Phase Separation of the B2 Structure Accompanied by an Ordering in Co Al and Ni Al Binary Systems

Phase Separation of the B2 Structure Accompanied by an Ordering

in Co-Al and Ni-Al Binary Systems

Hiroshi Ohtani

, Ying Chen

and Mitsuhiro Hasebe

1

Department of Materials Science and Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan

2_{School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan}

The phase separation of the B2 structure in the Co-Al and Ni-Al binary systems has been studied by combiningab initioenergetic calculations with the CALPHAD approach. The total energies of the ordered phases based on the bcc lattice were obtained using first-principle band-energy calculations. The cluster expansion method was applied to the results, and the free energies at finite temperatures were calculated for the bcc solid solution. The Co-Al and Ni-Al binary systems were analysed thermodynamically by considering the estimated metastable free energy of the bcc phase. The descriptions of the lattice stability parameters for each pure element were obtained chiefly from the Scientific Group Thermodata Europe (SGTE) datafile. The optimized parameters reasonably reproduced the characteristic features of these binary phase diagrams. The metastable (A2+B2) two-phase field forms in the Co-Al phase diagram, and this equilibrium is closely related to the anomaly in the phase boundaries of the binary system. On the other hand, the phase separation of the A2 and B2 structures are hindered by the presence of the D03phase in the Ni-Al system. Ground state analysis was performed to clarify the difference in the behaviour of the B2 phase. (Received November 20, 2003; Accepted February 3, 2004)

Keywords: phase diagram, thermodynamic analysis, ab initio energetic calculations, metastable phase separation, order-disorder transition

1. Introduction

Various changes in the physical properties of alloys are indirectly reflected in the equilibrium phase diagrams. For example, it is well known that a magnetic transition is manifested in an abnormal behaviour at the phase boundaries in some alloy systems. The phase boundaries in the Co-Al binary system may be classified as such a category. The Co-Al phase diagram consists of liquid, two fcc solid solutions

(Co) and (Al), hcp Co ("),with CsCl structure, and the intermetallic phases Co2Al5, CoAl3, Co4Al13, and Co2Al9. There exist one eutectic reaction concerning the liquid phase and six invariant reactions between the solid phases. The B2 () phase has a large homogeneity range at higher temper-atures, but the range decreases remarkably in the lower temperature region. It has been confirmed experimentally that the two-phase region for the A1 ()/B2 () phases extends over a wide composition range with decreasing temperature.1)Owing to a sudden drop in the homogeneous region, these experimental values are debatable, and are not adopted in the assessment of the binary phase diagrams.2) This judgement originates in the unusual shape of the phase boundary. Our preliminary analysis has shown that a phase separation, accompanied by an order-disorder transition of the B2 phase, plays an essential role in the anomaly at the phase boundary.

The Ni-Al binary system has almost the same phase constitutions as the Co-Al system. The system is composed of liquid, two fcc solid solutions(Ni) and(Al),with CsCl structure, and the intermetallic phases L12 (0), Ni5Al3, Ni2Al3, NiAl3. However, the features of the two phase diagrams are quite different, as shown in Fig. 1. In Fig. 1, the stable0_{phase (L1}

2structure) is omitted in the Ni-Al binary system to allow for a direct comparison with the Co-Al system. Although the presence of an order-disorder transition of the bcc structure is known, the phase equilibria for the A1 ()/B2 () composition appears differently for the two binary

systems.

Thus, in the present study, the difference in the behaviour of the B2 structure in the Co-Al and Ni-Al binary systems was studied using a thermodynamic analysis incorporatingab initioenergetic calculations. Furthermore, the phase separa-tion of the B2 structure is discussed in connecsepara-tion with the interaction energies between unlike atoms.

2. Calculation Procedures

2.1 Ab initiocalculations of the Gibbs free energy

To clarify the thermodynamic properties of the bcc-based structure over the entire composition range, the formation energy was estimated by incorporating band-energy calcu-lations using the cluster variation method. An outline for deriving the Gibbs free energies of the bcc-based structures is described as follows.

First, a set of superstructures {A-A2, A3B-D03, AB-B2, AB-B32, AB3-D03, and B-A2} was selected to be represen-tative of a series of bcc-based ordered phases. The total energies were calculated using the WIEN2k3) software package, based on the Full Potential Linearized Augmented Plane Wave (FLAPW) method within the generalized gradient approximation (GGA).4) _{Muffin-tin radii of 2.0 au}

(0.106 nm) for Co, Ni, and Al were assumed, and RKmaxwas

fixed at 9.0, which almost corresponds to the 20 Ry (270 eV) cut-off energy.

With the total energies known, the formation energy of the bcc-based superstructures,E_form, was defined by averaging the total energy of the elements with chemical compositions up to the segregation limit, as follows:

E_formðaÞ ¼EtotðaÞ x

ME bccM tot ðaÞ

ð1x_AlÞE_totbccAlðaÞ; ð1Þ

wheredenotes the type of superstructure andais the lattice constant. M denotes an element, Co or Ni.

The resultant formation energy can be further fitted using a

Special Issue on Advances in Computational Materials Science and Engineering III

Lennard–Jones type potential function, as shown in eq. (2).

E_form ðaÞ ¼ p

a 2n

q

a

n

þr ð2Þ

Here, the exponentn¼3:5is used to reproduceE_form to the highest accuracy.

Then, the effective cluster interaction energies,fviðaÞg, can

be extracted from these formation energies using the Cluster Expansion Method (CEM) developed by Connolly and Williams.5) _{This leads to a set of composition-independent}

parameters from which, the energy of the set of super-structures can be reproduced in terms of a set of correlation functionsf_ig

E_form ðaÞ ¼X

i¼0

viðaÞ

i; ð3Þ

where,viðaÞis the effective interaction energy of thei-point

cluster, and_i is the correlation function for clusteriin the phase ._i is defined as the ensemble average of the spin operator,ðpÞ, which takes values of1, depending on the atomic occupancy of the lattice site,p. The values of_i for the superstructures considered in this study are summarized in Table 1.

The upper limit of the summation in eq. (3),, specifies the largest cluster participating in the expansion. In this study, we considered a tetrahedron as being the largest cluster. Since in the bcc structure the tetrahedron is an irregular shape that

contains both the first- and second-nearest neighbour dis-tances, the second pair interactions are also taken into account, which leads to the consequence that mathematically, five should represent six types of cluster.

From Table 1, it can be seen that f_igis a regular66

matrix. Then, the matrix inversion of eq. (3) yields the effective interaction energies

viðaÞ ¼

X5

i¼0

ð_iÞ1E_formðaÞ: ð4Þ

The formation energies to the segregation limit for the Co-Al and Ni-Co-Al systems in the D03, B2, B32, and A2 structures in the ground state are summarized in Table 2. The Gibbs free energies of the metastable bcc-based phase in the Co-Al and the Ni-Al binary systems were evaluated using the effective cluster interaction energies up to the tetrahedron cluster, including the second pair interactions

E¼X

5

i¼0

vii: ð5Þ

At a finite temperature,T, the free energy of a phase of interest,G, can be obtained by adding the configurational entropy term,S, to the internal energy, as follows6,7) 600

800 1000 1200 1400 1600 1800 2000

20 40 60 80

Co

Al

γ(Co)

_β

L

γ(Al)

ε(hcp)

600 800 1000 1200 1400 1600 1800 2000

20 40 60 80

_Al

Ni

γ(Ni)

β

L

γ(Al)

Al (mol%) Al (mol%)

T

e

mperature

,

T

/ K

T

e

mperature

,

T

/ K

γ

'

NiAl

3

Ni

2

Al

3

Co

2

Al5

CoAl

3

Co

4

Al

13

Co

2

Al

9

Fig. 1 Characteristic features of the phase equilibria for the A1 ()/B2 () phases in the Co-Al and Ni-Al binary systems. The stable0 phase (with the L12structure) is omitted in the Ni-Al binary system.

Table 1 Values ofi for the tetrahedron cluster.

Ordered

0

1 2 3 4 5

structures point n.n.pair n.n.n.pair triangle tetrahedron

A-bcc 1 1 1 1 1 1

A3B-D03 1 1/2 0 1 1 1

AB-B2 1 0 1 1 1 1

AB-B32 1 0 0 1 0 1

AB3-D03 1 1=2 0 1 1 1

B-bcc 1 1 1 1 1 1

Table 2 Formation energies to the segregation limit for Co-Al and Ni-Al in the D03, B2, B32, and A2 structures in the ground state.

Alloy system Molar fraction of Al Structure Formation energy (kJ/mol)

0.25 D03 12:7

Co-Al 0.5 B2 62:3

B32 23:7

0.75 D03 18:3

0.25 D03 40:7

Ni-Al 0.5 B2 69:5

B32 34:0

G¼ETS: ð6Þ

We used the cluster variation method (CVM) with the tetrahedron approximation to calculate the configurational entropy. For the bcc-structure, the entropy formula is

S¼kBln

Q

i;j;k

ðNzijkÞ!

( )12

Q i

ðNxiÞ!

Q

i;j;k;l

ðNwijklÞ!

( )6

Q

i;j

ðNyijÞ!

( )4

Q

i;j

ðNyijÞ!

( )3;

ð7Þ

wherexi;yij;yij;zijk andwijkl are the cluster probabilities of

finding the atomic configurations specified by the subscript(s) at a point, the nearest neighbour pair, the second-nearest neighbour pair, a triangle cluster, and a tetrahedron cluster, respectively, andN is the number of lattice points.

By minimizing the grand potential with respect to all the correlation functions, the Gibbs energy of mixing can be obtained as a function of composition at a constant temper-ature,T.

2.2 Thermodynamic models

The Co-Al binary system is composed of the liquid (L), fcc ( Co, Al), B2 (), hcp ("Co), Co2Al5, CoAl3, Co4Al13, and Co2Al9 phases. On the other hand, the Ni-Al binary system consists of the liquid (L), fcc (Ni,Al), L12(0), B2 (), Ni5Al3, Ni2Al3, and NiAl3 phases.

The B2 structure () in the Co-Al and Ni-Al binary systems exhibits a range of non-stoichiometric alloys, and it was modelled using a multi-sublattice model,8)_by

consider-ing its orderconsider-ing behaviour. The general formula of the model was

Al1 M1

y1

Al y1M !

u1

Al2 M2

y2

Al y2M !

u2

: ð8Þ

M denotes Co or Ni in eq. (8). The subscriptsu1,u2. . .are the numbers of the sublattice sites that are within the brackets, and Al1_{, M}1_{. . .and Al}2_{, M}2_{. . .denote the elements} in the corresponding sublattice sites. The quantitiesy1

i andy

2

j

are the site fractions of the elementsiandjin their respective sublattice, designated by the numbers 1 and 2. If the relationship u1_þu2 _{¼1 holds, then the thermodynamic} quantities are referring to one mole of the sites. For each sublattice,s, the site fraction of speciesiis equal to

ys_i ¼ns_i=X j

ns_j ¼ns_i=ns with X

k

ns_k¼ns and X

s

ns¼n; ð9Þ

wherens

jis the number of speciesjin sublattices,nsis the number of sites in sublattices, andnis the total number of sites. The

quantityns _{is related tonbyn}s_¼nðus_=ðu1_þu2_{þ ÞÞ.}

Thephase was described using a four-sublattice model, so as to reflect the results of theab initiocalculations on the phase diagram computation as follows:

Al1 M1

y1

Al y

1

M !

0:25

Al2 M2

y2

Al y

2

M !

0:25

Al3 M3

y3

Al y

3

M !

0:25

Al4 M4

y4

Al y

4

M !

0:25

: ð10Þ

The number of each sublattice site is 0.25, and the four sites are therefore equivalent. The disordered state is described when the site fractions of the different species are the same in the four sublattices. For the ordered structures, if two sublattices have the same site fractions, as do the two others, but are different, then the model describes the B2 and B32 phases. If three sublattices have the same site fractions, and are different from the fourth, then D03ordering is described.

The Gibbs energy of the phase is expressed by the following equation:

Gm¼

X

i X

j X

k X

l

y1_iy2_jy3_ky4_lGi:j:k:l

þRT 4

X4

s¼1

X

i

ys_ilnðys_iÞ

þX

4

s¼1

X

i X

j>i

X

k X

l X

m

ys_iys_jyr_kyt_lyu_mLi;j:k:l:m; ð11Þ

whereGi:j:k:ldenotes the Gibbs energy of a compoundijkl,

and terms relative to the same stoichiometry are identical, whatever the occupation of the sublattice. Li;j:k:l:m is the

interaction parameter between unlike atoms on the same

sublattice.

The following regular solution approximation was applied to the disordered fcc (Co,Ni,Al), hcp ("Co), and liquid phases

G_m ¼xAlGAlþxMGM

þRTðxAllnxAlþxMlnxMÞ þxAlxML

Al,MþGmag;

ð12Þ

whereG_i denotes the Gibbs energy of elementi in the

phase. This quantity is called the lattice stability parameter, and is described by the formula

_G

i

_Href

i ¼AþBTþCTlnTþDT

2

þET3þFT7þIT1þJT9; ð13Þ

where_Href

i denotes the enthalpy of the pure elementiin its

stable state at 298.15 K. The parameter L_Al,M denotes the interaction energy between Al and M, and has a composi-tional dependency following the Redlich–Kister polynomial as

L_Al,M¼0L_Al,Mþ ðxAlxMÞ 1L

Al,M þ ðxAlxMÞ22L

Al,M:

ð14Þ

The magnetic contribution to the Gibbs free energy,Gmag,

Gmag ¼RTfðÞlnðþ1Þ; ð15Þ

where

fðÞ ¼11

A

( 791

140p þ

474 497

1

p1

3

6 þ 9 135þ

15 600

)

; for <1

ð16Þ

and

fðÞ ¼ 1

A

5

10 þ 15

315 þ 25

1500

; for 1; ð17Þ

and

A¼ 518

1125þ 11692 15975

1

p1

: ð18Þ

The variableis defined asT=Tc, where TC is the Curie

temperature andis the mean atomic moment expressed in Bohr magnetons,B. The value ofpdepends on the structure,

andp¼0:28for the fcc phase.

The L12 (0) phase in the Ni-Al system is an ordered structure based on the fcc phase. Therefore, it is also possible to treat the =0 _{equilibrium as an order-disorder transition.}

However, to avoid a complication of the parameters, a different thermodynamic model was applied to the0_phase,

and the Gibbs energy was represented using the two-sublattice model as

Al1 Ni1

y1

Al y

1

Ni !

0:75

Al2 Ni2

y2

Al y

2

Ni !

0:25 :

The Gibbs energy for one mole of the phase is given by the following equation.

Gm¼y1Aly

2

Al

_G0

Al:Alþy

1

Aly

2

Ni

_G0

Al:Niþy

1

Niy

2

Al

_G0

Ni:Al þy1_Niy2_NiG_Ni:Ni0 þRT0:75 ðy1_Allny1_Alþy1_Nilny1_NiÞ þ0:25 ðy2_Allny2_Alþy2_Nilny2_NiÞþy1_Aly2_Aly2_NiL_Al:Al,Ni0

þy1_Niy2_Aly2_NiL_Ni:Al,Ni0 þy1_Aly1_Niy2_AlL_Al,Ni:Al0

þy1_Aly1_Niy2_NiL_Al,Ni:Ni0 ð19Þ

Ni2Al3 is an intermetallic compound with an extension away from stoichiometry to the Al-rich side. The two-sublattice model given by

Al1 Ni1

y1

Al y

1

Ni !

0:4 Al2

y2

Al !

0:6 :

was applied, consideringy2

Al¼1.

Gm¼y1Al

_GNi2Al3

Al:Al þy

1

Ni

_GNi2Al3

Ni:Al

þRTðy1_Allny1_Alþy1_Nilny1_NiÞ þy1_Aly1_NiLNi2Al3

Al,Ni:Al ð20Þ

The Co2Al5, CoAl3, Co4Al13, and Co2Al9phases, as well as the Ni5Al3and NiAl3phases, with nil homogeneity ranges were treated as stoichiometric compounds.

Thermodynamic parameters of the Co-Al binary and Ni-Al systems were derived by analysing the reported experimental data and the free energies of the bcc structure derived by the

procedures described in Section 2.1. In the actual optimiza-tion process, the parameters for the bcc phase were first evaluated at 1000 and 2000 K using band-energy calcula-tions, and subsequently, the parameters for the other phases were obtained from the experimental data on the phase boundaries and thermochemical properties. Finally, all the parameters were optimized simultaneously, and a set of parameters that mostly agreed well with the experimental data was obtained.

3. Results and Discussion

3.1 Thermodynamic analysis of the Co-Al and Ni-Al systems

3.1.1 Co-Al binary system

According to the critical assessments of McAlister9)and Dupin and Ansara,10) _{there exist one eutectic reaction}

concerning the liquid phase and six invariant reactions between the solid phases. The peritectic reactions of Co2Al5, CoAl3, Co4Al13, and Co2Al9 were determined by thermal analysis and metallographic studies,11)_{and there were small}

discrepancies within experimental error. Several liquidus studies using melt analysis as well as thermal analysis have been reported, and the experimental data11–16)_{were adopted}

in this analysis. The B2 () phase that formed congruently from the melt at an almost equiatomic composition has a homogeneity range extending from 22%Al to almost 50%Al at higher temperatures, but the range decreases remarkably in the lower temperature region. The phase boundary on the Co-rich side has been studied metallographically and using X-ray diffraction,14) _EPMA,17) _{electromotive force}

measure-ments,18) _{lattice constant measurements,}19) _{and EPMA and}

lattice parameter measurements.1) _{The present analysis}

attempted to reproduce the experimental data of Takayama et al.,1) _{which locates the phase boundary nearest to the}

stoichiometric composition among the above measurement values.

The descriptions of the lattice stability parameters for each pure element were obtained chiefly from the Scientific Group Thermodata Europe (SGTE) data file,20) _{and are shown in}

Table 3. The evaluated parameters for the Co-Al binary system are shown in Table 4. The calculated Gibbs free energy of the bcc phases at 1000 K was compared with the results from theab initioenergetic calculations in Fig. 2. The calculated Co-Al binary phase diagram is compared with the experimental data1,12–15,21,22) _{in Fig. 3. The characteristic}

features of the binary phase diagram are well reproduced by the calculations. The dotted line shows the metastable two-phase separated region between the A2 (bcc Co) and the B2 () phases.

3.1.2 Ni-Al binary system

With regard to this binary system, a large number of experimental data have been reported, and a critical assess-ment has been carried out by Nashet al.23)_{The liquidus lines}

58%Al, and it melts congruently at 1911 K. The phase boundary of the B2 ()/L12 (0) composition was initially established using electron probe microanalysis.21,24,26)_{On the}

other hand, a slight difference in the experimental data can be seen for the phase boundary of the B2 ()/Ni2Al3phase, and the results obtained by X-ray and the microstructure observation24) _{are usually referred. Although the phase}

boundary of Ni5Al3 has not been clarified, an approximate position has been derived.27,28)The Ni phase dissolves Al up to 21.2% at the Ni-rich eutectic temperature of 1658 K.24) Since the L12(0) phase is an ordered structure based on the fcc structure, there is some scatter in the experimental data originating in the coherency with theNi matrix. Nashet al. adopted this experimental data,24,29) _{among the several}

reported values, and we have also adopted this strategy. With regard to the thermodynamic properties, the enthalpy of formation of each of the solid phases has been evaluated,30)

and the mixing enthalpy of the liquid phase has been reported.31)_{In addition, the enthalpies of formation of the}0_,

Ni2Al3, NiAl3phases have been reported.32)Thermodynamic analysis of this binary system has been carried out by several researchers,33–36)and even an attempt to calculate the phase diagram by applying the first-principle method has been carried out.37)

The procedure used in our thermodynamic analysis was the same as used for the Co-Al binary. The evaluated thermodynamic parameters based on the experimental data are given in Table 4. The calculated Gibbs free energy of the bcc phase is compared with the results of the ab initio energetic calculations in Fig. 4. The calculated Ni-Al binary

phase diagram is compared with the experimental da-ta24–26,28,38–42)_{in Fig. 5.}

3.2 Effect of the phase separation of the B2 structure on the phase boundaries

A thermodynamic analysis of the Co-Al system shows the occurrence of the metastable (A2 + B2) two-phase region in the Co-rich side of the phase diagram, as can be seen in Fig. 3. The two-phase separation, based on the bcc structure, closely relates the anomaly in the phase boundaries in the Co-Al binary system. This situation is schematically illustrated in Fig. 6, where the Gibbs free energies of the B2 () and the A1 ( Co) phases are drawn. The two-phase separation of the

phase does not occur at higher temperatures, T1, mainly owing to the effect of the entropy of random mixing on the atomic arrangement. On the other hand, the tendency towards ordering becomes stronger at lower temperatures,T2. In the B2 structure, the interaction between unlike atoms strength-ens around the 50%Al composition, since the degree of order reaches a maximum at the equiatomic composition. The decrease in the enthalpy term due to the attractive interaction results in the acute point in the free energy curve in the vicinity of the 50%Al composition, and consequently, a two-phase separation in thephase forms. This situation yields a significant shift of the phase boundary for the ( Co +)/

composition to the equiatomic composition.

3.3 Effective interaction energies of clusters

The variation in the effective interaction energy, viðaÞ,

with the lattice constant calculated using eq. (4) is shown in

Table 3 Lattice stability parameters for Al, Co, and Ni.

Element Phase Lattice stability parameter, J/mol Temperature range, K Reference

L _GL

Al _Gfcc

Al

11005:02911:841867Tþ7:9341020_T7 _298<T<933:6

10482:38211:253974Tþ1:2311028_T9 _933:6<T<6000

hcp _Ghcp

Al _Gfcc

Al 54811:8T 298<T<2900 20)

Al (bcc, B2) _G Al:Al:Al:Al

_Gfcc

Al 100834:813T 298<T<2900

0_(L1

2) G

0

Al:AlGfccAl 0 298<T<6000

Ni2Al3 GNiAl:Al2Al3 _Gfcc

Al 54811:8T 298<T<2900

Present work

L _GL

CoG hcp Co

15085:0378:931932T2:198011021_T7 _298<T<1768

16531:0569:683796T9:34881030_T9 _1768<T<6000

(fcc)

_Gfcc CoG

hcp

Co 427:590:615248T _298<T<6000

Co Tc¼1396,¼1:35

(bcc, B2) _G

Co:Co:Co:CoG hcp

Co 29380:7138T _298<T<6000

Tc¼1450,¼1:35

0_(L1

2) G

0

Co:CoG hcp

Co 427:590:615248T 298<T<6000

L _GL

Ni _Gfcc

Ni

16414:6869:397T3:823181021_T7 _{298<T<1728 20)}

18290:8810:537T1:127541031_T9 _1728<T<3000

hcp

_Ghcp

Ni GfccNi 1046+1.255T _298<T<3000

Ni Tc¼633,¼0:52

(bcc, B2)

_G

Ni:Ni:Ni:NiGfccNi 8715:0843:556T _298<T<6000

Tc¼575,¼0:85

0_(L1

2)

_G0

NiNiGfccNi 0 _298<T<6000

Table 4 The optimized thermodynamic parameters for the Co-Al and Ni-Al binary systems.

System Phase Thermodynamic Parameter, J/mol Reference

L LL

Al,Co¼3440102:7T ðxAlxCoÞ 65500 ðxAlxCoÞ216700 ðxAlxCoÞ31950

(fcc) L

Al,Co¼ 11750033:2T ðxAlxCoÞ ð9900þ25:45TÞ

Tc¼ 2300

_G0

Co:Al:Al:Al0:75 _Gbcc

Al 0:25 _Gbcc

Co _{¼ 24200þ1:56T}

_G

Al:Co:Al:Al0:75GAlbcc0:25GbccCo

(bcc, B2)

_G

Co:Co:Co:Al0:25 _Gbcc

Al 0:75 _Gbcc

Co _{¼ 13600þ13:2T}

Co-Al _G

Co:Co:Al:Co0:25GbccAl 0:75GbccCo Present work _G

Co:Co:Al:Al0:5 _Gbcc

Al 0:5 _Gbcc

Co ¼ 67700þ0:477T

0_L

Co:Co:Al,Co:Al¼ 12700þ7:84T Co2Al5 GCoAl:Co2Al50:714

_Gfcc Al0:286

_Ghcp

Co ¼ 60200þ8:91T CoAl3 GCoAlAl:Co30:75GfccAl 0:25G

hcp

Co ¼ 54900þ7:78Tþ5:14104TlnT Co4Al13 GCoAl:Co4Al130:765

_Gfcc Al0:235

_Ghcp

Co ¼ 52800þ7:5T Co2Al9 GCoAl:Co2Al90:818

_Gfcc

Al0:182G hcp

Co ¼ 45400þ8:065Tþ0:0026TlnT

L LL

Al,Ni¼ 11980039:4Tþ ðxAlxNiÞ ð5000053:44TÞ ðxAlxNiÞ231400

(fcc) L_Al,Ni¼ 178700þ8:91Tþ ðxAlxNiÞ ð7700þ13:3TÞ þ ðxAlxNiÞ2 ð3650017:5TÞ _G

Ni:Al:Al:Al0:75GbccAl 0:25GbccNi ¼ 33000þ1:113T _G

Ni:Ni:Al:Al0:5 _Gbcc

Al 0:5 _Gbcc

Ni ¼ 70300þ1:176T

(bcc, B2)

_G

Ni:Ni:Ni:Al0:25GbccAl 0:75GbccNi ¼ 406001:34T 0_L

Al,Ni:Al:Al:Al¼4500 0_L

Ni:Al,Ni:Al:Al¼ 24000

Ni-Al 0_L

Ni:Ni:Al,Ni:Al¼ 10400þ6:33T Present work

_G0

Ni:Al0:25 _Gfcc

Al0:75GfccNi ¼ 42400þ1:125T

0_(L1

2) G

0

Al:Ni0:75 _Gfcc

Al0:25 _Gfcc

Ni ¼ 22900 0_L0

Ni:Al,Ni¼2170016:8T

Ni5Al3 GAl:NiNi5Al30:375GfccAl0:625GfccNi ¼ 53300þ0:1T Ni2Al3

_GNi2Al3

Ni:Al 0:6 _Gfcc

Al 0:4 _Gfcc

Ni ¼ 72800þ9:82T 0_LNi2Al3

Al:Al,Ni¼ 5740 NiAl3 GNiAlAl:Ni30:75

_Gfcc Al 0:25

_Gfcc

Ni ¼ 89900þ42:75T

0.2 0.4 0.6 0.8

-70 -60 -50 -40 -30 -20 -10 0

Gibbs free ener

gy

,

∆

G

/ kJ·mol

-1

Molar Fraction, xAl

Co _Al

ab-initio calculation Thermodynamic analysis

Fig. 2 The calculated Gibbs free energy of the bcc phase in the Co-Al binary system at 1000 K compared with the results from theab initio

energetic calculations.

400 600 800 1000 1200 1400 1600 1800 2000

20 40 60 80

Gwyer Fink

Takayama Koester

Schramm

Ettenberg

Co

Al

β

γ(Co)

L

ε Co

2

Al5

CoAl

3

Co

4

Al

13

Co

2

Al9

γ(Al)

Al (mol%)

T

emperature

,

T

/ K

Fig. 7. The positive numbers define the attractive force working between unlike atoms. The i-point effective inter-action energy at the equilibrium lattice constant in these two binary systems can be compared using the data in Table 5. The equilibrium lattice constant of the B2 structure was 5.38 au (0.285 nm) for the Co-Al system, and 5.46 au (0.289 nm) for the Ni-Al system. There seems to be no remarkable difference in these values between the two binary systems.

Ino43)proposed a pairwise interaction model to explain the coexistence of the ordering and the phase separation. This model requires an attractive first-nearest neighbour interaction and a repulsive next nearest neighbour interaction. However, the sign of the interaction energies may not play an essential role in the systems under investigation in this study, since both parameters show positive numbers in the binary systems.

3.4 Origin of the different behaviour of the B2 phase in the Co-Al and Ni-Al systems

The formation energies to the segregation limit for Co-Al and Ni-Al in the D03, B2, B32, and A2 structures in the ground state, as listed in Table 2, are plotted versus Al concentration in Fig. 8. In comparing the two systems, it can be seen that in the Co-Al system, the energy of the D03 structure at the 25%Al composition is located slightly above the straight line connecting the energy values at the 0%Al and 50%Al compositions, which results in a two-phase separation of the A2 (bcc Co) and B2 () structures in Co-Al that is more stable than the formation of the Co3Al-D03 structure in the ground state. On the other hand, the energy plot of the Ni-Al system shows the stabilization of the Ni3Al-D03 structure compared with the two-phase separation of the A2 (bcc Ni) and B2 () structures, since the formation energy of the D03 Gγ

Gβ

Gγ

Gβ

Τ=Τ₁ Τ=Τ1

Τ=Τ₂

Τ=Τ₂

400 600 800 1000 1200 1400 1600 1800 2000

20 40 60 80

Co

Al

β

γ(Co)

ε

Al (mol%) Al (mol%)

Gibbs free ener

gy

Gibbs free ener

gy

T

e

mperature

,

T

/ K

Fig. 6 The schematic Gibbs free energies of the B2 () and A2 (Co) phases in the Co-Al system.

400 600 800 1000 1200 1400 1600 1800 2000

20 40 60 80

Alexander

Verhoeven Jia Hilpert Taylor Fink

Phillips

Robertson Nash

Ni Al

L

β

γ(Ni)

γ'

NiAl

3

Ni

2

Al3

Ni

5

Al3

γ(Al)

Al (mol%)

T

e

mperature

,

T

/ K

Fig. 5 A comparison of the calculated Ni-Al binary phase diagram with previous work.

-70 -60 -50 -40 -30 -20 -10 0

0.2 0.4 0.6 0.8

Ni Al

ab-initio calculation Thermodynamic analysis

Molar Fraction, xAl

Gibbs free ener

gy

,

∆

G

/ kJ·mol

-1

phase is lower than the straight line connecting the energy values at the 0%Al and 50%Al compositions. This energetic analysis suggests that a two-phase separation of the bcc-Co and B2 () phases occurs on the Co rich side in Co-Al system, while such a separation is not realized in the Ni-Al system.

Figure 9 shows the density of states (DOS) of the B2 () structure in the Co-Al and Ni-Al alloy systems. In Fig. 9,Ef

denotes the Fermi energy, where no electrons occupy the

electronic states above this energy level. It can be seen that the distribution of the DOS of the B2 phase in the two systems is similar, with both showing a low DOS at the Fermi level. However, when observing the DOS of the D03 phase for theses two systems, as shown in Fig. 10, we observe a marked difference between these two systems, in that the Fermi level is located near the peak of the DOS for the Co3Al-D03 phase but decreases in a region with a very low DOS in the Ni3Al-D03phase. This fact indicates that, from an energetic point of view, the stable D03structure of the Ni3Al phase is highly preferred, while in the Co3Al-D03 phase is unstable with respect to separation of the A2 and B2 phases in the Co-Al system.

This difference in the density of states at the Fermi level might be considered to be the result of the extra d-electron in Ni versus Co. From the point of view of the rigid band approximation,44)this difference in the number of electrons for two neighbouring elements shifts the Fermi level towards the higher energy side in the Ni-Al system versus the Co-Al system. This raises the different relative positions of the Fermi level in the DOS curve, which consequently leads to the different structural stabilities.

The ground state analysis of the Ni-Al system suggests that the phase separation concerning the D03 structure forms at absolute zero. When assuming that such a two-phase separation forms at a finite temperature, however, the agreement between the calculated phase boundaries and experimental data is insufficient. For example, Fig. 11 shows the phase diagram in the case that a two-phase separation between the metastable D03and B2 structures occurs around

-80 -70 -60 -50 -40 -30 -20 -10 0 10

0.2 0.4 0.6 0.8

Formation ener

gy

,

∆

E

/ kJ·mol

-1

Co Al 0.2 0.4 0.6 0.8

-80 -70 -60 -50 -40 -30 -20 -10 0 10

Ni Al

CoAl-B2 Co3Al-D03

CoAl-B32

CoAl₃-D0₃ Co-bcc Al-bcc

NiAl-B2 Ni₃Al-D0₃

NiAl-B32 NiAl₃-D0₃ Ni-bcc Al-bcc

Molar Fraction, xAl Molar Fraction, xAl

Fig. 8 Variation of the formation energies to the segregation limit for Co-Al and Ni-Al with concentration of Al.

Table 5 Effective interaction energy of the B2 structure at the equilibrium lattice constant for the Co-Al and Ni-Al binary systems.

Effective cluster Co-Al Ni-Al interaction (kJ/mol) (kJ/mol)

v1 15.6 33.9

v2 52.5 63.2

v3 5.7 9.1

v4 4.8 8:6

v5 2:9 6:6

Al-Co

4

1 2

3

ν ν

ν ν

ν

ν5

0

-50.0 -25.0 0.00 25.0 50.0 75.0

-50.0 -25.0 0.00 25.0 50.0 75.0

5.30 5.55 5.80 6.05 6.30

Lattice Constant (au)

5.30 5.55 5.80 6.05 6.30

Lattice Constant (au) Al-Ni

4

1 2

3

ν ν

ν ν

ν

ν₅

0

νi

Eff

ective Interaction / kJ·mol

-1

νi

Eff

ective Interaction / kJ·mol

-1

Fig. 7 Variation of the effective interaction energy,vi, of the Co-Al and

773 K. The homogeneity range of the B2 single phase is comparatively reduced owing to the influence of the metastable miscibility gap with decreasing temperature, and accordingly, the calculated phase boundaries deviate from the experimental data. Therefore, the phase separation of the D03structure is not likely in the experimentally observable

temperature range.

4. Conclusions

The phase separation behaviour of the B2 structure in the Co-Al and Ni-Al binary systems was studied using first-principle band-energy calculation and thermodynamic analy-sis, yielding the following results.

(1) The Co-Al and Ni-Al binary systems were analysed thermodynamically by considering the free energy of the bcc phase estimated fromab initiocalculations. The optimized parameters reasonably reproduced the char-acteristic features of these binary phase diagrams. The metastable (A2 + B2) two-phase field forms in the Co-Al binary system, and this metastable equilibrium relates closely to the anomaly seen in the phase boundaries in this system. On the other hand, the phase separation of the A2 and B2 structures is not likely in the Ni-Al system.

(2) The effective interaction energies for clusters were obtained using ab initiocalculations on the formation energies for the bcc-based ordered structures. However, there seems to be no marked difference in the values between these two binary systems.

(3) Ground state analysis showed that the D03 structure forms in the Ni-Al system, while two-phase separation between the ordered B2 and the disordered A2 phase occurs in the Co-Al system. The difference in the 400

600 800 1000 1200 1400 1600 1800 2000

20 40 60 80

Ni

Al

L

β

γ(Ni)

γ'

γ(Al)

D03+B2

Alexander

Verhoeven Jia Hilpert Taylor Fink

Phillips

Robertson Nash

Al (mol%)

T

emperature

,

T

/ K

NiAl

3

Ni

2

Al

3

Ni

5

Al

3

Fig. 11 The calculated Ni-Al phase diagram assuming that the critical temperature of the two-phase separation between the metastable D03and B2 structures was around 773 K.

0 5 10 15 20 25

-20 -15 -10 -5 0 5

-20 -15 -10 -5 0 5

0 5 10 15 20 25 30 35

Co3Al (D03)

Ni3Al (D03)

Energy, E / eV Energy, E / eV

To

tal density of stat

e

To

tal density of stat

e

Ef

Ef

Fig. 10 The density of states of the D03structure in the Co-Al and Ni-Al alloy systems.

0 2 4 6 8

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4 CoAl(B2)

0 2 4 6 8 10 12

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4 NiAl(B2)

Energy, E / eV Energy, E / eV

To

tal density of state

To

tal density of state

Ef

Ef

stability of the D03 phase in these binary systems corresponds to a difference in the number of d electrons between Co and Ni.

Acknowledgments

The authors are grateful to Prof. Tetsuo Mohri, Hokkaido University, and Prof. Marcel Sluiter, Tohoku University, for helpful discussions about the CVM calculations.

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