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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