CALPHAD Vol. 11, No. 2, pp. 253-270, 1987
Printed in the USA.
0364-5916,'87 $3.00 + .OO
(c) 1987 Pergamon Journals Ltd.
FORMULATION OF THE A2/B2/D03 ATOMIC ORDERING ENERGY AND
A THERMODYNAMIC ANALYSIS OF THE Fe-Si SYSTEM
Byeong-joo Lee, Seh Kwang Lee* and Dong Nyung Lee
Department of Metallurgical Engineering
Seoul National University
San 56-1 Shillim-2dong Kwanak-gu Seoul 151, Korea
* Presently with Division of Materials Engineering, Korea Advanced
Institute of Science and Technology, P.O.Box 131 Chongnyang Seoul Korea
ABSTRACT : The Fe-Si phase diagram is composed of three solutions and five
intermetallic compounds. The three solutions can be described as
sub-regular solutions unless the magnetic and A2/B2/D03 atomic
ordering reactions take place in the bee solution. The AZ/BZ/D03
atomic ordering energy has been formulated on the basis of a Cp-
integration method. The miscibility gap along the B2/D03 transi-
tion temperature below lOOOK is attributed to the magnetic and
atomic ordering reactions. All the intermetallic compounds except
the high temperature FeSi, phase are described as stoichiometrie
compounds. The free energy of the high temperature FeSi, phase
has been formulated based on a two-sublattice model. The thermody-
namic data for all the phases have been evaluated and used to
calculate the Fe-Si phase diagram.'
1. Introduction
The ferro-para magnetic transition, and the A2/B2 and B2/D03 transitions
take place in the bee solid solution of the iron-silicon binary system. The
strukturbericht symbols A2, B2 and DO3 stand for the random bee, CsCl type and
BIF, type ordered structures, respectively. These transitions should be eon-
sidered in calculation of the Gibbs free energy of the bee solution.
The magnetic ordering energy has been formulated by integrating empirical
formula of a specific heat contributed by the magnetic ordering reaction(l)
and has been used to calculate phase diagramsinvolving the magnetic transition.
The atomic ordering energy has been calculated using an extended Bragg-Williams
-Gorsky mode1(2-4). The BWG model cannot express the atomic ordering energy
in an analytical form. Therefore it is inconvenient to use the BWG model in
Calculation of complete phase diagrams. In order to circumvent the drawbacks of
the BWG model, Inden suggested to use a Cp-integration method by expressing
the specific heat contributed by the atomic ordering reaction in the same form
as used in the magnetic ordering reaction(5).
The purpose of this work is to formulate the A2/B2/D03 atomic ordering
energy in an analytical form based on the Cp-integration method and to analyze
the iron-silicon system in consideration of the ordering energy and others.
2. Formulation of the A2/B2/D03_atomic ordering energy
2.1. Formula for the specific heat contributed by magnetic or atomic ordering
When a pure metal undergoes a ferro-para magnetic transition at the Curie
Received 2 September 1986
253
254 B-J. LEE, S.K. LEE and D.N. LEE
temperature Tc, the specific heat contributed by the magnetic ordering reation
can be accurately expressed as the following empirical formula after Inden(6).
= Krn,lro R ln for r, < 1
= Kmasro R ln
1 t T1;;5
1 - r;;;5
for em > 1
where R is the gas constant, rm is defined as T/Tc with T being absolute tem-
perature, and the superscripts m, lro and sro represent magnetic ordering
reaction, magnetic long range order in the ferromagnetic temperature range and
magnetic short range order in the para magnetic temperature range respectively.
The coefficients Km,lro and Kmjsro are constants which depend on metals and
lattice structures.
The temperature dependence of the specific heat for the alloy Fe
shown in Fig.1. The curve shows the ferro-ParamaFneti~transition a Et'
Sils is
about
600~~ and the B2/D03 atomic ordering transition at about 1000°C. The atomic
ordering contribution to the specific heat may be expressed as in the magnetic
contribution(5).
Thus the specific heats contributed by the atomic ordering reations can
be expressed as the following formula.
CB,lro
a
= KB*lro R ln for _ =B' 1
B ,sro
cP
= KB,Sro R In
1 + TB5
TB5
for ?B 5 1
1 -T
CD,lro = KD'lro R ln
1 + $)
P 1
for 'CD < 1
(2)
CD,sro = KD,sro
1 t 755
P
R In
1 - $
for 'D ? 1
Here as in Eq.(l) TB and TD are defined by T/TB and T/TD respectively. The
suoerscripts and subscriots. B. D. lro and sro renresent A2 + 82 transition.
B2-+ DO3 transition, atomic.long range order and short range order below and
above the transition temperatures TB and TD respectively.
The order-disorder transition can occur in a solid solution. The tran-
sition temperatures TB and TD in the A-B binary bee solution may be expressed
as a function of composition.
TB = x(1-x) (TB,o + (l-2x)TB,l + (l-2xJ2TB,2
t (1-2~)~ TB,3 + (1-2xj4 TB,4
TD = x(1-2x)( TD,o + (l-4x)TD,l +
+ (l-2xJ3 TD,3 + (l-4& TD,4
3
(1-4x
3
1’ TD,2
(31
255
i ti - J
t’ 1
256 B-J. LEE, S.K. LEE and Q.M. LEE
where x is the atomic fraction of element B in the A-B system, and TB,i and
TB i are coefficients of the polynomials. The above formula indicate that
th& A&B2 transition occurs in x=0 and x=1, and the B&D03 transition in x=0
and x=0.5 at O°K in accordance with the BWG model(2).
Equation(Z) can be expanded into power series as follows(l) :
B,lro
cP = 2KB'lro R( T; + r;/3 + +5) for "B c 1
(4)
D,lPO
cP
= *KDJro R( T$ t ?;/3 + TD '95) fOF TD < 1
The coefficients K's will be evaluated in the following sections.
2.2. Formulation of the atomic ordering enthalpy
2.2.1. Evaluation of the ordering enthalpy from specific heat data
As temperature rises, energy is added to a system and the configurational
state changes from IJO3 to B2 and then to A2 state which In turn will increase
enthalpy of the system. The atomic ordering enthalpy can be evaluated from
specific heat data. The atomic ordering contribution to the specific heat is
schematically shown in Fig.2.
(RB2(=)
alpies absorbed above and below the A2/B2 transition temperature,
The_e~~~(TB)~ and ORBS - HB2(0)) , and those above and below the B2/
DO
a
transition temperature, {RD03(=) - HD03(TB)) and ~~~03(T~~ - H~~~(T~) -
RB 3(O) f can be evaluated using Eqs.(ii) as follows :
HB2(m) - HB2(TB) = i
TB
($"'" dT zz -$$ RTB KBjsro
?.?
H'*(T,) - HB2fO) =
B lro dT = cp' & RTB KB,lPO
0
m
HDo3(,) - RDa3(TB) =
I
,;,sro dT = 4 RTB KDysro
TD
I
TD
HDQ3(TD) - Hoo3(0) =
Q
(5)
Let fg and fy) be the fractZon of the total atomic ordering enthalpies which
are absorbed above TB and TB, Eqs.(s) then yield.
2 RTB KBpSro : & RTB KB'lro =: fg : (I-fg)
2 RTB KDYSro : &!j RTD KD,lro = fD : fl-fB)
(6)
When analyzing the specific heat data for the bee solution in the Fe-S1
system, Inden(7) found fB=fB=O.Z.
A2/B2/D03 ATOMIC ENERGY ORDERING ENERGY 257
2.2.2. Evaluation of the ordering enthalpy from Interchange energies
Inden et a1.(2,7) formulated the ordering enthalpy in terms of interchange
energies. Their model is briefly described in this section. The ground
states of the B2 and DO3 ordering are shown in Fip.3. The DO3 lattice (bee)
may be considered to be composed of four fee sublattices I, II, III, and IV as
shown in Fig.4. The atoms located in f1 + II) are nearest neighbours with
respect to the atoms in (III + IV) and vice versa ; the atoms in I or III are
second nearest reighbors with respect to the atoms in II or IV, respectively,
and vice versa.
The B2 and DO3 ordering become perfect at x=0.5 and x=0.25 or 0.75, res-
pectively. The number of A-B bonds per gram-atom of the A-B solution varies
with composition x. Therefore the number of A-B bonds can be expressed as a
function of composition x. The A-% bond can be formed not only between nearest
neighbor atoms A and B but also between second nearest neighbor atoms A and B.
Therefore two interchange energies W and w may be defined.
W= -2VAB + VAA + 'B%
(81
w = -&A% + vAA + vB%
where Vi. and vi.
3 4
are bond energies between nearest neighbors 2 and j and
between econd n arest neighbors i and j, respectively. By definition, -W/2
and-w/2 are the Internal energy changes when one A-B bond is formed between
the nearest neighbor atoms and between the second nearest neighbor atoms,
respectively.
When random mixing occurs at composition x in 1 gram-atom of solution, the
number of A-% bonds %A% is given by
NAB = No Z x (1 - x)
where No is the hvogadro number and Z is the coordination number. For bee
solution, Z=8 for the nearest neighbor atoms and Z=6 for the second nearest
neighbor atoms. Therefore the mixing enthalpy of random bee solution(A2) &Hi2
becomes.
8HA2 m = -No(4W t 3~) x(1 - x) (10)
If the solution changes from the disordered A2 state to the ordered B2 state,
the number of A-E bonds between nearest neighbor atoms as well as the number
of A-% bonds between second nearest neighbor atoms will change. For x being
less than 0.5, as in the Fe-Si system, the sublattice sites I and II will be
filled with only A atoms and the sublattic sites III and IV will be occupied
by A and B atoms randomly in the ratio (f-2x) : 2x. And the number of A-%
bonds between the nearest neighbor atoms, NAB nn and the number of A-% bonds
between the second nearest neighbor atoms, NAh,nnnbecome
NAB,nn =N,Zx
(11)
NAB,nnn = No Z x (1 - 2x)
It follows that the mixing enthalpy in the B2,AH$2,and the A2+%2 ordering
enthalpy, AHh2+32 ,are expressed as
&HE2 = -No {4Wx t 3wx(l - 2x)) (121
&%A2+B2 = &HE2 - &Hi2 = -No(4W - 3w)x' (13)
258 8-J. LEE, S.K. LEE and D.N. LEE
When the B2+D03 transition takes place in the solution, the number of A-B bonds
between the nearest neighbor atoms remains constant, whereas that between the
second nearest neighbor atoms changes, and N~B,nnn becomes
N AB,nnn= N, 2 x for x < 0.25
and No 2(1-2x)/2 for x > 0.25
(14)
DO3 Therefore the mix ng enthalpy of the DO3 state, AH,
enthalpy, ,HB2jDo 3
, and the B2+D03 reaction
,are given by
cHio3 = -No(4W t 3w)x for x < 0.25
(15)
and -N,4Wx - No3w(1-2x)/2 for x > 0.25
&Q+D03 = aHD03 _ AHi2
m
= -6Nowx 2 for x c 0.25
and -6Nowt0.5 - xl* for x > 0.25
(16)
Thus the ordering enthalpies have been expressed in terms of the interchange
energies W and w.
2.3. Evaluation of the coefficients K
B,sro
> K
B,lro
, K
D,sro and .KD,lro
It follows from Eqs.(5), (13) and (16) that
TB P
_&2+B2 = HB2(-) - BB2(0) = Cp'
i
B 1" dT t
i
C;ys'o dT
0 TB
= ( 71KB,lro + 79 ~~3~~‘) RTB
120 140
(17)
= No (4W - 3w)x2
Similarly,
TD m
+HB2+DO3 = BDO3(,) _ HDo3(o) =
i
CD,lro
P dT t i
C;ysro dT
0 TD
= ( 71 KD,lro + 79 KD,sro) RTD
120 i-&F
= 6 Nbwx* for x < 0.25
and ~N,w(O.~-X)~ for x > 0.25 (18)
The coefficients K
B,sro KB,lro KD,sro
(6), (7), (17) and (18)'as foll:ws :
and KDylro can be evaluated using Eqs.
KB,sro = (4W/k - 3w/k) x /TB/AB (19)
A2/B2/D03 ATOMIC ENERGY ORDERING ENERGY
KB,lro = $ (l/fB - 1)(4W/k - 3w/k)x2/TB/AB
KD,sro = (6w/k)x2/TD/AD for x < 0.25
and (6W/k)(0.5-X)2/TD/AD for x > O.;, 75
(20)
(21)
259
KD,lro = 474 m (l/f,-1)(6W/k)X*/TD/AD for X < 0.25 (22)
and $$ (l/fD-1)(6W/k)(O.5-x)'/TD/AD for x > 0.25
where k is the Boltzman constant
AB=g+ B (l/f, - 1) (23)
AD=& $$ (l/fD - 1) (24)
2.4. The A2/B2/D03 atomic ordering free energy
The short range order in the B2 and DO3 states contributes to the atomic
oredering energy in the A2 state. The atomic ordering energy for the A2 state
Cord,A2 can be calculated as follows :
T T
Gord,A2 =
i
I (t-T)/t 1 C;'=' dt t
I
{(t-T)/t I C;"" dt
m m
= -K Bysro RTB (rib/l0 •t .rg14/315 + 7,24/1500) (25)
-KDJsro RTD (t;4/10 t ri14/315 t T;*'+/1500)
at T > TB > TD
Here it should be noted that the ordering energy is zero at infinitely high
temperature.
Similarly the atomic ordering energies for the B2 and DO3 states can be
caluclated as follows :
Cord,B2 =
TB
I
I(t-T)/tI C;"" dt + jT { (t-T)/t) C;ylro dt
m TB
T
t
I
{(t-T)/t} C;"" dt
m
= _KB,sro RTB (g _ $ TB )
_ KB,lro RTB ( ‘I~ 4/6 + $0/135 + 9600 + & - g TD )
- KD'sro RTD ( +/lo t $4/315 t .li*4/1500)
260
(26)
B-J. LEE, S.K. LEE and D.N. LEE
at TB > T > TD
and
B
Gord,D03 = {(t-T)/t] C;,Sro
rT
dt t
J
{(t-T)/tl CpB,lro dt
m TB
TD
t
I
{(t-T)/t) C;ySro dt t i'
V-
{(t-T)/t] +lro dT
m ‘LJ
= -KBysro RTB ( $ - 1125 TB 518)
-KBylro RTB ( $16 t ~hO/135 t +'600 t 71/120
_KD,sro RTD (79/140 - 51&D/1125)
_KD,lro RTD ( ~:/6 t ~bO/135 t +/600 t 71/120
at TB > TD > T
- 518r,/675)
- 518T,/675)
(27)
Substitution of Eqs.(l9) through(22) into Eqs.(25) through(27) gives us an
expression for the atomic ordering free energy.
G ord = AHA2+B2 f(rB) + AHB2*D03 f(.rD) (28)
Here AH A2+B2 and AH B2+D03 are given in Eqs.(l3) and (16), and f(r) is expressed
as
f(T) = (T -4/1O t r-14/315 t ~-*~/1500)/A for Z > 1
and 1 - {518~/1125 - (474/497)(1/f-l)(T4/6 + T"/135 t ~'7600
- 518~/675)) /A for T < 1 (29)
where A is defined in Eqs.(23) and (24).
3. Thermodynamic analysis of the Fe-X system
The Fe-Si phase diagram is composed of five intemtallic compounds and
three solution(Fig.5), structures of which are listed in Table 1(B). In this
study the reference states of Fe and Si were fee and diamond cubic states,
respectively, and the following lattice stability data(9,lO) were used.
OGl Fe
_ oG;e = -11274 t 163.87811 t O.O041755T* - 22.031 In T
(J/g-atom)
oa G Fe - OGY Fe
= 1462.4 - 8.282T - o.00064T2 t 1.15T In T t 'Gmae
OG1 Si - 'G& = 50626 - 30.OT (30)
oa G
si - oGd si
= 44350 - 19.5411
AZ/B2/D03 ATOMIC ENERGY ORDERING ENERGY 261
Fig.5. Experimental phase
diagram of the Fe-Si
system(l8).
1
0 Xsi
Fig.6. Calculated excess free
energy of mixing compared
with experimental data
for liquid alloys in the
Fe-Si system at 1873X(11),
I I $ I I
0.1 0.2 0.3 0.4
Xsi
Fig.7. Illustration of composi-
tional dependence of
magnetic ordering energy
(M) and atomic ordering
energy(A), and sum of the
both ordering energies(M
+A) in the bee Fe-X alloys
at 900 K.
262 B-J. LEE, S.K. LEE and D.N. LEE
0 Y
GSI - si OGd = 50626 - 17.87T
where superscipts 1, y, a and d stand for liquid, fee, bee and diamond cubic
phases respectively, and magnetic ordering energy 'Gmag is given in Ecr.(46).
Table 1. The Structure of Soid Phases in the Fe-Si System
Formular
o or 6Fe
yFe
. . .
Fe,Si
Fe,Si
Fe,Si,
FeSi
FeSih
1 FeSi2
Si
Crystal structure
bee (AZ)
fee
Ordered bee (B2)
Cubic, BiF type (DO3)
Uncertain, but an ordered
structure related to bee
Hexagonal, Mn,Si3 type (DO8)
Cubic, B20
Tetragonal
Orthorhombi~
Cubic, A4 structure
3.1. The FeSi phase
Since the composition range of the Fe.71 phase is uncertain and negligibly
narrow, it was treated as a stoichiometric compound. Some measured data of
the Gibbs free energy of the FeSi phase are compiled by Hultgren at al.(ll).
Since the data were estimated with respect to a-Fe and diamond structure Si,
they were modified to values with respect to T-Fe and diamond Si as follows :
OG 1 OGY 1 oGd FeSi - 2 Fe - 2 Si = ( 'GFeSi - ; 'Gle - $ 'G&)
t 2 ( ‘Gze - 'Gie f (31)
The measured data and the lattice stability in Eq.(30) gave us the following
result,
OG
lOY
FeSi - F GFe - F Si
1 OGd = -4310.86 4 5.1453 T (J/g*atom) (32)
3.2. The liquid phase
The Gibbs free energy of liquid phase of the Fe-X system was represented
by a subregular solution.
G1 = (l-x) 'G;, 4 x 'G&
+ RT {(l-x) In(l-x) + x In x 1
4 x(1-x) {L;(T) + (I-2x) L;(T)1 (33)
The interaction parameters and LL were calculated from phase equilibrium
between FeSi and liquid phases {Eq. t 34)I and phase equilibrium between liquid
phase and pure silicon fEq.(35)} .
A2/B2/D03 ATOMIC ENERGY ORDERING ENERGY 263
and 81 1 Si xl2 = oGd Si (35)
-1 where G 1 -1 Fe xl1 andG 1 Si x11 represent the partialmolalfree enrgies of Fe and Si
at liquidus compositions between FeSi and liquid phases and cill 1
x2
represents
the partial molal free energies at liquidus compositions between liquid phase
and pure silicon.
follows :
The calculated values of the parameters Lo and Ll are as
L;(T) = -192251 + 53.8654T
L;(T) = 2148.8 - 24.7336T
(36)
The excess free energy of mixing at 1873K calculated using the parameters in
Eqs.(36) is in very good agreement with the measured data(l1) as shown in Fig.
3.3. The FeSi2 phase
There are two different phases in FeSi2, that is, the high temperature
phase FeSip and the low temperature phase FeSiJ. The FeSi$ phase has some
composition range, which is attributed to vacancies in Fe sites(l1) and may be
expressed as a formula unit(Fe, Va)Si2 where Va stands for vacancy. The Gibbs
free energy for the FeSig phase can be obtained using a two-sublattice model.
The first sublattice is composed of Fe and Va, and the second sublattic is
composed of Si. The formula suggested by Sundman and Agre (12) based on the
sublattice model yields the Gibbs free energy for the FeSi, phase, Gh, R
Gh = (l-YVa) OG~e:Si + YVa OGh Va:Si
t RT (1-yVa) ln(l-YVa) ' YVa In YVa
' YVa(l-YVa) LFe Va.Si > . (37)
where yVa is the site fraction of vagancy in the sublattice composed of Fe
atoms and vacancies. The parameter GRe:Si is the Gibbs free energy of for-
mation of one mole of formula units of a FeSi2 phase with Fe in the first and
Si in the second sublattice. The parameter OGVa:Si is the Gibbs energy of for-
mation of one mole of formula units of a FeSi2 phase with vacancy in the first
and Si in the second sublattice. The parameter LFe Va:Si is an interaction
parameter, where a colon ":" separates components id different sublattices and
a comma It," separates components in the same sublattice. The relative number
of atoms in the two sublattices are one in the first and two in the second
sublattice. It should be noted that OGFe.Si
of FeSit phase, whereas 'GVa.Sl
is associated with 3 gram-atoms
is associated with 2 gram-atoms of FeSi$ phase.
Therefore the number of gramlatoms of Gh in Eq.(37) varies with composition.
In order to calculated compositions in equilibrium with other phases using the
common tangent method, the Gibbs energy of a solution should be given in a
same unit throughout the composition.
press the Gh
Therefore Eq.(37) was modified to ex-
value in the unit of 3 gram-atoms. The modified form of Gh may
be obtained by multiplying Eq.(37) by the factor 3/(3-y,,).
Gh 3 =p
3-YVa
rt WyVa) oG;e:SI + yVa 'Giazsi 1
t RT {(l-yVa) ln(l-YV,) t yVa In YVa 1
264 B-J. LEE, S.K. LEE and D.N. LEE
+ YVa(l-YVa) LFe Va*Si' f *
It is noted that the'factor 3/(3-yv,) varies from
from 0 to 1.
1 to 1.5 when yVa varies
(38)
oh Since no measured values of GEe:Si, OG$a:Sj, and LEe Va.Si are available,
they may be calculated from the measured phase diagram data using the following
equilibrium conditions between the FeSib 2 phase and liquid solution phase.
-h
GFeSi21xh
-h
GVaSi21xh = 3'B11xl
Here
Gh FeSi2 = Gh
_ (x-2) aGh
3 ax
Bh VaSi2
= Gh + (l-x) a$
(39)
(40)
(41)
(42)
where x is the mole fraction of Si and is related with yVa as follows :
x = 2/13 - yVa) (43)
The three unknown parameters, oGh
oh
GV*
ated from the two equations(39) &%lSt40).
. can not be evalu-
~~~~~~~r~F~~~a~~~~ phase was at
First, assumed to be an ideal solution (LFe,va:Si
ameters, OGhFe.Si and *Gba. .
= 0)and the two unkown'par-
were roughly evaluated from Eqs.(39) and (ilO)using
the tie-line variation me&d(l3).
roughly estimated OGh
The parameter L was evaluated using the
Eqs.(39) and (40). %?$s of the
and oGFa&i2 values and the equilibrium conditions of
ree parameters were refined by trial and
error method. The calculated values of the parameters are summerize
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