thermodynamic properties. The Helmholts

free energyFis minimized with respect to the configurational

degrees of freedom such as site occupancies and correlation functions. The enthalpy and entropy terms in CVM are given by eq. (1) using a cluster expansion (CE) as:

F ¼X¡max ¡

J¡²¡T X¡max

¡

£_¡S¡ ð1Þ

where¡represents a cluster,Jan effective cluster interaction

(ECI), ² a correlation function, £ a Kikuchi­Barker

coefficient,12) _{and S}

¡ the entropy contribution from the

cluster ¡.¡max represents the maximal cluster considered in

an expansion. For CE and CVM, we used the code developed

by Sluiteret al.13)

Once an accurate CE has been obtained, the total energy of

any Mg­RE­Zn arrangement can be calculated, and

there-fore, the free energies at afinite temperature can be obtained.

To make a CE for Mg, RE, and Zn configurations, we assign

an occupation variable·_ito each possible site i, which takes

the value+1 if Y or Zn resides at that site and 0 if Mg is at

that site. It has been shown that the dependence of any Special Issue on Long-Period Stacking Ordered Structure and Its Related Materials (I)

property on a configuration can be exactly expanded in terms

of polynomials ²_¡ of these discrete occupation variables ·_i.

²_¡ is defined as a product of occupation variables

·i;·j;. . .;·k, where the indices i, j, +, k correspond to a

collection of sites that form a cluster¡, such as a pair cluster,

a triplet cluster.

The maximal clusters were generated using an algorithm

that searches for the most compact clusters.14) Maximal

clusters are defined by two values, one is maximum distance between two vertexes, another is maximum number of vertex consisting clusters. As a result of several trial, we found the following optimum values; the maximum distance is 5.0 and

7.0¡, and the maximum number of vertex is 4 and 2, for

Mg­Y­Zn and Mg­La­Zn, respectively. A singular value

decomposition algorithm was used to extract values for 16 ECI out of a set of 244 using 133 enthalpies of formation for

Mg­Y­Zn. The same procedure was applied to extract values

for 22 ECI out of a set of 71 using 143 enthalpies of

formation for Mg­Zn­La. The formation enthalpy of a

structure is defined as the difference between its enthalpy and the concentration weighted enthalpies of pure Mg, RE and Zn with each stable structure. Figure 1 shows the hcp structure and Table 1 lists the number of points consisting each

clusters. As shown in the Table 1, for the Mg­Y­Zn system,

an optimal set of clusters consists of point, pair, triangle, and

four-point clusters, whereas for the Mg­La­Zn system, the

clusters consist of point and pair clusters. The values of the ECI for a particular system must be determined using an

accuratefirst-principles method. This is typically obtained by

calculating the energy of a series of configurations, and then

fitting a truncated form of the first term of eq. (1) to the

values of these energies. In this work, we disregard the electronic and vibrational degrees of freedom and consider only the contributions to the free energy of the configura-tional degrees of freedom associated with Mg, Y and Zn distributions. Free energy of a disordered state was calculated

from the obtained ECI energyJ_¡. Thermodynamic properties

such as tendency to phase separation can be investigated by this method.

Total energy calculations were performed using the VASP

code,15,16) _{which is based on the density functional theory.}

We employed Blochl’s projector augmented wave (PAW)

method as implemented by Kresse and Joubert.17,18)

Ex-change and correlation functions were given by the generalized gradient approximation as proposed by Perdew

et al.19) We used the standard version of the PAW­GGA potential for Mg, Y, La and Zn. A kinetic energy cutoff

was set to 345.9 eV. The k-point meshes of Brillouin zone

sampling in a primitive cell were¥centered grids with Fermi

broadening of 0.1 eV. The convergence criterion for

elec-tronic self-consistency and ionic relaxation loop are 10¹4_and

10¹3_{eV, respectively. In the present calculations, the unit}

cells of the Mg­RE­Zn ordered structures are fully relaxed

with respect to the volume, shape of the unit cell and atomic positions.

y

x

z

B

O A

C

0 1 2

3 4 5

6 7

8 9

10 11

Fig. 1 Side view of the a hexagonal cell with 11 atoms constituting a

cluster. The numbers denote atomic sites and correspond to the fourth column of Table 1. For pure magnesium, the a, b and c lattice parameters

[image:2.595.84.257.71.186.2]

are 3.15, 3.15 and 5.14¡, respectively.

Table 1 Effective cluster interaction for the multibody clusters. Site

numbers correspond to the numbers denoted in Fig. 1.

RE=Y

N type elements site ECI (meV/atom)

1 Nul ¹16.9

2 point Zn 0 ¹26.9

3 point Y 0 ¹184.5

4 pair Zn­Zn 0,6 ¹39.3

5 pair Y­Zn 0,6 ¹385.6

6 pair Zn¹Zn 0,7 ¹113.0

7 pair Y¹Y 0,7 ¹97.9

8 pair Y¹Zn 0,1 ¹197.5

9 four-point Y­Y­Zn­Y 0,1,4,6 133.7

10 pair Y­Y 0,1 385.7

11 triangle Y­Zn­Zn 0,3,4 ¹233.6

12 triangle Zn­Zn­Zn 0,1,6 212.5

13 four-point Zn­Zn­Zn­Y 0,1,4,9 135.4

14 triangle Y­Zn­Zn 0,1,4 ¹60.8

15 four-point Y­Y­Y­Y 0,1,4,6 ¹303.8

16 four-point Y­Y­Y­Y 0,1,6,9 224.3

RE=La

N type elements site ECI (meV/atom)

1 Nul 23.5

2 point Zn 0 ¹257.0

3 point La 0 ¹303.2

4 pair Zn­Zn 0,6 62.6

5 pair La­Zn 0,6 ¹65.0

6 pair Zn­Zn 0,8 ¹81.2

7 pair La­La 0,1 233.0

8 pair La­La 0,8 ¹116.1

9 pair La­Zn 0,11 ¹138.9

10 pair La­Zn 0,5 ¹48.3

11 pair La­Zn 0,10 ¹115.5

12 pair Zn­Zn 0,5 44.4

13 pair Zn¹Zn 0,11 238.9

14 pair Zn¹Zn 0,10 ¹44.2

15 pair La­La 0,10 20.0

16 pair La­Zn 0,7 ¹67.1

17 pair Zn­La 0,5 54.6

18 pair La­La 0,2 ¹67.9

19 pair La­La 0,11 76.8

20 pair La¹Zn 0,8 ¹57.4

21 pair La­Zn 0,2 ¹91.5

[image:2.595.303.548.92.625.2]

[image:3.595.50.293.66.314.2]

3. Results and Discussion

Figure 2 shows the calculated free energy surfaces of

Mg­Y­Zn hcp phase at (a) T=1000 K and (b) 500 K, and

of Mg­La­Zn hcp phase at (c) 1000 K and (d) 500 K. CVM

calculations were performed for various (x, y) values of

Mg1¹y(RExZn1¹x)y, and safely converged results are denoted

by small crosses in the figure. The free energies have the

lowest values around Y0.4Zn0.6 or La0.5Zn0.5 at 1000 K.

Around this compositional region, the system becomes much more stable compared to pure Mg. These results clearly show a strong attractive interaction between the Y and Zn atoms, or La and Zn atoms. These behaviors did not change at 500 K;

however, we could not obtain information around Y0.4Zn0.6

or La0.5Zn0.5because the CVM calculation did not converge

well. In a highly ordered state in which several correlation functions are nearly 0 or 1, CVM calculation becomes difficult. This situation was also caused by the strong attractive interaction between the RE and Zn atoms. The shapes of the free energy surfaces look similar in the

Mg­Y­Zn and Mg­La­Zn systems; however there is a

difference in their phase separation behaviors. Figure 3

shows the calculated free energies of Mg1¹y(YxZn1¹x)y, and

Mg1¹y(LaxZn1¹x)y. At the RE-rich corner forx=0.9, namely

Mg1¹y(Y0.9Zn0.1)yand Mg1¹y(La0.9Zn0.1)y, the sectional view

of the free energy curves has a simple parabolic form, as shown in Figs. 3(b) and 3(d). Conversely, the free energies

of Mg1¹y(Y0.5Zn0.5)y and Mg1¹y(La0.5Zn0.5)y, shown in

Figs. 3(a) and 3(c), respectively, show a monotonous

decrease with an increase in the y value. This is a clear

result from the attractive interaction energy between Y and Zn or La and Zn, and the system becomes more stable with a decrease in the Mg content. Here we note that the calculated

free energies of Mg1¹y(Y0.5Zn0.5)y at T=400 and 600 K

exhibit a slight additional positive contribution to the curves

around y=0.3. This represents a less stable condition in a

randomly mixed state, and directs the system towards a phase

separation between the Mg-rich corner and the Y­Zn-rich

side, as pointed out in our previous paper.7) Our previous

work was based on the calculation of phase diagrams (CALPHAD) method, in which the experimental data and

the enthalpy of formation calculated using thefirst-principles

method were analyzed, and thermodynamic properties were

simply predicted at a finite temperature. In this paper, we

confirm phase separation by first-principles calculations

combined with CVM, explicitly considering the finite

temperature effect of configuration.

Here, we will discuss spinodal decomposition accompa-nied by the phase separation. In this study, spinodal

temperatureTspiwas obtained as described below. Instability

of the system can be investigated by the the second derivative of the free energy with regard to the correlation functions. In a stable state, the free energy has the lowest value, and consequently the second derivative of the free energy with

regard to its variables are all positive. Therefore, we defined

Tspi as the temperature at which the smallest eigenvalue of

the second derivative of the free energy changes its sign from positive to negative. The calculated contours of the spinodal

(a) (b)

Mg Zn

Y

0 _{10 20 30 40 50 60 70 80 90}

100% 90 8₀ 70 60 50 40 30 20 10 0 10 0% 0 10 20 30 40 50 60 70 80 90 100%

-8 -10

-12 -12 -14 -14 -14 -16 -16 -16 -18

-18 -20

-20 -20 -22 -22 -24 -24 -26 -28 -30 -30 -32

T = 1000 K

0 ₁0 ₂0 _{30 40 50 6}0 ₇0 _{80 9}0

100% 90 80 70 60 50 40 30 20 10 0 100% 0 10 20 30 40 50 60 70 80 90 100% Mg Zn Y

-4 _{-6 -8} -10 -12

-12 -12 -14 -14 -14 -14 -16 -16 -18 -18 -18 -20 -22 -24

T = 500 K

(c) (d)

0 ₁0 _{20 30 40 50 60 70 80 9}0

100% 90 80 70 60 50 40 30 20 10 0 100% 0 10 20 30 40 50 60 70 80 90 100% Mg Zn La -6 -8 -10 -12 -14 -14 -16 -16

-16 -18

-18 -20 -20 -22 -22 -22 -24 -24 -26 -26 -28 -30 -32

T = 1000 K

Mg Zn

La

-4

-6 -8 -10 -12

-12 -14

-14 -14

-16

-18

-18 -20

-20 -20 -22 -22 -24 -24 -2 6

-26 -28

T = 500 K

Fig. 2 (a) Calculated free energy surfaces of Mg­Y­Zn at 1000 K, (b) at

500 K (upper right), (c) Mg­La­Zn at 1000 K, and (d) 500 K. Small

crosses indicate compositions where the CVM calculations are converged.

The contour lines of the free energy are shown up to ¹35 kJ/mol in

decrements of 2 kJ/mol.

-35 -30 -25 -20 -15 -10 -5 0 F (kJ/mol)

Mg1-y-(Y0.5Zn0.5)y

600 K 800 K 1000 K

(a)

-30 -20 -10 0 F (kJ/mol) 1.0 0.8 0.6 0.4 0.2 0.0 y Mg1-y-(La0.5Zn0.5)y

600 K 400 K 1000 K

(c)

-30 -20 -10 0 F (kJ/mol) 0.8 0.4 0.0 y

Mg1-y-(La0.9Zn0.1)y 600 K 800 K 1000 K

(d)

-30 -20 -10 0 F (kJ/mol) 0.8 0.4 0.0 y 600 K 800 K 1000 K

Mg1-y-(Y0.9Zn0.1)y

(b)

Fig. 3 (a) Calculated free energies of Mg1¹y(YxZn1¹x)y for x=0.5,

[image:3.595.330.520.71.427.2]

temperatures of (a) Mg­Y­Zn, and (b) Mg­La­Zn are shown

in Fig. 4. For the Mg­Y­Zn system, the spinodal temperature

reaches 1000 K and above at around Mg0.5Y0.2Zn0.3 and

exceeds 1100 K near Y0.5Zn0.5, and the spinodal region

extends to a broader compositional region. In contrast, for

Mg­La­Zn, the maximum spinodal temperature is 350 K, and

the spinodal region is limited to a small compositional region near the Mg-rich side. This comparison clearly shows the difference in phase separation behavior of these two systems. Taking account of the fact that an LPSO structure is formed

in the Mg­Y­Zn system and not observed in the Mg­La­Zn

system, it is strongly suggested that phase separation plays a key role in the formation mechanism of an LPSO structure.

Recent studies using transmission electron microscopy have shown that alloying elements such as Zn and RE form

local L12-type clusters in the fcc stacking layers in a hcp

matrix, and proposed plausible space groups ofP63/mcm for

the 14H type andC2/m,P3112 orP3212 for the 18R type.10)

The energies of formation for these structures were estimated

using the first-principles method, as shown in Table 2. All

LPSO structures have a negative values, and a definite

difference in the formation energies could not be found

between RE=Y and RE=La. These values are compared

with the calculated free energies of the hcp phase of the Mg­

RE­Zn system at T=300 and 500 K, as shown in Fig. 5.

The dashed line shows a tangent plane touching the free

energy surface of the Mg­Y­Zn system at 500 K. This

line represents the total energy of the two-phase mixture

composed of the Mg-rich hcp and Y­Zn-rich hcp phases. It

should be noted that the formation energies of these LPSO structures locate above the energy of the two separated hcp

phases. According to our previous calculations,7) the LPSO

structures become slightly stable at finite temperatures

because of the lattice vibration effect. However, the energy

gain is less than 1 kJ/mol and the situation depicted in Fig. 5

does not change. If the energy of formation plays a key role in the formation of an LPSO structure, as in the case of precipitation in the solution phase, an LPSO structure could

also be expected in the Mg­La­Zn system, but actually there

are no such reports to the best of our knowledge. Conversely, there is a pronounced difference in the spinodal temperatures

of the two systems above 1000 K and³350 K for the Mg­Y­

Zn and Mg­La­Zn systems, respectively. This contrast in the

behaviors of phase separation and spinodal decomposition may suggest a dominant factor governing the formation of an LPSO structure.

According to the information concerning the

thermody-namic aspect of the Mg­RE­Zn alloys, it would be denoted

that whether the occurrence of spinodal decomposition in the temperature range where the alloying elements diffuse sufficiently enough is closely related to a key factor of the formation mechanism of an LPSO structure. This spinodal

(a)

(b)

0 _{10 20 30 40 50 60 70 80 90}

100% 90

80 70

60 50

40 30

20 10

0

100%

0 10 20 30 40 50 60 70 80 90 100%

Mg Zn

Y

mol% Mg

mol%

Y

mol% Zn 1100

1000

1000

900 900

900

800

800

700

700

700

700

600

600

500

500

500

0 _{10 20 30 40 50 60 70 80 90}

100% 90

80 70

60 50

40 30

20 10

0

100%

0 10 20 30 40 50 60 70 80 90 100%

Mg Zn

La

mol% Mg

mol% La

mol% Zn

300

200

200

100

100

Fig. 4 Calculated spinodal temperatures of (a) Mg­Y­Zn and (b) Mg­La­

[image:4.595.77.269.72.449.2]

Zn. The contours of the spinodal temperatures are shown up to 1200 K in increments of 100 K.

Table 2 Calculated energies of formation of 14H RE4Mg35Zn3(P63/mcm),

18R RE4Mg29Zn3(C2/m) and 18R Y4Mg29Zn3(P3112). The unit is for

the mole of atoms in each of the structures.

RE Energy of formation (kJ/mol)

14H (P63/mcm) 18R (C2/m) 18R (P3112)

Y ¹6.1 ¹7.0 ¹7.0

La ¹5.8 ¹6.7 ¹6.7

-14 -12 -10 -8 -6 -4 -2 0 2

F (kJ/mol)

0.12 0.08

0.04 0.00

x in Mg1-7/3xRE4/3xZnx

14H 18R

RE = Y RE = La RE = Y

RE = La

300 K

500 K

Free energy of separated hcp in RE = Y (500 K)

Fig. 5 Comparison of the formation energies of two-phase mixtures of the

[image:4.595.324.526.113.391.2]

decomposition may result in the segregation of Y and Zn on the (0001) plane of a hcp matrix, similar to the plate-like G.P.

zone observed in, for example, Mg­Zn alloys. Because the

radius of a rare earth element is, in general, 17­22% larger

than that of Mg, the hcp lattice is expanded in the vicinity of these enriched layers. Our previous calculation showed that the introduction of large-radius atoms causes structural transformation from hcp to other long periodic structures in

the finite temperature region.8) The transformation brings

about the periodic introduction of SFs in the hcp lattice. Then, the RE and Zn atoms will segregate to the SFs, mainly because of the difference in chemical potentials for each

element between fcc and hcp.20)_{This is the so-called Suzuki}

effect.21)_{Consequently, the LPSO structures form through the}

spinodal mechanism and structure transformation in the hcp matrix.

4. Conclusions

To investigate the formation mechanism of long period stacking ordered (LPSO) structures, the thermodynamic

properties of the hcp phase of the Mg­RE­Zn system were

examined by means of the first-principles calculations

combined with the cluster variation method. As RE elements, Y and La were selected because an LPSO structure has been

observed in the Mg­Y­Zn system, while no such formation

has been reported in the Mg­La­Zn system. The keyfindings

in the present study are as follows:

(1) The calculated free energy surfaces showed a

down-ward slope with an increase in the yvalue of Mg1¹y(RE0.5

-Zn0.5)y. This is mainly because of the strong attractive

interaction energy between the RE and Zn atoms. Both systems show a tendency to phase separation between the

Mg-rich corner and the RE­Zn-rich side.

(2) The spinodal temperatures in the ternary miscibility gaps were examined from a second derivative of the free energies. The critical temperature of the spinodal

decom-position in the Mg­Y­Zn system reaches 1000 K and above.

In contrast, it is ³350 K at most for the Mg­La­Zn system

and the spinodal region is limited to the Mg-rich side. (3) The formation enthalpies of the 18R-type LPSO

structure with C2/m, P3112 and 14H-type with P63/mcm

were calculated at ground state. All computed structures have negative values, and there seems to be no significant difference in the formation energies of these two ternary systems. It is noteworthy that the evaluated values locate

above the free energies of the Mg-rich and RE­Zn-rich

two-phase separated microstructure.

(4) On the basis of the results obtained in the present study, the LPSO structure in Mg alloys is formed by spinodal

decomposition between the Mg-rich and Y­Zn-rich regions

in the hcp matrix as well as structure transformation from the hcp to other structure in which periodic stacking faults are introduced.

Acknowledgment

We are grateful to Prof. M. H. F. Sluiter for helpful discussions about the CVM calculations. This work was supported by MEXT KAKENHI Grant Number 23109005 and JSPS KAKENHI Grant Number 24656416.

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