Effect of the Order Disorder Transition of the bcc Structure on the Solubility of Be in the Fe Be Binary System

Effect of the Order-Disorder Transition of the bcc Structure

on the Solubility of Be in the Fe-Be Binary System

Hiroshi Ohtani, Yoshiko Takeshita

*

_{and Mitsuhiro Hasebe}

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

In the Fe-Be binary system, the solubility of Be in the-Fe phase deviates significantly from the so-called Arrhenius equation near temperatures of 600_{C. The metastable ordering of the bcc structure in this binary system is expected to play a key role in the phase boundary} anomaly. Thus, a thermodynamic analysis of the Fe-Be binary system has been performed considering the ordering behaviour of the bcc phase. The total energies of the ordered structures based on the bcc lattice were obtained usingab initioenergetic 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 formation energy of thephase was also calculated using band-energy calculations. The results were analysed together with some experimental data using the sublattice model, and the equilibrium phase diagram was calculated. The results support the formation of a metastable (bcc + B2) two-phase region accompanied by an ordering of the bcc structure. This metastable ordering of the bcc phase was the dominant factor governing the anomalous change in the solubility of Be in the higher temperature range.

(Received November 20, 2003; Accepted February 3, 2004)

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

1. Introduction

It is well known that the solubility of Be in bcc Fe ()

deviates significantly from the Arrhenius equation; i.e., a

proportional relationship exists between the logarithm of the

solubility and the reciprocal of the temperature. Takayamaet

al.1,2) detected this anomaly experimentally in the Fe-Be

binary system and showed that the behaviour could be attributed to the magnetic transition of bcc Fe. According to their thermodynamic analysis, the solubility limit of the

FeBe2() phase in the binary system was approximated using

the following expression

x=¼exp Q

p_þQf

RT

; ð1Þ

where Qp _{and Q}f _{are the paramagnetic and ferromagnetic}

terms of the free energy change associated with the

dissolution of the FeBe2 phase into the phase, and are

described by

Qp¼ 1 2½G

form FeBe2

p

þ ½_FeBep;

Qf ¼1 2½G

Fe

f _{þ ½S}

Fe

f_TBe:

ð2Þ

In eq. (2), ½Gform

FeBe2

p_{, ½}

FeBe

p_{, ½G}

Fe

f_{, and ½S}

Fe f

represent the Gibbs energy of formation for the FeBe2phase,

the interaction energy between Fe and Be atoms in the

paramagneticphase, the ferromagnetic free energy, and the

ferromagnetic entropy of pure Fe, respectively. In terms of eqs. (1) and (2), it can be understood that the solubility limit

deviates from the Arrhenius relation whenTBe, the descent

rate of the Curie temperature with an addition of Be to pure Fe, is highly negative. However, the interpretation of the anomaly in the solubility line on the basis of eq. (1) requires

TBe¼ 1800K,2)which is much more negative than the

experimental result of1250K.3,4)_{A thermodynamic}

analy-sis5) _{assuming T}

Be¼ 1250K cannot reproduce the

solubility of Be in thephase with sufficient accuracy.

On the other hand, the age hardening of this alloy has been investigated experimentally in relation to the anomaly in the

solubility of Be.6–11)The results are summarized in Ref. 12),

and a brief outline of the ageing process of this alloy is as follows. The disordered bcc structure forms in the initial stage, and consequently, the B2-type ordered structure

separates in the bcc phase. Ah100imodulated structure with

changes in concentration was observed in some samples using electron microscopy. This ordering of the bcc structure possibly influences the solubility of Be.

Therefore, in the present study, a thermodynamic analysis of the Fe-Be binary system was carried out considering a metastable ordering of the bcc structure, to clarify the origin of the anomaly in the solubility.

2. Calculation Procedures

The construction of phase diagram calculations based on experiments and thermodynamic analysis are generally referred to as the Calculation of Phase Diagrams (CAL-PHAD) approach. This method provides a very accurate understanding of the properties originating in the macro-scopic character of the material under study. However, a shortcoming of this approach is that it is hard to obtain information on metastable equilibria, or on undiscovered phases, since the thermodynamic parameters from this method can only be evaluated from experimental data. Such serious problems, which are intrinsic to the CALPHAD approach, may be solved with the assistance of the first-principle energetic-calculation method. Thus, in the present study, some thermodynamic quantities for bcc structure

obtained by theab-initioapproach will be introduced in the

conventional CALPHAD-type analysis of the Fe-Be binary system.

*_{Undergraduate Student, Kyusyu Institute of Technology, Present address:}

Hitachi HBM Co. Ltd., Tokyo 140-8502, Japan.

2.1 Ab initiocalculation of the Gibbs free energy

The bcc phase is a metastable phase located in the central part of the phase diagram, and thus, the thermodynamic properties in this region are unknown. To clarify the uncertainty of the thermodynamic properties, the formation energies of some bcc-based ordered structures were calcu-lated using the Full Potential Linearized Augmented Plane

Wave (FLAPW) method. The formation energy of the

(C14) phase was also derived in the same way.

An outline for deriving the Gibbs free energies of the

bcc-based structures incorporating theab initiocalculations is 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 WIEN2k13) procedure,

based on the FLAPW method. The generalized gradient

approximation (GGA)14) _{was used for the exchange and}

correlation potentials. Muffin-tin radii of 2.0 au for Fe and Be

were assumed, and RKmax was fixed to 9.0, which almost

corresponds to the cut-off energy of 20 Ry.

With the total energies known, the formation energy of the

bcc-based superstructures, E_form , is defined by averaging

the total energy of the elements with chemical composition to the segregation limit, as shown in the following equation

E_formðVÞ ¼E_totðVÞ ð1x_BeÞEbccFe_tot ðVFeÞ

x_BeEbccBe_tot ðVBeÞ;

ð3Þ

where denotes the type of superstructure and V is the

volume.

Then, the effective cluster interaction energies, fviðVÞg,

can be extracted from these formation energies using the Cluster Expansion Method (CEM) developed by Connolly

and Williams.15) _{This leads to a set of}

composition-independent parameters, from which the energy of the set of superstructures can be reproduced in terms of a set of

correlation functionsf_ig,

E_formðVÞ ¼X

i¼0

viðVÞ _i; ð4Þ

where,viðVÞis 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

atom occupancy of the lattice site,p. The values of_i for the

superstructures considered in this study are summarized in Table 1. The formation energies to the segregation limit for

the Fe-Be binary system in the D03, B2, B32, and A2

structures in the ground state are summarized in Table 2. The

formation energy of the(C14) phase is also listed in Table 2.

The upper limit of the summation in eq. (4),, specifies the

largest cluster that participated in the expansion. In this study, we considered a tetrahedron as being the largest cluster, since in the bcc structure, the tetrahedron forms an irregular shape containing both the first- and second-nearest neighbour distances. The second pair interaction was also taken into account, which leads to the consequence that

mathematically, five should represent six types of cluster.

From Table 1, we can see thatf_igis a66matrix, and

that it is a regular matrix. The matrix inversion of eq. (4) yields the effective interaction energies as

viðVÞ ¼X

5

i¼0

ð_iÞ1E_formðVÞ: ð5Þ

The Gibbs free energies of the metastable bcc-based phase in the Fe-Be binary system were evaluated using the effective cluster interaction energies up to the tetrahedron cluster, including second pair interaction as

E¼X

5

i¼0

[image:2.595.303.549.95.204.2]

vii: ð6Þ

Table 1 The values ofi for the bcc superstructures {A-A2, A3B-D03, AB-B2, AB-B32, AB3-D03, B-A2}.

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

[image:2.595.47.551.636.786.2]

B-bcc 1 1 1 1 1 1

Table 2 The formation energies to the segregation limit for the Fe-Be binary system in the D03, B2, B32, and A2 structures in the ground state. The formation energy of the(C14) phase is also listed.

Basic lattice Composition of Be Strukturbericht Formation enthalpy (kJ/mol)

0 Fe A2 0.0

0.25 Fe3Be D03 12:3

BCC 0.5 FeBe B2 31:4

B32 9:3

0.75 FeBe3 D03 18:6

1 Be A2 0.0

Space Group Composition of Be Strukturbericht Formation enthalpy (kJ/mol) (bcc Fe, hcp Be standard)

P63=mmc 0.667 FeBe2 C14 31:5

At a finite temperature,T, the free energy of a phase of

interest, G, can be obtained by adding a configurational

entropy term,S, to the internal energy, as follows16,17)

G¼ETS: ð7Þ

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;

ð8Þ

wherexi;yij;yij;zijk, and wijkl 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, the triangle, and the tetrahedron clusters,

respectively, andN is the number of lattice points.

Minimizing the grand potential with respect to all the correlation functions allows for the Gibbs energy of mixing to be obtained as a function of composition at a constant

temperature,T.

2.2 Thermodynamic analysis

The Gibbs energy expression for the terminal solid

solutions (bcc), (fcc), -Be (hcp), and -Be (bcc), the

intermediate phases (C14), (C15), and", and the liquid

phases will be described in this section.

The regular solution approximation was applied to the

liquid, (fcc), -Be, and -Be phases. The molar Gibbs

energy was calculated using the following equation Gm¼xBeGBeþxFeGFe

þRTðxBelnxBeþxFelnxFeÞ

þxBexFeLBe,FeþGmag;

ð9Þ

where _{Gi denotes the Gibbs energy of element i. This}

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

_G

i0H

ref

i ¼AþBTþCTlnTþDT

2

þET3þFT7þIT1þJT9;

ð10Þ

where0_Href

i denotes the enthalpy of the pure elementiin its

stable state at 298.15 K. The parameter LBe,Fe denotes the

interaction energy between Be and Fe, and has a composi-tional dependency following the Redlich-Kister polynomial as

LBe,Fe¼0LBe,Feþ ðxBexFeÞ 1LBe,Fe

þ ðxBexFeÞ22LBe,Fe:

ð11Þ

To describe the bcc-B2 transition in the thermodynamic modelling, the Gibbs free energy of the bcc phase was

represented according to the two-sublattice model.18) The

Gibbs energy for one mole of phase, denoted as

(Fe,Be)m(Fe,Be)n, was represented by the two-sublattice

model as shown in eq. (12).

G¼y1_Fey_Fe2 G_Fe:Feþy1_Fey2_BeG_Fe:Beþy1_Bey2_FeG_Be:Fe

þy1_Bey2_BeG_Be:BeþRT

_m

mþnðy 1 Felny

1 Feþy

1 Belny

1 BeÞ

þ n

mþnðy 2 Felny

2 Feþy

2 Belny

2 BeÞ

þexGþGmag

ð12Þ

The termys

i denotes the site fraction of element i in the

sublattices. The termsmandnare variables denoting the size

of the sublattices, and straightforwardly, the relationships of

m¼0:5andn¼0:5hold for the B2 structure._Gi:jdenotes

the Gibbs energy of a hypothetical compoundij, and terms

relative to the same stoichiometry are identical, whatever the occupation of the sublattice. The excess Gibbs energy term,

ex_{G, contains the interaction energy between unlike atoms,}

and is expressed using the following polynomial. ex_G¼y1

Fey 1 Bey

2

FeLFe,Be:Feþy1Fey 1 Bey

2

BeLFe,Be:Be þy2_Fey2_Bey1_FeLFe:Fe,Beþy2Fey

2 Bey

1

BeLBe:Fe,Be ð13Þ

Li;j:k(orLi:j;k) is the interaction parameter between unlike

atoms on the same sublattice.

The Gibbs energies of the(C14) and(C15) phases were

described using the two-sublattice model assuming m¼

0:333,n¼0:667andm¼0:25,n¼0:75, respectively. The "phase was represented as the stoichiometric line compound, Fe0:1Be0:9.

The magnetic contribution to the Gibbs free energy,Gmag,

was given by the expression

Gmag¼RTfð Þlnðþ1Þ ð14Þ

where

fð Þ ¼11

A

(

79 1 140p þ

474 497

1 p1

3

6 þ 9

135þ 15

600

)

; for <1

ð15Þ

and

fð Þ ¼ 1

A

5

10 þ

15

315 þ

25

1500

; for 1; ð16Þ

and

A¼ 518

1125þ 11692

15975 1

p1

: ð17Þ

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

temperature, andis the mean atomic moment expressed in

Bohr magnetons, B. The parameter p depends on the

structure, andp¼0:4for the bcc phase andp¼0:28for the

fcc phase.

3. Discussion

3.1 Thermodynamic analysis of the Fe-Be binary system

An intensive critical assessment of this binary system has

been carried out.5)According to the assessed diagram based

on the work by Aldinger and Petzow,19) the Fe-Be binary

system is composed of the(bcc),(fcc),(C14),(C15),

The liquidus boundaries of the binary system have been

investigated.3,4,20–22)_{Three invariant reactions concerning the}

liquid phase occur, L+, L+-Be, and L +,

as described in detail by Okamoto.5)_{In addition, the eutectoid}

reactions of-Be_{-Be +, and-Be +"are known}

at lower temperature. The solidus boundary for thephase is

not well defined, while the solvus boundary has been

investigated.1,2)_{The observed trend was in good agreement}

with the earlier work.23)_{The Curie temperatures of Fe-rich}

Fe-Be alloys have been determined,3,4) _{and the Curie}

temperature of thephase was changed to1250K based

on these results. The age hardening process of the phase

was studied because of the anomalous change of the solubility line. Several intermediate stages were found before the final equilibrium condition was attained in the age hardening process. The initial stage of the process was a modulation of the homogeneous bcc structure, and the modulations developed into metastable precipitates with structures related to the matrix. It is generally agreed that the modulation gives rise to two phases: the disordered bcc and

the B2 type of metastable ordered phase. The stablephase

then finally precipitates. A metastable phase boundary for the

(bcc + B2) two-phase region has been proposed12) _using

several sets of experimental data,24–26) _{and Ino}12) _has

proposed a pairwise interaction model to explain the metastable phase relationships.

The information on the experimental data on the phase boundaries and other thermodynamic quantities were ther-modynamically analysed together with the estimated meta-stable quantities of the bcc phase. The evaluated parameters of the Fe-Be binary system are shown in Table 3. The descriptions of the lattice stability parameters for each pure element were obtained chiefly from the Scientific Group

Thermodata Europe (SGTE) data file.27)_{The Gibbs energies}

of mixing in the bcc solid solution at 227 and 727_{C are}

denoted by the square symbols in Fig. 1. For comparison, the values obtained by the present assessment are drawn using the solid lines for the ordered state (BCC B2) and the disordered state (BCC A2), respectively. The convex curva-ture of the free energy in the vicinity of the equiatomic composition corresponds to the formation of the B2 structure. The calculated results of the Fe-Be phase diagram are

compared with the experimental data1,3,4,20,21,23,28,29)in Fig. 2.

The shaded area shown in Fig. 2 is the metastable (bcc+B2) two-phase region, which is accompanied by the ordering of the bcc structure on formation. The dotted line shows the order-disorder transition line, along which the two-phase field expands into the higher temperature range.

3.2 Effect of the order-disorder transition on the solubility limit

The solubility of Be in thephase is shown in Fig. 3. The

solubility would be denoted by the broken line if there were neither an order-disorder transition nor a magnetic transition in the bcc Fe phase. This is approximated by the straight line following the Arrhenius law for dilute solutions. Thus, the deviation of the solubility from the ideal Arrhenius law is represented by the shaded areas. The dashed lines show the order-disorder transition temperature and the magnetic transition temperature of the bcc Fe phase. The solubility

of Be decreases in the lower temperature region below

650_{C, while it increases in the higher temperature region.}

The decrease in solubility in the lower temperature region can be explained from the viewpoint of the change in

magnetism.1,2) _{On the other hand, the solubility increases}

owing to the order-disorder transition of the bcc phase in the higher temperature range. Figure 4 shows the change in the

order parameter along a solubility line for thephase. The

order parameter was defined by the following equation

¼y1_Bey2_Be: ð18Þ

The solubility line in Fig. 4(a) intersects the order-disorder

transition line indicated by the dotted line at about 650C. As

can be seen in Fig. 4(b), the order parameter along the

solubility line increases at a higher temperature than 650_C,

yielding the progress of the B2 ordering in the bcc disordered phase.

The solubility of Be can be approximated from the equilibrium condition as follows. The chemical potentials of Fe and Be are derived from eq. (12) as eqs. (19) and (20), respectively,

_Fe¼GGFe:Fe¼GFe:Feþy1Bey 2 BeGþ

RT

2 ðlny 1 Feþlny

2 FeÞ

þy1_Beðy1_Fey_Be2 þy1_Bey2_FeÞ LFe,Be:Fe

þy1_Bey2_Beðy1_Bey1_FeÞ LFe,Be:Be

þy2_Beðy1_Fey_Be2 þy1_Bey2_FeÞ LFe:Fe,Be

þy1_Bey2_Beðy_Be2 y2_FeÞ LBe:Fe,Be ð19Þ

and

_Be¼GGBe:Be¼GBe:Beþy1Fey 2 FeGþ

RT 2 ðlny

1 Beþlny

2 BeÞ

þy1_Fey2_Feðy1_Fey1_BeÞ LFe,Be:Fe

þy1_Feðy1_Fey_Be2 þy1_Bey2_FeÞ LFe,Be:Be

þy1_Fey2_Feðy2_Fey2_BeÞ LFe:Fe,Be

þy2_Beðy1_Fey2_Beþy1_Bey2_FeÞ LBe:Fe,Be; ð20Þ

where the relationship,G¼_G

Fe:BeþGBe:FeGFe:Fe

_G

Be:Be holds. In deriving eqs. (19) and (20), the magnetic

free energy term,Gmag, was excluded, because a temperature

range higher than the magnetic transition point is considered here. The atomic fractions of the elements on each sublattice are expressed in terms of the order parameter as eq. (21).

y1_Fe ¼1xBe 2

y1_Be¼xBeþ 2

y2_Fe ¼1xBeþ 2

y2_Be¼xBe

2 ð21Þ

The equilibrium condition between theand phases is

shown in eq. (22), assuming that the ideal stoichiometry of

thephase is FeBe2.

_G¼

Feþ2

Be ð22Þ

Substitution of eqs. (19) and (20) in eq. (22) and assuming LFe,Be:Fe ¼LFe:Fe,BeandLFe,Be:Be¼LBe:Fe,Be, and in addition,

eliminating terms larger than the second power inxBeyields

Element Phase Lattice stability parameter, J/mol Temperature range,_C

L _GL

FeG bcc Fe

12040:176:55843T3:67515511021_T7 _25<T<1538

14544:7518:01055T2:29603051031_T9 _1538<T<5727

_G

Fe

_Gbcc Fe

1462:4þ8:282T1:15TlnTþ6:4104_T2 _25<T<1538

1713:815þ0:94001Tþ0:49250951031_T9 _1538<T<5727

Fe Tc¼1043,¼2:22 25<T<5727

Be _GBe Fe G

bcc Fe

3705:78þ12:591T1:15TlnTþ6:4104_T2 _25<T<1538

3957:199þ5:24951Tþ4:92511030_T9 _1538<T<5727

_G

Fe:FeG bcc

Fe 100000þ34:218T(present work) 25<T<2727

_G

Fe:FeGbccFe 225631.931 (present work) 25<T<2727

16065:48917:197431Tþ1:1542962TlnT1:974197103_T2_þ0:576371106_T3_12646T1 _25<T<1254

L _GL

BeG hcp

Be 128817:697652:043056Tþ83:9345962TlnT25:899997103T2þ1:535023106T326970699T1 1254<T<1287

126670:571615:444703Tþ78:4982999TlnT22:135851103_T2_þ1:117948106_T3_27235823T1 _1287<T<2727

7477:59428:148507Tþ4:0312159TlnT5:82533710

3_T2_þ1:12184106_T3_51381T1 _25<T<1254

Be

Be

_G; Be Be G

hcp

Be 114335:48575:994154Tþ73:9842999TlnT21:078651103T2þ1:119065106T327251743T1 1254<T<1287

118695:885594:274122Tþ76:201923TlnT21:182279103_T2_þ1:059735106_T3_28502591T1 _1287<T<2727

_G

BeG hcp

Be 63491:085T 25<T<2727

_G

Be:BeG hcp

Be 57001:114T(present work) 25<T<2727

_G

Be:BeG hcp

Be 70333:42T(present work) 25<T<2727

System Phase Thermodynamic parameter, J/mol Reference

L LL

Be,Fe¼ 733001:8T ð8500þ5:57TÞ ðxBexFeÞ G_Be:Fe0:5_Gbcc

Be0:5G bcc Fe

¼ 37100þ9T

G

Fe:Be0:5GbccBe0:5

_Gbcc Fe

0_L

Be,Fe:Be¼0L

Be:Be,Fe¼ 4T

0_L

Fe:Be,Fe¼0L

Be,Fe:Fe¼ 3804T

0_L

Be,Fe¼ 71900þ8:1T

Be LBe

Be,Fe¼ ð28450þ27:9TÞ ðxBexFeÞ 340 ðxBexFeÞ

2

Fe-Be _G

Be:Fe0:6667G hcp

Be 0:3333GbccFe ¼ 38000þ7:18T Present work

_G

Fe:Be0:3333G hcp

Be 0:6667GbccFe ¼143700þ25:9T

LBe,Fe:Be¼ 213450þ68:4T118400 ðy1Bey1FeÞ

LBe:Be,Fe¼L

Fe:Be,Fe¼151010:4Tþ5030 ðy2Bey2FeÞ L_Be,Fe:Fe¼ 216250þ67:3T112000 ðy1

Bey1FeÞ

" _G" Be:Fe0:9

_Ghcp

Be 0:1GbccFe ¼ 12350þ0:38T

_G

Be:Fe0:75GbccBe0:25GbccFe ¼ 33050þ5:65T

_G

Fe:BeG hcp

Be þ0:75GbccBe0:75GbccFe ¼2657009:07T

L

Be:Be,Fe¼ 60901760 ðy2Bey2FeÞ

Effect

of

the

Order-Disor

der

Transition

of

the

bcc

Structure

on

the

Solubility

of

Be

in

the

Fe-Be

Binary

System

[image:5.595.44.778.63.550.2]

-35 -30 -25 -20 -15 -10 -5 0

0.2 0.4 0.6 0.8

Fe

Be

BCC_B2 BCC_A2

227

°

C

-35 -30 -25 -20 -15 -10 -5 0

0.4 0.6 0.8

0.2

BCC_A2

BCC_B2

Fe

Be

ab-initio calculation

Molar Fraction, x

Be

Molar Fraction, x

Be

Gibbs free ener

gy

,

∆

G

/ kJ·mol

-1

Gibbs free ener

gy

,

∆

G

/ kJ·mol

-1

727

°

C

Fig. 1 The Gibbs energies of mixing in thebcc solid solution at 227_{C and 727C based onab initioenergetic calculations.}

200 400 600 800 1000 1200 1400 1600

Fe 10 20 30 40 50 60 70 80 90 Be

γ

L

bcc+B2

Be (mol%)

T

emperature

,

T

/

°

C

16Oes

Oesterheld 29Wev Wever 48Gor Gordon 48Tei Teitel 60Gel Geles 63Heu Heubner 65Ham Hammond 75Ko Ko 16Oes Oesterheld 29Wev Wever

α

ζ

ε δ

Fig. 2 Data for the calculated Fe-Be phase compared with the exper-imental data.

100 101 102

Solubility of Be in

α

-Fe

, (mol%)

6 7 8 9 10 11 12 13

order-disorder transition

magnetic transition

Reciprocal Temperature, 104_{/T / K}-1

Fig. 3 The solubility of Be in thephase.

0 0.1 0.2 0.3 0.4 0.5 0.6

Or

der P

arameter

Fe 10 20 30 40 50 60

Be Contenet in α phase (mol%)

830 °C

650

1000 1190

200 400 600 800 1000 1200 1400

5 10 15 20 25 30 35

Fe

bcc+B2

Be (mol%)

T

emperature

,

T

/

°

C

γ

_α

Fig. 4 (a) Enlarged Fe-rich portion of the calculated Fe-Be binary phase diagram. The dotted line shows the order-disorder transition line. (b) Change in the order parameter along the solubility line for thephase. The increase of the value at a higher temperature than 650_C yields the progress of the B2 ordering in the bcc disordered phase.

[image:6.595.97.504.74.285.2] [image:6.595.54.288.329.498.2] [image:6.595.330.522.330.517.2] [image:6.595.110.489.560.747.2]

the following equation.

fðxBe; Þ ¼ ðGGFe:Fe2GBe:BeÞ

2G4LFe,Be:Feþ 1

4ð3Gþ4LFe,Be:Fe 10LFe,Be:BeÞ2þ ð4Gþ16LFe,Be:Fe

8LFe,Be:BeÞxBe

2RTlnxBeþRT 2xBe

2

¼0 ð23Þ

Defining the solubility of Be to be x0

Be when ¼0, transforms eq. (23) into

fðx0_Be;0Þ ¼ ðGFeBe2

_G

Fe:Fe2GBe:BeÞ

2G4LFe,Be:Feþ ð4Gþ16LFe,Be:Fe

8LFe,Be:BeÞx0Be2RTlnx 0

Be¼0: ð24Þ

When eq. (24) is subtracted from eq. (23), then the difference is

xBe

x0 Be

¼exp

ð3Gþ4LFe,Be:Fe

10LFe,Be:BeÞ=8RTþ 1 8x2 Be

2

:

ð25Þ

According to the phase diagram, the solubility of Be at

727C can be estimated asxBe¼0:15. Under this condition,

xBe=x0_Be¼1:15when¼0:2, andxBe=x0_Be¼1:7when¼ 0:4. Thus, it can be understood that the solubility of Be increases with an increase in the magnitude of ordering in the

phase.

3.3 Relationship between ordering accompanied by phase separation and interatomic interactions

The coexistence of ordering and phase separation, as seen in the Fe-Be system, has been explained using pairwise

atomic interactions.12) _{That is, the ordering occurs in the}

crystal lattice owing to the attractive interaction between nearest neighbour atoms. The difference in size of unlike atoms results in a strain energy, which leads to an instability in the system. The atoms change their positions to lower the total energy of the crystal, and consequently, a separation of the ordered and disordered structures develops. The intera-tomic interactions were estimated by expanding the total energies of the ordered structures obtained from the band-energy calculations.

Using the Murnaghan equation of states,30)_{as shown in eq.}

(26), the total energies of the A2, B2, B32, and the D03

structures were expressed as a function of the volume

EðVÞ ¼ BV

B0_ðB0_1Þ "

B0

1V0

V

þ _V

0

V

B0 1

#

þEðV0Þ;

ð26Þ

where B, B0, and V0 are the bulk modulus, its pressure

derivative, and the equilibrium volume, respectively, at normal pressures. Table 4 shows the coefficients in eq. (26)

for each ordered structure. In Table 4,EðVÞis the total energy

of each ordered structure in the equilibrium volume of the B2

structure;i.e.,V0¼124:0438(au3_{). The formation energy of}

each structure is represented as E_form, by defining the

average concentration of the total energy of the bcc Fe and the bcc Be phases at the segregation limit. By applying eq. (5)

to these formation energies, the effective interaction,fviðVÞg,

[image:7.595.306.547.94.195.2]

corresponding to each cluster was calculated, as shown in Table 5. The positive numbers define the attractive force working between unlike atoms.

In Table 5, v2 represents the interaction energy between

the nearest neighbour atoms, while v3 denotes the

next-nearest neighbour interaction. This qualitative explanation concerning the coexistence of the ordering and phase

separation12) _{seems to be supported by our calculations.}

However, the interaction energy at the next-nearest neigh-bour position is extremely small, and is almost 1/6 that at the nearest neighbour position. Furthermore, according to an-other paper we have published on a thermodynamic analysis

of the Co-Al system,31) the coexistence of the A2 and B2

structures is possible, even in the case where the first- and second-nearest neighbour interactions have positive num-bers. Based on this knowledge, the attractive interaction at the nearest neighbouring position is the predominant factor in the coexistence of the ordering and phase separation, and the contribution of the chemical next-nearest neighbour inter-action may be rather small.

4. Conclusions

A thermodynamic analysis of the Fe-Be binary system was performed to examine the relationship between the anom-alous change in the solubility and the metastable ordering of

the bcc phase. The total energies of the A2, B2, B32, and D03

structures in the ground state were calculated using the FLAPW method. The cluster expansion method was applied to these results, and the effective interaction energies for a tetrahedron cluster were obtained, and the free energies at

Table 4 The coefficient of the Murnaghan equation of states for the bcc superstructures.

Structures B B

0 _EðV

[image:7.595.304.548.233.319.2] [image:7.595.103.237.666.730.2]

0Þ V0 GPa Ry/au3 _{Ry au}3 Fe (bcc) 334.2450 5.5525 5091:1796 155.5298 Fe3Be (D03) 75.3890 0.7851 3833:1698 144.3524 FeBe (B2) 180.8774 3.6243 2575:1689 124.0438 FeBe (B32) 87.6360 5.9438 2575:1367 122.6823 FeBe3(D03) 81.5496 3.6914 1317:1225 112.6662 Be (bcc) 244.0350 3.0866 59:0657 105.6658

Table 5 The effective interactionfviðVÞgcorresponding to the clusters.

Effective cluster interactions (kJ/mol) Tetrahedron CVM

v1 4:0

v2 28.8

v3 4:8

v4 4.0

finite temperatures were calculated for the metastable bcc

solid solution. The formation energy of thephase was also

calculated using band-energy calculations. The results were analysed together with some experimental data using the sublattice model, and the equilibrium phase diagram was calculated. The results are summarized as follows.

(1) The calculated binary phase diagram was consistent with the experimental data concerning the phase boundaries, thermodynamic quantities, and the Curie point, and also with the metastable Gibbs energy for the bcc phase calculated using the first-principle technique. (2) The solubility of Be decreases at temperatures lower

than 650_{C from the magnetic transition, while it}

increases at higher temperatures owing to the order-disorder transition of the bcc structure.

(3) The predominant factor in the coexistence of the ordering and phase separation, as observed in the Fe-Be binary system, is supposed to be the attractive interaction at the nearest neighbouring position.

Acknowledgments

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

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