ACTA POLYTECHNICA SCANDINAVICA

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1 Ch 292 ACTA POLYTECHNICA SCANDINAVICA CHEMICAL TECHNOLOGY SERIES No: 292 A Thermodynamc Analyss of the System Fe-Cr-N-C-O Rauno Luoma Helsnk Unversty of Technology Department of Materals Scence and Rock Engneerng Laboratory of Metallurgy FIN HUT, Fnland Current address: OMG Harjavalta Nckel Oy Teollsuuskatu 1 FIN Harjavalta, Fnland Dssertaton for the degree of Doctor of Technology to be presented wth due permsson for publc examnaton and debate n Audtorum V1 at Helsnk Unversty of Technology (Espoo, Fnland) on the 5 th of December, 2002, at 12 o clock noon.

2 2 Luoma, R., A Thermodynamc Analyss of the System Fe-Cr-N-C-O. Acta Polytechnca Scandnavca, Chemcal Technology Seres No. 292, Helsnk 2002, 91 pp. Publshed by the Fnnsh Academes of Technology. ISBN , ISSN Keywords: C-Cr-Fe-N-O-system, phase dagrams, actvtes, thermodynamc modellng ABSTRACT A thermodynamc database for the system Fe-Cr-N-C-O has been bult usng prevously assessed bnary and ternary systems. Sx ternary systems, Fe-Cr-O, Fe-C-O, Fe-N-O, Cr-N-O, Cr-C-O, and N-C-O, have been assessed. Quaternary and qunary systems were calculated usng only nterpolaton models. Ths method of buldng a database s known as the Calphad method and t s wdely used n modern thermodynamcs. An assocated soluton model wth a non-deally nteractng speces, namely Fe, Cr, N, C, FeO, FeO 1.5, Cr 2/3 O, and NO was used for the lqud phase. The sold metallc phases were descrbed usng the sublattce model wth carbon and oxygen on the second sublattce, and sold oxde phases were descrbed usng the compound energy model. The carbde phases were treated as stochometrc or semstochometrc phases. The optmsaton was performed usng the Parrot module ncluded n the Thermo-Calc program. The model parameters for the lqud phase n metal-oxygen systems were transformed from the parameters optmsed wth the onc lqud model by other authors. Because of the new assessments of the bnary systems, all the ternary systems ncludng oxygen were optmsed. Only n the N-C-O system could the parameters not reproduce the expermental data. The calculated quaternary systems are n good agreement wth the expermental data wthout usng any quaternary parameters. The model parameters assessed n ths work descrbe the system Fe-Cr- N-C-O well accordng to the expermental nformaton from ts sub systems. The complete Gbbs energy expressons for the alloy phases were presented, allowng the calculaton of the phase dagrams and thermodynamc mxng propertes of the mxture phases. All rghts reserved. No part of the publcaton may be reproduced, stored n a retreval system, or transmtted, n any form or by any means, electronc, mechancal, photocopyng, recordng, or otherwse, wthout the pror wrtten permsson of the author.

3 3 PREFACE Ths work was carred out n the Laboratory of Metallurgy at Helsnk Unversty of Technology, Outokumpu Research Oy, Outokumpu Harjavalta Metals Oy, and OMG Harjavalta Nckel Oy durng the years I would lke to express my deep grattude to my supervsor, Professor Laur Holappa, for hs nvaluable encouragement and nterest n my work. I want to thank all the co-workers I have had durng these years for the encouragng workng envronment they helped to create. Specal thanks are due to Professor Mats Hllert, Professor Bo Sundman, and Larry Kaufman PhD, for ther valuable advce and skll n thermodynamcs. I want to thank Outokumpu Research Oy and Docent Pekka Tasknen for the opportunty to use the Thermo-Calc program. I want to thank OMG Harjavalta Nckel Oy for the opportunty they gave me to fnsh ths work. I would lke to express my thanks to all those nnumerable people who have helped me collect all the papers for ths work. Por, November 2002 Rauno Luoma

4 4 CONTENTS Lst of symbols Introducton Thermodynamc background Gbbs energy Descrpton of mxture phases Random substtutonal soluton models Sublattce model Compound energy model Partally onc lqud Assocate model Interpolaton models Calculaton practce Database programs The Calphad prncple Optmsaton practce Descrpton of the system Gas phase Lqud Sold metallc phases Sgma phase Wüstte and bunsente Hematte and chromum oxde Cr 2 O Spnel Graphte Carbde phases Bnary systems System Fe-Cr System Fe-N System Fe-C System Fe-O System Cr-N System Cr-C System Cr-O System N-C System N-O System C-O Ternary systems System Fe-Cr-N System Fe-Cr-C System Fe-Cr-O System Fe-C-O System Fe-N-C System Fe-N-O System Cr-N-C System Cr-N-O System Cr-C-O System N-C-O...58

5 7 Quaternary systems System Fe-Cr-N-C System Fe-Cr-N-O System Fe-Cr-C-O System Fe-N-C-O System Cr-N-C-O System Fe-Cr-N-C-O Dscusson Conclusons References Appendx. Thermodynamc data

6 6 LIST OF SYMBOLS A Anon spece a, b, c, d, a 0, a 1, a n and q Parameters of thermodynamc functons B Neutral spece C Caton spece C p Heat capacty f Actvty coeffcent G Gbbs energy G magn Gbbs energy of magnetc term G Gbbs energy of pure spece d G Gbbs energy of deal mxng ex G Excess Gbbs energy H Enthalpy L n Interacton parameter n Amount of moles N s Total number of stes on sublattces p Pressure p Structure-dependent parameter of magnetc orderng P Number of stes on caton sublattce Q Number of stes on anon sublattce R Gas constant, J/K mol S Entropy SER Stable element reference state T Temperature unt (n K) T c Cure temperature Va Vacancy Wp Number of permutatons x Mole fracton y Ste fracton β Magnetc momentum τ T/T c

7 7 1 INTRODUCTION Phase dagrams are commonly used to represent the thermodynamc propertes of multcomponent, multphase systems. They have become the most mportant tool especally for chemsts, physcal metallurgsts etc. Phase dagrams of complex thermodynamc systems also provde a means of understandng the phenomena that occur n several pyrometallurgcal processes, helpng those workng n the feld of extractve metallurgy. Thermodynamc calculatons n multcomponent systems are very complex. Even though the basc equatons are smple, they are almost mpossble wthout the ad of computers. Ths has led to large computersed databank systems that contan thermodynamc nformaton for multcomponent systems n the form of mathematcal functons. Usng such an expert system, t s possble to calculate any phase dagram for the systems ncluded n the databank and also make predctve calculatons and other useful dagrams n order to solve a specfc problem. Durng the last decade thermodynamc assessments of many oxde systems have been carred out. These works were manly concerned wth slag chemstry and therefore they dd not nclude carbon. Metal melts are stll descrbed usng smple models for dlute systems, whch, n most cases, they are. In the producton of ferrochromum or stanless steels melts contan so many components other than ron that the systems can no longer be treated as dlute systems. Therefore there s a need for a databank that deals comprehensvely wth such metals systems. The present work deals wth the system Fe-Cr-N-C-O whch s a basc system n descrbng stanless steels. The system s descrbed usng models that allow calculatng thermodynamc propertes over whole concentraton range. The database does not nclude slag-formng components as slca or calcum oxde and therefore calculated dagrams does not represent stuatons where there s slag as stable phase. The man applcaton for ths database wll be n steel makng processes at hgh temperatures and therefore the low temperature behavours are not taken account. Nevertheless, ths database s a full descrpton of the fve-component system and can be used at any concentratons. Whle usng ths database n other areas, one should notce that the uncertanty of the descrpton s much greater n the areas where no expermental nformaton s avalable.

8 8 2 THERMODYNAMIC BACKGROUND The thermodynamc descrpton of equlbrum can be made usng several functons of state. For multcomponent systems t s practcal to use a model that allows the system to be descrbed unambguously and as a whole. Ths can be done most smply by choosng the Gbbs energy as the fundamental property functon. 2.1 Gbbs energy The Gbbs energy of a phase s defned as a functon of temperature, pressure and amounts of speces n the phase as: or by mole fractons as: G = G T, p, n ) ( 1) ( G = G T, p, x ) ( 2) ( These defntons are satsfactory n cases that they do not nclude surface phases or changes n strong electromagnetc felds /1/. The defnton G = H T * S ( 3) where H s enthalpy T s temperature S s entropy s vald for all equlbra, chemcal reactons and changes between equlbra /2/. The temperature dependence of Gbbs energy can be descrbed usng the heat capacty, whch s often descrbed usng an emprcal formula, such as: c C p + dt 2 T 2 = a + bt + ( 4) Enthalpy and entropy can be calculated from heat capacty by: C p H = ( 5) T C p T S = ( 6) T

9 9 Accordng to the equatons 3-6, the Gbbs energy for a pure element, compound or mxture phase as a functon of temperature s: 2 3 T c T G = a0 a1t + at (1 lnt ) b d ( 7) 2 2T 6 where a,b,c,d are the same constants as n equaton ( 4) a 0 and a 1 are constants that can be calculated from the constants a,b,c and d and standard values H(298) and S(298) Equaton 7 s an adequate descrpton for stochometrc compounds and elements f pressure dependence and magnetc orderng are not taken nto account. For ferromagnetc phases the magnetc orderng wll gve addtonal terms for the Gbbs energy for pure elements and consttuents: ( β ) f ( τ ) G mag = RT ln 1+ ( 8) where τ = T/T c β s the average magnetc momentum n Bohr magnetons f(τ) s structure-dependent and gven, based on the modellng of the c p functon by Inden /3/ and Hllert /4/ as for τ < 1 f ( τ ) = τ 140 p τ + 1 p A 9 15 τ τ for τ > 1 f ( ) τ τ τ + + τ = ( 9) A where A = p p s a structure dependent parameter: for fcc and hcp metals p=0.28 and for bcc metals p= Descrpton of mxture phases The Gbbs energy for a mxture phase can be expressed as: G ( 10) = n G where G s the Gbbs energy of the phase n s the amount of spece G s the partal Gbbs energy of speces

10 10 The Gbbs energy for a speces n a mxture phase can be dvded n three parts as: G + d ex = 0 G + G G ( 11) where G s the partal Gbbs energy for pure speces d G s the partal Gbbs energy of deal mxng ex G s the excess Gbbs energy Combnng these two equatons, the Gbbs energy for a mxture phase wll be: G= 0 d n G + n G + n ex G ( 12) Descrptons for the Gbbs energy of deal mxng and the excess Gbbs energy are dfferent for each model used n calculaton Random substtutonal soluton models The random substtutonal soluton model s a pure mathematcal model wdely used for many smple systems. For a speces n a random substtutonal soluton the Gbbs energy s expressed as: 0 G = G + RT ln x + RT ln f ( 13) where x s the mole fracton of component f s the actvty coeffcent of component R s the gas constant T s the temperature For a phase we wll get, accordng to equatons 12 and 13 0 G = x G + RT x ln x + Ex G ( 14) where Ex G s the excess Gbbs energy for the phase The excess Gbbs energy s descrbed usng polynomal functons. In the smplest case Margules polynomals /5/ are used, whch are, for a spece n bnary phase: k ln f = ak x ( 15) k Usng the Gbbs-Duhem relaton, t can be seen that parameters a 0 and a 1 are zero. For a bnary phase the followng functon can then be expressed: Ex k G = RTx x q x ( 16) j k k j where q k are parameters that can be calculated from expressons for the actvty coeffcents of components

11 11 The problem wth these Margules polynomals s that the parameters are not transformed drectly to hgher order systems. The Redlch-Kster polynomals /6/, whch are mathematcally more effcent and can be easly enlarged to hgher order systems, are wdely used nowadays. These polynomals do not have an dentcal form for each term, whch was also a problem for the Margules polynomals. The general polynomal for the bnary mxture phase s: Ex G = x x n k j k= 0 L j ( x x ) j k ( 17) where k L,j s a pressure- and temperature-dependent parameter Partal excess Gbbs energes for the bnary mxture phase are then defned as /7/: Ex Ex G G j n 2 0 = RT ln f = x j Lj + k = 1 k L j k 1 ( x x ) ( 2k + 1) x x ) j k 1 ( x x ) ( x ( 2k + 1) x ) n 2 0 k = RT ln f j = x Lj + Lj j j ( 18) k= 1 j For multcomponent systems these functons can be calculated smlarly. If there s a need to defne a ternary nteracton, t can be expressed by the equaton: where Sublattce model 0 1 ( V L + V L + V L) G = x x x ( 19) Ex 2 j k j k V V V j k 1 x x j xk = x x x j xk = x j x x j xk = xk + 3 For sold phases that have an nternal structure t s better to use a model that can descrbe the structure more specfcally. In the sublattce model the elements of the phase are dvded nto separate lattces accordng to the crystallographc structure of the phase /8/. For example, n a smple body-centred cubc structure stes at the body s centre and n corner postons can be descrbed as dfferent sublattces, as presented s Fgure 1.

12 12 Fgure 1. Smple body-centred structure. In the sublattce model, nstead of the overall composton x, the ste-fracton y s used as a coeffcent when defnng the Gbbs energy of the phase. The ste-fracton s defned as the fractonal ste occupaton of each of the components on the specfc sublattce /8/,.e. n y ( 20) S S = S n j where n S s the number of atoms of component on sublattce S summaton j s performed for all components on sublattce S If there are vacances on the sublattce they are taken nto account as components. The overall composton gven n mole fractons s drectly related to ste fractons by the followng relatonshp /9/: x = S S N y S S S N ( 1 yva ) S ( 21) where N S s the total number of stes on sublattce S y S Va s the number of vacances on sublattce S The deal entropy of mxng s made up of the confguratonal contrbutons by components mxng on each of the sublattces. The number of permutatons that are possble, assumng deal nterchanges wthn each sublattce, s gven by the equaton /9/: W P = S S N! n! S ( 22) and the molar Gbbs energy of deal mxng s /8/: d S S S G = T S = RT N y ln y ( 23) d S

13 13 The end members generated when only pure components exst on the sublattce, effectvely defne the Gbbs energy reference state. For a sublattce phase wth the followng formula (A,B) 1 (C,D) 1 t s possble for four ponts of complete occupaton to exst where pure A exsts on sublattce 1 and ether pure C or D on sublattce 2 or, conversely, pure B exsts on sublattce 1 wth ether C or D on sublattce 2. The composton space of the phase can then be consdered n Fgure 2. below as consstng of four compounds, the so-called end members, at the corners of the square. The composton of the phase s then encompassed n the space between the four end-member compounds and the reference energy surface wll look lke Fgure 2. Fgure 2. Composton space encompassed by the system (A,B) 1 (C,D) 1 /9/. The surface n Fgure 2 can be represented by the equaton ref G = y y G + y y G + y y G + y y G ( 24) A C AC B C Usng the same sublattce phase (A,B) 1 (C,D) 1 as an example, the Gbbs excess energy of mxng s defned as the A-B and C-D nteractons nsde the sublattces. The parameters can usually have Redlch-Kster-type polynomals for composton dependence. Thus the excess Gbbs energy for ths phase can be represented by the equaton /8/: BC A D AD B D BD Ex n k 1 1 k G = y A yb yc LA, B: C ( y A y B ) + y A yb y D k = 0 n k= 0 k L A, B: D ( y 1 A y 1 B ) k + y 1 A y 2 C y n 2 k D k = 0 L A: C, D Compound energy model n 2 k D k= k k ( y y ) + y y y L ( y y ) ( 25) C D Sold oxde phases can be defned usng the sublattce model f onc consttuents are defned as beng on the sublattces. In that case the model s known as the Compound Energy Model (CEM) /10/. If all the end-members are neutral, the parameters and defntons are dentcal wth the sublattce model. In cases where there are many charged end members, the model easly becomes very complex and s dffcult to get balanced. The model can be most smply presented through examples. If we have a phase that s formed from two compounds, AX and BY, where A and B are catons and X and Y are anons, all possble compounds can be descrbed by means of an exchange reacton: AX + BY = AY + BX ( 26) Whose Gbbs energy change s expressed as: B C B: C, D C D

14 G = G + G G G ( 27) AX : BY AY BX If there s a need for more parameters to defne the system, the anon-anon and caton-caton nteractons can be used /10/. In cases where the charges of anons or catons are not the same, some of the end members are charged. For example, n the system ron-oxygen the wüstte phase (FeO) can be defned wth two sublattces, wth catons Fe 2+ and Fe 3+ and vacances on the frst sublattce and oxygen anons on the second sublattce.e. wth a halte structure /18/, formulated as: AX BY (Fe 2+,Fe 3+,Va) 1 (O 2- ) 1 ( 28) The end members are then FeO, FeO +, and O 2-, of whch only the frst one s neutral. The structure can be represented as a consttutonal trangle, as shown n Fgure 3. Each corner of the trangle represents one end member. Only the sold lne trough the trangle has physcal sgnfcance, as t represents the neutral combnatons of Fe 3+ and Va. (Fe 3+ :O 2- ) Fe 2 O 3 (Fe 2+ :O 2- ) (Va:O 2- ) Fgure 3. Structural model of wüstte /18/. The compound energy model gves the followng expresson for the Gbbs energy: mx G = y G + RT 2 2: O + y G 3: O + y G Ex ( y y + y ln y + y ln y ) + G 2 3 ln Va Va Va: O Va ( 29) where the varables y 2 and y 3 are the ste-fractons of dvalent and trvalent ron respectvely, and y Va s the ste-fracton of a vacancy.

15 15 The three G terms descrbe the Gbbs energes of the three end members n Fgure 3. Because two end members have a net charge, t s possble to evaluate only a neutral combnaton of these. The parameter G VaO represents the halte phase wth only oxygen, and ths parameter wll be used n other oxygen systems where halte s also stable. Therefore t s convenent to set t to zero as G VaO =0. The term preceded by RT s the deal entropy of mxng on the metallc sublattce. The sublattce wth oxygen has no entropy of mxng, as t s completely flled wth O 2- /18/. The excess Gbbs energy, mx G ex, for the wüstte can be descrbed wth Redlch-Kster expressons as: Ex G = y + y y 2 + y y y3 L V V 0 0 L L 2,3: O 2, Va: O 3, Va: O k j ( y y ) 2 ( y y ) 2 Va k k ( y y ) L 3 3 Va L 2,3: O j j L 2, Va: O 3, Va: O ( 30) There are sets of Redlch-Kster terms n each sde of the trangle n Fgure 3. However, as the vacances are needed only to mantan electroneutralty, the parameters nvolvng vacances should be set to zero. One may otherwse have problems when combnng assessments of halte phases from dfferent subsystems because the same parameter wll appear n other systems /18/. A more complcated phase structure s used to descrbe the spnel phase. In the Fe-O system the magnette has a spnel structure, wth oxygen ons on the fcc sublattce and dvalent and trvalent metallc ons on the octahedral and tetrahedral ntersttal sublattces. A normal spnel has the trvalent ons on the octahedral stes and the dvalent ons on the tetrahedral stes. However, at low temperatures, magnette s an nverse spnel, wth the tetrahedral stes flled wth Fe 3+ -ons and the octahedral stes flled wth both Fe 2+ and Fe 3+ -ons. At hgher temperatures magnette transforms gradually nto a normal spnel, and before meltng, t s almost random. The structure of the stochometrc magnette s thus /10/: (Fe 3+,Fe 2+ ) 1 (Fe 3+,Fe 2+ ) 2 (O 2- ) 4 ( 31) Only one of the end members s also neutral n ths case. Ths s a normal spnel. Fgure 4 shows the consttutonal dagram for a stochometrc spnel, ncludng the neutral lne. On the neutral lne all ponts have the same composton, but only one pont s a stable compound and defnes the degree on nverson.

16 16 EE EA y E AE y A AA Fgure 4. Consttutonal dagram of spnel /10/. The Gbbs energy for a normal spnel s defned as: G ( 32) 0 n = G Fe : Fe At the other end of the neutral lne the Gbbs energy for an nverse spnel s: G ( 33) 0 0 =.5 G G RT ln L Fe : Fe Fe : Fe Fe : Fe, Fe In ths case, the phase s defned by eght parameters: 0 G L Fe 2+ 2+, Fe Fe : Fe : Fe 2+ 0 G 3+ Fe Fe : Fe L, Fe 3+ : Fe 0 3+ G 2+ Fe 2+ Fe : Fe L 2+ : Fe 2+ 0 G, Fe Fe : Fe L Fe : Fe, Fe 3+ A condton of electroneutralty bnds these parameters so that on the neutral lne there are only four ndependent parameters, whch can be expressed as: G n ( G L L L ) Fe : Fe, Fe Fe, Fe : Fe Fe : Fe, Fe ( 2 L L L L ) Fe, Fe G : Fe Fe, Fe : Fe G s the Gbbs energy for the exchange reacton and can be defned as beng zero. The model can also be used by defnng the nteracton parameters as zero. Hence, there are only two parameters to be optmsed. The dependence for the other parameters s then: Fe : Fe, Fe Fe : Fe, Fe G G G G Fe Fe Fe Fe : Fe : Fe : Fe : Fe = G n + G = G Fe G Fe 3+ : Fe Fe 3+ : Fe 3+ : Fe 3+ = 2 G 3+ = G n + 4RT ln G + 4RT ln 2 ( 34)

17 17 The last equaton s the defnton for the standard state for charged components and therefore the value for G Fe 3+:Fe3+ can be set to any numercal value, ncludng zero. If the phase s formed from two components, whch have a common dvalent caton, such as a soluton of magnette, Fe 3 O 4, and chromte, FeCr 2 O 4, the structure of the phase s /10/: (Fe 2+,Fe 3+,Cr 3+ ) 1 (Fe 2+,Fe 3+,Cr 3+ ) 2 (O 2- ) 4 ( 35) The consttutonal dagram for ths phase s shown n Fgure 5. The number of end members s seven and two of them are neutral. The neutral plane defnes all compostons between Fe 3 O 4 and FeCr 2 O 4 and also all possble degrees of nverson. The Gbbs energy of ths phase wll be defned n the same way as earler defned for a smple spnel and t can be expressed usng four ndependent parameters /18/. Fgure 5. Consttutonal dagram for the soluton of two spnels. Corners AF and AE descrbe two normal spnels and the shaded plane s the neutral plane /10/. Magnette also has a consderable devaton from ts deal stochometry at hgher oxygen potentals and temperatures. Ths can easly be accommodated n the structure by allowng an excess of Fe 3+ on the octahedral stes and at the same tme ntroducng vacances to mantan electroneutralty. The expermental methods used have not shown whether the vacances are on octahedral or tetrahedral stes. Fnally, magnette has a small devaton towards excess ron n equlbrum wth wüstte and lqud at hgher temperatures. Ths s due to some Fe 2+ enterng as ntersttals nto the remanng octahedral stes, whch are normally empty. To model ths, we add one more sublattce wth two stes that are manly flled wth vacances. The fnal expresson for magnette wll be /10/: (Fe 2+,Fe 3+ ) 1 (Fe 2+,Fe 3+,Va) 2 (Fe 2+,Va) 2 (O 2- ) 4 ( 36) Ths descrpton wll then have 12 G functons but only four ndependent parameters, as presented by Sundman /18/.

18 18 Fgure 6 shows the consttutonal dagram for a spnel wth the composton of MgAl 2 O 4 and ts neutral plane. The corners of the neutral plane are, n ths case, the normal spnel MgAl 2 O 4 and the metastable compounds γ-al 2 O 3, γ-mgo, and AlMg 2.5 O 4. Ths phase can be defned usng four ndependent parameters. Fgure 6. Consttutonal dagram for the spnel MgO-Al 2 O 3. A phase comprsng two non-stochometrc spnels can be expressed wth the formulas shown above. The fnal descrpton of the phase wll then have eght ndependent parameters /10/ Partally onc lqud The lqud phase can be treated as a random substtutonal phase n many metallc systems. In cases where the phase ncludes oxygen, ths necesstates the defnng of thermodynamc functons for the phase more accurately n order to avod a rather complcated set of parameters that do not have any physcal meanng. One possblty s to use the sublattce model for the lqud phase. Ths model s also known as a partally onc lqud model /11/. In ths model we have two sublattces, wth caton consttuents on the frst sublattce and anon and neutral consttuents and vacances on the second. The sublattce formula for the model can then be wrtten as: +ν ν j 0 ( C ) ( A j, Va, Bk ) P Q where C represents a caton A an anon Va a hypothetcal caton B neutrals

19 19 The charge of an on s denoted by ν and the ndces, j, and k are used to denote specfc consttuents. The number of stes on the sublattces s vared, so that electroneutralty s mantaned, and the values of P and Q are calculated from the equatons: P = Q = j ν y j ν y A C j + Qy Va ( 37) Equaton 37 smply means that P and Q are equal to the average charge on the opposte sublattce, wth the hypothetcal vacances havng an nduced charge equal to Q. Mole fractons for the components can be defned as: x x C A = = P P + P P + Q y ( 1 y ) Qy C A Va ( 1 y ) Va ( 38) where C denotes a caton and A an anon The ntegral Gbbs energy for ths model s then gven by: G = k1 k2 j k1 y C k2 Aj k1k2 G C: Aj + RT P y C ln yc + Q + y y + 1 y 2 j L 12: j 1 2 j + y y j y + 1 j L 2 : j1 j2 j1 j2 + y y j yk L : jk + j k + y y L y + Qy j 1 2 k Va y Aj j y ln y y 1 y y y k C y y 2 y j G Aj y Va y + y Va Va L C L L : Vak + Q Va 1 2: Va : jva k ln y Va y Bk + G k Bk y Bk ln y Bk ( 39) where G C:Aj s the Gbbs energy of formaton for (ν 1 +ν 2 ) moles of atoms of lqud C A j and G C and G Bk are the Gbbs energy of formaton per mole of atoms of lqud C and B respectvely. The frst lne represents the Gbbs energy reference state, the second lne the confguratonal mxng term, and the last three lnes the excess Gbbs energy of mxng. In some papers /12/ recprocal terms are also used. The model s certanly complex, but the terms have some physcal meanng, representng dfferent nteractons n dfferent parts of the system. Selleby /13/ suggests that the number of terms needed to descrbe a ternary system such as Fe-Mn-S s qute smlar for both onc sublattce lqud and assocate models. The modellng of onc lquds s complex and the advantages of the varous technques only become apparent as they become more commonly used.

20 20 There are a number of mmedate advantages to the onc sublattce model. The frst s that t becomes dentcal to the more usual Redlch-Kster representaton of metallc systems when the caton sublattce contans only vacances. Ths mmedately allows data from an assessed metallc system to be combned wth data from an oxde system so that the full range of composton s covered. In bnary cases the model can be made equvalent to an assocate model. For example, the assocate model would consder a substtutonal soluton of Cu, S and Cu 2 S n the system Cu-S. If an onc sublattce model wth a formula (Cu +1 ) P (S -2,Va,S) Q s used, t s straghtforward to derve parameters to gve an dentcal result to the assocate model /9/. However, t should be noted that the Gbbs energy of the two models does not reman equvalent f they are extended nto ternary and hgher-order systems /11/ Assocate model The lqud phase n oxde systems can also be treated as a substtutonal soluton of metals and oxdes. In ths case oxdes are assocates or complexes. The thermodynamc propertes of the lqud phase then depend predomnantly on the Gbbs energy of the formaton of these assocates nstead of the nteracton between the components. Ths gves rse to the enthalpy of mxng dagrams, whch are charactersed by sharp changes at crtcal compostons where assocates exst, and also by markedly non-deal mxng entropes. The dervaton of Sommer /14/ can be used as an example. Ths consders the formaton of a sngle assocate wth the formula A B j wthn a bnary system, A-B. It s assumed that the lqud contans n A and n Bj moles of free A and B n equlbrum wth n ABj moles of assocate A B j. The mole fractons of A, B, and A B j n a bnary alloy contanng 1 mole of A and B atoms are then gven by the formula: x x x A B A B = n = n j A B = n + n + A B j n j A B j A B j ( 40) The excess Gbbs energy of mxng s then gven by the general formula: G + Ex ass reg mx = G G ( 41) where G ass s the Gbbs energy due to the formaton of the assocate defned as: ass G n A B = G ( 42) j A B j Here G ABj s the Gbbs energy of formaton of one mole of the assocate. G reg consders the Gbbs energy due to the nteractons between the components A and B themselves and wth the assocate A B j such that: G reg = reg A B reg j reg j G A, B + GA, A B + G j B, A B ( 43) j n A n n + n + n B A B J n A n A + n n B A B + n A B J n A n B + n n B A B + n A B J The confguratonal entropy s gven smply by: S conf mx ( ln x + n ln x + n ln x ) = R n ( 44) A A B B A B j A B j

21 21 The model was appled to numerous onc melts and good agreement was found wth expermental results /14, 15/. One of the man crtcsms of assocate formalsm has been that, although the concept consders assocates or complexes to be dffuse n nature, mathematcal formalsm mples that dstnct molecules exst, whch s more dffcult to justfy. However, f ths strngent vew s relaxed, t can be seen that the model merely mples some underlyng structure to the lqud, whch s qute reasonable, and t does provde functons that allow a temperature dependence for the enthalpy of mxng. A more serous crtcsm s that some knowledge of the relevant assocate s necessary before the model can be appled. The most approprate assocate can be selected by the fttng of expermental results for enthalpes of mxng, whch s suffcent n a large number of cases. However, n some systems there may be a number of dfferent assocates and t s not always obvous whch types actually exst. The man advantage of the assocate model s that t allows a smple strategy to be adopted for optmsaton. It s easy to further defne ternary and hgher-order assocates and extend the model to multcomponent systems. Ths was shown by Rannkko /15/ n hs study on the system Cu-O-CaO-SO 2. It s easy to show equvalence between the onc sublattce model and the assocate model, as demonstrated by Hllert /11/. For the system (A +ν A ) P (B -ν B,Va -ν A,B 0 ) Q where +ν A =-ν B equaton 39 can be smplfed for one mole of atoms as: G = y B G 0 A B νb 1 + νa + RT ( y ln y + y ln y + y ln y ) B B b y Bν B 1 ν A b Va Va G + ν Ex mx A ( 45) where y b s the ste fracton occupaton of the neutral B 0 on sublattce Q. Equatons wll yeld, for one mole of atoms: x G = A B j G Ass + RT ( x ln x + x ln x + x ln x ) A B j 1+ x A B A B j j A A ( + j 1) The two models become dentcal n the case where =j=ν A =-ν B =1 and t s possble to show the followng denttes: x ABj = y B x B = y b x A = y Va If y A = 1 the G ass term has the followng equvalence B B + G Re g ( 46) x A B j G Ass = y y G ( 47) A B 0 +νa νb A : B and the excess terms of the assocate show the followng equvalences x x x A A B x x x B G A B j A B j reg A, B G G = y reg A, A B j reg B, A B j A y Va = y = y A A y B y y Va b L + νa 0 νa A : B, Va y y B B L L + νa νb A : B, Va + νa νb 0 A : B, B ( 48)

22 22 They can also be made dentcal n a general case f the condtons = -ν B /ν A and j = 1. Ths has been used by Kowalsk et al. /16/ for the systems Fe-O, Cr-O, and N-O. They selected the assocates CrO 1.5, FeO, FeO 1.5, and NO for the lqud phase. These assocates correspond to the (Cr +3 ) 2/3 (O -2 ) 1, (Fe +2 ) 2 (O -2 ) 2, (Fe +3 ) 2/3 (O -2 ) 1, and (N +2 ) 2 (O -2 ) 2 phase compounds resultng from the onc lqud descrpton adopted by Taylor et al. /17/ for the Cr-O and N-O lqud and by Sundman /18/ for the Fe-O lqud. In the recent remodellng /19/ of the Fe-O lqud the Fe +3 speces, and n consequence the (Fe +3 ) 2 (O -2 ) 3 phase compound, has been replaced by the neutral FeO 1.5 speces. The numercal values for the Gbbs energy of formaton for these assocates can be calculated straght from the correspondng values for these phase compounds. Only the change n the number of atoms per assocate should be taken nto account. Smlarly, the excess Gbbs energy can be transformed. Kowalsk et al. reassessed the parameters to gve a smpler model. The equvalence breaks down n ternary and hgher-order systems, as there s the ntroducton of more compostonal varables n the assocate model than for the sublattce model /9,11/. One problem arses when selectng the assocates. For example, Kowalsk et al. /16/ used the assocate CrO 1.5 n the system Cr-O. In ths case the actvty coeffcent of oxygen n nfnte dluton does not have a defned value. Ths s due to the fact that the actvty coeffcent can be calculated from the equaton: Cr + 1.5O Ö CrO 1.5 ( 49) For ths reacton the Gbbs energy change G s defned. From ths we can calculate the equlbrum constant K, whch s defned as: From ths we can calculate: ΛG a RT K = e = = a CrO CraO f f x CrO1.5 CrO Cr xcr fo xo ( 50) f x CrO1.5 CrO1.5 f 1.5 O = ( 51) 1.5 K fcr xcr xo whch does not have defned lmtng value when x CrO1.5 goes to zero. But by selectng Cr 2/3 O as the assocate we wll get the reacton: 2/3 Cr + O Ö Cr 2/3 O ( 52) and ths wll gve the actvty coeffcent by the equaton: f x = ( 53) Cr2 / 3O Cr O f 2 / 3 O 2/3 2/ 3 K fcr xcr xo whch wll have a defned value when x Cr2/3O goes to zero and t s f f O = ( 54) K f Cr2 / 3O 2/3 2/3 Cr xcr

23 23 For ths reason assocates should be selected so that the assocate contanng the least oxygen has stochometrc number one for oxygen, as does assocate FeO n the system Fe-O. Ths problem does not arse wth the onc sublattce model, when there s always oxygen on the anon sublattce n the form O Interpolaton models The molar excess Gbbs energy of a bnary system usng Redlch-Kster formalsm wth components 1 and 2 s expressed as: Ex ( x x ) k k G = x1 x2 L ( 55) where x 1 and x 2 are mole fractons and k L 12 s the nteracton parameter. When usng these bnary parameters to calculate ternary systems, several geometrc models are proposed. In these models the excess Gbbs energy n a ternary soluton at a composton pont p s estmated from the excess Gbbs energes n three bnary subsystems at ponts a, b, and c by the equaton: Ex G = x1x2g12( a) + x2 x3g23( b) + x1x3g13( c) + ( ternary terms) ( 56) where G 12(a), G 23(b), and G 13(c) are the bnary G-functons evaluated at ponts a, b and c. The ternary terms are polynomal terms and may be chosen n order to ft ternary expermental data. The dfferences between four models are shown n Fgure 7 /20/. The Kohler/Toop /21/ and Mugganu/Toop /22/ models are asymmetrcal, as one component s sngled out, whereas the Kohler /23/ and Mugganu /24/ models are symmetrcal ones. For systems wth strong nteractons, the four models can gve qute dfferent results when bnary G ex functons are composton-dependent. Usng Thermo-Calc as the optmsaton program, the only avalable geometrc models are Mugganu and Kohler, whch are both symmetrcal /36/. Most of the systems that are already optmsed and avalable n databases were made usng the Mugganu model. Therefore, t was obvous to select ths model n the present work. However, a few words wll be addressed to the choce between these models.

24 24 Fgure 7. Geometrc nterpolaton models for estmatng ternary thermodynamc propertes from optmsed bnary systems. Suppose that the data for the three bnares of a ternary system have been optmsed to gve bnary coeffcents G xy n equaton ( 56) and the Ex G s estmated n the ternary system usng the Kohler model. Along the lne a-3 n Fgure 7(a) the rato x 2 /(x 1 +x 2 ) s constant and the rato s equal to x 2 at pont a. Therefore functon G 12(a) n equaton ( 56) can be wrtten as: k k x1 x2 G = 12 ( a) L12 ( 57) k x1 + x2 where k L 12 are the bnary coeffcents. That s, G 12(a) as gven s constant along the lne a-3. Smlarly, the functons G 23(b) and G 13(c) may be wrtten n terms of the ratos x 3 /(x 2 +x 3 ) and x 3 /(x 1 +x 3 ) /20/.

25 25 In the Mugganu model, we note that (x 1 -x 2 ) s constant along the lne ap n Fgure 7(c). Hence equaton ( 55) can be substtuted drectly nto equaton ( 56) wth no change. Ths dfference means that the Kohler model assumes that the energy change of the par exchange reacton s constant at a constant x 1 /x 2 rato, whereas the Mugganu model sets ths energy as equal to ts value at the geometrcally closest bnary composton. From a physcal standpont, the former assumpton s perhaps more justfable. Nevertheless, for more concentrated solutons, the Kohler and Mugganu models wll gve qute smlar results /20/. However, n dlute solutons, the Kohler model s to be preferred for the reason llustrated n Fgure 8. In a ternary soluton dlute, n component 1, the Kohler model estmates values G 12(a) and G 13(c) from the values n the bnary systems at compostons that are also dlute n component 1, as seems reasonable. On the other hand, the Mugganu model uses values from bnary systems at compostons that are far from dlute /20/. Fgure 8. The Kohler (a) and Mugganu (b) models appled at a composton dlute n component 1. In the asymmetrcal models llustrated n Fgure 7 (b) and (d), the energes of the 1-2 and 1-3 par nteractons are assumed to reman constant at constant x 1. An asymmetrcal model s thus more physcally reasonable than a symmetrcal model f components 2 and 3 are chemcally smlar whle component 1 s chemcally dfferent, as for example n metal-metal-oxygen systems /20/.

26 26 3 CALCULATION PRACTICE 3.1 Database programs There are some programs that were developed to calculate chemcal equlbrum n multcomponent systems. These programs calculate the equlbrum by mnmsng the Gbbs energy of a system. In multcomponent systems ths mnmsaton s such a large calculaton process that t cannot be done wthout modern computers. Therefore development n ths area has been very heavly dependent on the mprovement of computers calculaton capacty. In ths work the Thermo-Calc program s used. It was developed n Sweden, n Royal Insttute of Technology (RIT). Ths program has an optmsaton module called Parrot, whch allows the usng of dfferent knds of expermental thermodynamc data to assess parameters. The Thermo-Calc program can handle a large varety of models that are used to descrbe the thermodynamcs of dfferent types of soluton phases. It also has a flexble nterface that enables the user to calculate many dfferent types of equlbra and dagrams of varous types. The software can handle sngle equlbra, thermodynamc property dagrams or phase dagrams, wth up to twenty elements or 200 speces /25/. The Thermo-Calc program can be used to calculate many knds of stablty dagrams, for example phase dagrams, speces of phases, lqudus surfaces and thermodynamc property dagrams. Although ths program s delberately restrcted to equlbrum calculatons, t can also be used for smple non-equlbrum smulatons, for example, soldfcaton smulatons based on the Schel- Gullver model. Thermo-Calc can also be used as a subroutne package n other models to calculate the socal equlbrum n some more complcated systems /26/. Other programs that can be used for optmsng the systems are BINGSS, MTDATA, F*A*C*T and Chemsage. BINGSS was developed by H. L. Lukas n Stuttgart and t s a very powerful optmser for bnary systems. It can easly use thousands of expermental values of dfferent knds. It does not nclude any database and ts only purpose s the optmsaton. Usng a parameter fle made by BINGSS one can calculate dagrams usng the BINFKT program. H. L. Lukas has also created optmsaton programs for ternary systems, called TERGSS, and quaternary systems, called QUAGSS, but those programs do not nclude very many soluton models /27/. MTDATA /28/ s a large database program, whch has ts own database. It can be used for many dfferent calculatons, partcularly for multcomponent systems. F*A*C*T /29/ and Chemsage /30/ have bascally the same optmsaton routne, because they are both based on developments by G. Erksson. The latest versons of these programs are dentcal and called FactSage. 3.2 The Calphad prncple Whle buldng a database for mult-component systems t s essental to take nto account some prncples.

27 27 Unary systems (.e. elements) must have the same standard state. A useful selecton for the standard state s SER, n whch the enthalpy of the most stable state of an element s zero at 25 C. The entropy of the most stable crystal structure at zero Kelvn s zero. The enthalpy and entropy of an element at any temperature can then be calculated usng equatons 5 and 6 and the Gbbs energy of an element usng equaton 3. Thermodynamc propertes for all speces are then calculated from the Gbbs energy of the formaton of those speces. All calculated parameters must work wth both hgher order and lower order systems. Ths also means that f there s a phase n a ternary system that does not exst n bnary systems as a stable phase but can mathematcally be calculated as beng able to exst, the parameters of that phase should be defned wth values that keep the phase metastable. These basc rules are also known as the Calphad method. The optmsaton accordng to ths method s descrbed n Fgure 9. Scheme of Calculaton Analytcal Expressons Optmsaton Procedure Unary Systems Elements, Phase Stabltes Optmsed Bnary Systems All Bnary Expermental Values Extrapolated formulas No Measured Values Analytcal Excess Terms Optmsaton Procedure Analytcal Excess Terms Optmsaton Procedure Extrapolated Ternary Systems Optmsed Ternary Systems Hgher Order Multcomponent Systems Few Ternary Experemental Values Few Expermental Values Fgure 9. Schematc presentaton of the Calphad method /31/.

28 Optmsaton practce The general practce of optmsaton s performed through a process of tral-and-error combnatons of the adjustable parameters. Optmsaton programs, such as Parrot, reduce the error usng mathematcal algorthms, whle a manual procedure means that the user changes these parameters accordng to ther personal judgement. Although a mathematcal least-squares-type approach mght be consdered deal, a great deal of judgement s stll requred n ths process, partcularly when there s lttle expermental nformaton avalable. It must also be noted that these mathematcal programs do not take account of the physcal meanngs of parameters and therefore t s possble to fnd a combnaton of parameters that works wth a bnary system, but does not work n hgher order systems. The smplest way to explan the method of optmsaton s to look at t wth an example. The followng shows a smplfed optmsaton procedure for the system Fe-Cr-O /53/. To start wth, a lterature survey must be carred out. In the case of the example system the latest and largest lterature survey was carred out by Raghavan /32/. The THERMET lterature databank n Grenoble was also checked. All the data that were found were adopted for the optmsaton. On the other hand, the Parrot program can not handle any number of expermental data. Besdes, as there are hundreds of measured values for one phase they may gve too much weght on one study. Therefore t was necessary to crtcally analyse the data and ther errors before selectng the most consstent set from these data. The optmsaton of the ternary system Fe-Cr-O was performed usng the Parrot module. It was started from sold oxde phases. Frst the thermodynamc parameters for chromte were calculated, usng measured data of chromte-cr 2 O 3 -metallc-phase equlbrum. The oxygen partal pressures n these equlbra were also used. The values used at the start of ths optmsaton were the values of the Gbbs energy of the formaton of chromte that were found n the lterature. In the next step, the parameters of the spnel phase were optmsed, usng the oxygen pressures n ths phase. Then the spnel phase parameters were optmsed together wth those of a sesquoxde phase (Fe 2 O 3 -Cr 2 O 3 ), after fxng these parameters. The parameters of wüstte were optmsed usng frst only the measured oxygen potentals n dfferent compostons and at dfferent temperatures. Then they were optmsed together wth the parameters of the other sold phases. After all the sold phase parameters were optmsed the work wth the lqud phase was started. Frst the parameters of the metal-rch lqud were optmsed usng measured oxygen concentratons and actvtes. After that they were optmsed together usng measured equlbra oxdc lqud. When all the phases had been descrbed separately, calculatons were performed wth all parameters and all expermental data. Durng ths procedure the parameters dd not change very much.

29 29 4 DESCRIPTION OF THE SYSTEM The program used n ths work, Thermo-Calc, allows the use of a dfferent model for each phase. The reference state was chosen to be SER. 4.1 Gas phase The gas phase s an deal mxture phase. Its speces are O 2, CO, and CO Lqud The lqud phase s descrbed wth an assocate model usng Fe, Cr, N, C, FeO, FeO 3/2, Cr 2/3 O, and NO as assocates. Ths descrpton can handle both the metallc and oxde lqud. 4.3 Sold metallc phases These phases are body-centred and face-centred cubc phases (bcc and fcc) for ron, chromum, and nckel. Carbon and oxygen are on ntersttal stes, whch gves the second sublattce for these phases. In the bcc and fcc phases the ste numbers are 1:3 and 1:1, respectvely. These phases are descrbed usng Redlch-Kster polynomals and the nterpolaton model of Mugganu. These phases are formulated as: Bcc: (Fe,Cr,N) 1 (C,O,Va) 3 Fcc: (Fe,Cr,N) 1 (C,O,Va) Sgma phase In the bnary system ron-chromum there s an ntermetallc sold phase known as the sgma phase. Nckel has sgnfcant solublty n ths phase. The phase s descrbed wth three sublattces usng ste numbers 8:4:18. The phase s formulated as: (Fe,N) 8 (Cr) 4 (Cr,Fe,N) Wüstte and bunsente The wüstte and bunsente phases are descrbed wth the compound energy model usng two sublattces. The phase s formulated as: (Cr 3+,Fe 2+,Fe 3+,N 2+,N 3+,Va) 1 (O 2- ) 1 In ths work these phases are referred to as halte accordng to ts prototype phase. 4.6 Hematte and chromum oxde Cr 2 O 3 The hematte and chromum oxde phases are descrbed wth the compound energy model usng three sublattces. The phase s formulated as: (Cr 3+,Cr 2+,Fe 3+ ) 2 (Cr 3+,N 2+,Va) 1 (O 2- ) 3

30 30 In ths work ths phase s referred to as corundum accordng to ts prototype phase. 4.7 Spnel The spnel phase s descrbed wth the compound energy model usng four sublattces. The phase s formulated as: (Cr 3+,Cr 2+,Fe 3+,Fe 2+,N 2+ ) 1 ( Cr 3+,Fe 3+,Fe 2+,N 2+,Va) 2 (Fe 2+,Va) 2 (O 2- ) 4 Ths phase s magnette Fe 3 O 4, Cr 3 O 4, chromte FeCr 2 O 4, and nckel chromte Cr 2 NO 4. The propertes of the spnel phase are based on those of ts component spnels Fe 3 O 4, Fe 2 NO 4, FeCr 2 O 4, and Cr 2 NO 4. Magnette has been descrbed by Sundman /18/ usng four ndependent parameters, as presented earler n Chapter Taylor used the same model for magnette and also enlarged the model to present chromte. Nckel ferrte s defned by Luoma /47/ usng the same model. Combnng these three descrptons, the fnal model wll have ten ndependent parameters combned to the 50 parameters needed to descrbe ths phase. 4.8 Graphte The graphte phase s a stochometrc pure carbon phase. 4.9 Carbde phases The system chromum-carbon contans three carbdes, descrbed as M 3 C 2, M 7 C 3, and M 23 C 6 n ths work. The M 3 C 2 phase s a stochometrc chromum carbde phase wth no solublty of ron or nckel and s descrbed wth two sublattces. The M 7 C 3 phase has two sublattces and the M 23 C 6 has three sublattces. The phases are formulated as: M 3 C 2 : (Cr) 3 (C) 2 M 7 C 3 : (Cr,Fe,N) 7 (C) 3 M 23 C 6 : (Cr,Fe,N) 20 (Cr,Fe,N) 3 (C) 6

31 31 5 BINARY SYSTEMS 5.1 System Fe-Cr The system ron-chromum has been assessed by Andersson and Sundman /33/. The assessed phase dagram s n very good agreement wth the dagram publshed n /34/ except for the lqudus lne at hgher chromum contents. The reasons are the expermental dffcultes and the hgh affnty of chromum to oxygen. The assessed phase dagram s shown n Fgure 10. Lqud fcc bcc Sgma Fe Cr Fgure 10. The assessed phase dagram for the system Fe-Cr /33/. 5.2 System Fe-N At hgher temperatures the system ron-nckel s rather well known. Publcaton /35/ presents a lterature survey up to 1989 carred out by Swartzendruber. In the M verson of the Thermo-Calc database system /36/ there s a set of thermodynamc parameters that are based on the unpublshed work of Dnsdale /37/. These parameters gve an assessed phase dagram shown n Fgure 11. A remarkable dfference between the present and the earler assessed dagrams s that the ntermetallc, low-temperature FeN 3 phase, found by many authors, s gnored from the present dagram.

32 32 Lqud bcc fcc bcc Fe N Fgure 11. The assessed phase dagram for the system Fe-N. 5.3 System Fe-C The system ron-carbon s very well known. Publcaton /35/ presents a lterature survey conducted by Okamoto, whch also ncludes the metastable phase dagram Fe-Fe 3 C. The assessed parameters are gven by Gustafson /38/. The calculated phase dagram shown n Fgure 12 s n very good agreement wth the expermental data. Lqud Lqud + graphte bcc fcc fcc + graphte bcc bcc + graphte Fe Fgure 12. The assessed phase dagram for the system Fe-C.

33 System Fe-O The ron-oxygen system has recently been optmsed by three workers. Sundman /18/ made the frst contrbuton, usng an onc sublattce model for the lqud phase and also a reassessment modfyng the descrpton of the lqud phase /13/. Kowalsk et al. /16/ optmsed ths system usng the assocate model for the lqud phase. In ths work Sundman s later descrpton of the lqud /19/ was adopted. Ths descrpton was converted to that of the assocate model as shown n Secton The parameters obtaned are gven n Table 1. These parameters gve very good agreement wth the expermental data. The model for sold ron phases has been taken from Kowalsk et al. /16/. In ther model oxygen occupes the ntersttal sublattce of both the bcc and fcc structures of ron. Also n ths case the agreement wth the expermental data s good as can be seen n Fgure 14. Table 1. Interacton parameters for the lqud phase of the system Fe-O. Parameter Value (J) 0 L(Fe FeO) *T 1 L(Fe FeO) *T 0 L(Fe FeO 1.5 ) L(FeO FeO 1.5 ) L(FeO FeO 1.5 ) L L bcc fcc FeO bcc Fe3O4 Fe2O3 Fgure 13. The assessed phase dagram for the system Fe-O.