Evaluation Model for Viscosity of Fe Ni Cr Alloys Using Gibbs Free Energy of Mixing and Geometric Methods

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1 ISIJ Internatonal, Vol. 57 (27), ISIJ Internatonal, No. 8 Vol. 57 (27), No. 8, pp Evaluaton Model for Vscosty of Fe N Cr Alloys Usng Gbbs Free Energy of Mxng and Geometrc Methods Yanhu LIU, Xuewe LV* and Chenguang BAI School of Materals Scence and Engneerng, Chongqng Unversty, Chongqng, 444 Chna. (Receved on Aprl, 26; accepted on October 3, 26) Purpose of ths nvestgaton s to predct the vscosty of lqud Fe N, Fe Cr, N Cr and Fe N Cr alloys at 873 K. The actvaton energes of bnary alloys were calculated by the mxng Gbbs energes, and the proposed actvaton energes of pure Fe, N and Cr are , and J/mol. Geometrc models (Kohler, Toop and Chou models) were used to predct the cess actvaton energes of ternary alloys. The so-vscosty curves of Fe N Cr alloy are calculated by three geometrc models (Kohler, Toop and Chou models) from sub-bnary systems. N can decrease the vscosty n the whole range, whle Fe has a twosded effect. Cr can result n the decrease of the vscosty at the N-rch corner and Fe-rch corner. The effects of N and Cr on vscostes of Fe N Cr alloys are consstent wth the measured results. But when Cr ceeds 2 n mole%, the addton of Cr wll cause the rse of the vscosty. The evaluated values by the Chou model are the bggest and have a much more reasonable low vscosty regon. Comparson between the evaluated results and permental values of the Fe N Cr ternary alloys were nvestgated. The average errors between the measured results and predcted values by Kohler, Toop and Chou model are below 5%, ndcatng that evaluated vscostes of Fe N Cr ternary alloys by three models (Kohler, Toop and Chou) can reproduce the measured results qute well. KEY WORDS: vscosty; actvaton energy; mxng Gbbs free energy; geometrc models.. Introducton Thermo-physcal propertes of these lqud metals, such as heat capacty, thermal conductvty, surface tenson, densty and vscosty are of partcular nterest because these propertes play mportant roles n heat and mass transport processes. Knowledge of the vscostes 7) of pure lqud metals and respectve mxtures s mportant for practcal and theoretcal purposes. Furthermore, the vscosty of multcomponent 8,9) lqud mxtures s an nvaluable type of data for the chemcal engneer n the desgn and optmzaton of ndustral processes, partcularly for the smeltng, castng and weldng processes of Fe-based alloys and stanless steels. Therefore, t s very mportant to obtan a precse and relable vscosty, especally for hgh meltng pont alloys from the ndustral pont of vew. Proposed methods for the measurement of the vscosty nclude: capllary, oscllatng vessel, rotatonal bob or crucble, oscllatng plate, dranng vessel, levtaton usng the dampng of surface oscllatons and acoustc methods. It s well known that the measurement of the vscosty of molten metals, especally at hgh temperatures, s dffcult because of the reactvty wth the crucbles and atmosphere, and also the sensor lmtaton of the low vscosty. Complatons of the vscostes of pure lqud metals at dfferent temperatures, as well as those of bnary lqud mxtures through the whole composton range, * Correspondng author: E-mal: lvxuewe@63.com DOI: can be found n the lterature. 3,6, 4) Nevertheless, the stuaton changes n the case of mult-component lqud mxtures, and there s only a lmted number of vscosty data. Over the last years, a large number of estmaton models for vscosty avalable have been developed and mproved. In addton, molecular dynamcs 8,5 8) provdes a way to predct the vscosty of lqud mxtures wthout makng use of permental data. But the devatons between calculated and permental values are, n general, greater than those obtaned wth the other types of models. Vscostes of molten Fe N alloy were measured by usng an oscllatng cup vscometer over the entre composton range from lqudus temperatures up to 873 K wth hgh precson and cellent reproducblty by Sato et al. 4) However, there are few permental values avalable for Fe Cr and N Cr bnary systems, and even less for Fe N Cr ternary alloys. 9,9 2) So no systematc study ncludng permental measurement and model evaluaton has been carred out for Fe N Cr hgher-order metallc system. The am of ths nvestgaton s to predct the vscostes of lqud Fe N, Fe Cr, N Cr and Fe N Cr ternary alloys at 873 K. Vscostes of the sub-bnary alloys were calculated based on the mxng Gbbs energy model proposed by Seetharaman et al. 22,23) The actvaton energes of bnary alloys are calculated by the mxng Gbbs energes, and vscostes of the bnary alloys were evaluated based on the Arrhenus equaton, and can reach a good agreement wth the permental values. The polynomal equatons were used to descrbe the varatons of cess actvaton energes 27 ISIJ 296

2 of three bnary systems, and the geometrc models (Kohler, Toop and Chou models 24 26) ) were used to predct the cess actvaton energes of ternary alloys. The calculated results of the ternary alloys were compared wth the permental values by consderng the applcablty of the three geometrc models. 2. Vscosty Evaluaton of the Sub-bnary Alloy The absolute rate theory method of Eyrng 27) provdes the presson of the vscosty calculaton for a lqud mxture, whch has been appled successfully to varous metallc as well as onc systems. ρhn η = A p M... () RT Where η s the dynamc vscosty of the alloy, h s Plank s constant, N A s Avogadro s number, ρ s the densty of the lqud alloy, M s the molar mass of the alloy, s the actvaton energy for the flow process, and T s the absolute temperature. Actvaton energy s consdered to be a functon of both temperature and composton of the melt. Theoretcally, vscosty of one alloy can be predcted f ts actvaton energy s gven out, but t s not easy to obtan the value of actvaton energy. Seetharaman et al. 22) proposed the followng relatonshp between the actvaton energy and the mxng Gbbs energy: = x + Gmx + 3 RTxAxB... (2) The subscrpt stands for component. The frst term on the rght-hand sde of Eq. (2) represents the lnear varaton of the actvaton energy from the pure components n the absence of nteracton between dfferent components. The s the actvaton energy of pure component n lqud state. ΔG mx represents the mxng Gbbs energy of the alloy n a soluton. Where, x A and x B are the respectve mole fractons. In the case of hgh-order metallc systems, the molar mass of the alloy can be calculated by Eq. (3). In a smlar way, the densty of the melt may be estmated, as a frst approxmaton, by Eq. (4). ρ = ρx... (3) 2.2. Mxng Gbbs Energes of Bnary Alloys An assessment set of thermodynamc parameters for Fe Cr and Fe N alloy systems 28 3) can be used to calculate the mxng Gbbs energes of alloys. Sometmes, there happens some dscrepances between calculatons and permental data of hgher order systems, and these troubles can be traced back to the assessments of lower order systems. In order to obtan the accurate values, the revson of actvtes of the Fe Cr and Fe N lqud phase proposed by Lee et al. 3) has been adopted n ths study, whch can gve better agreement wth permental data. Mxng Gbbs energes of bnary systems can be calculated by the followng equaton and these values are as shown n Fg. and lsted n Table 2. As for the N Cr system, data concernng ts mxng Gbbs energy studed by Novakovc et al. 32) can reproduce the permental values farly well by enablng a more precse classfcaton of the N Cr system n terms of weakly compound-formng alloys. The mxng Gbbs energes of N Cr alloys are also shown n Fg. and lsted n Table 2. It should be ponted out that a wde composton range of sold soluton sts n Fe Cr and N Cr systems at 873 K, so these alloys are assumed to be super-cooled lqud n order to obtan a smooth lne of Gbbs free energy of mxng. G = RT( x lna + x ln a )... (5) mx A A B B 2.3. Evaluaton for Vscostes of Bnary Alloys at 873 K Wth the help of mxng Gbbs energy and the actvaton energes of pure metals proposed n ths study, the actvaton energes of bnary alloys at 873 K are easy to be Table. Data of the vscosty, 7) densty 3) and molar mass used n ths study (Note: Vscosty of pure Cr s estmated from the Fe Cr bnary systems). Metal Vscosty, mpa s Densty, g/cm 3 Molar mass, g/mol Fe N Cr M = M x... (4) 2.. Actvaton Energes for Pure Fe, N and Cr at 873 K Denstes and vscostes for pure metals are shown n Table. 4,3) Then the actvaton energes for Fe, N and Cr wll be obtaned wth combng Eq. (), and the proposed values for actvaton energy are , and J/mol, respectvely. It should be ponted out that the meltng pont of Cr s as hgh as 2 48 K, whch s qute hgher than 873 K. In order to predct the vscostes of lqud Fe N Cr ternary alloys, the densty and vscosty of lqud Cr should be provded. In ths study, the densty of pure Cr are usng the value at the meltng pont, and ts vscosty s selected as 5.77 mpa s, whch s estmated from measured values of Fe Cr bnary alloys. Fg.. Mxng Gbbs free energy of bnary A-B systems wth X B at 873 K ISIJ

3 Table 2. Mxng Gbbs free energes of the Fe N, Fe Cr and N Cr alloys at 873 K (J/mol). A, mole B, mole Fe N Fe Cr N Cr obtaned, and the values are shown n Table 3. By combnng Eqs. () (4), t s not hard to obtan the vscostes of bnary alloys. As shown n Fg. 2, the evaluated vscostes of Fe N bnary alloy are compared wth the measured results proposed by other researchers, 4,33 35) showng that the predcton values can reproduce the permental values qute well. Here another two dfferent evaluaton models for the vscostes of Fe N alloys are also studed and plotted n Fg. 2. An presson derved by Ida et al. 34) descrbes the cess vscosty of lqud bnary alloys by usng some basc physcal propertes. Hra 35) estmated the vscostes of lqud alloys by systematcally analyzng the vscosty data for lqud metals from n the lterature. By comparng the predcted values by three models and permental data, t can be found that the Hra model can not reproduce the permental data at the composton regons near pure ron and pure nckel, whle the evaluated values by Ida model seems a lttle hgher than the measured results. The predcted results by Seetharaman model can reach a good agreement wth the average of all the measured values. Fgure 3 shows the comparson between the evaluated vscostes and permental data of Fe Cr alloys. Two-sde effect can also be observed, whch s consstent wth the measured results by Kamaeva et al., 36) whch are caused by a concentraton change n the bndng energy between the atoms determned by the geometrc and compostonal short-range order. Expermental data of Fe Cr bnary alloys Table 3. Actvaton energes and cess actvaton energes of the Fe N, Fe Cr and N Cr sub-bnary alloys at 873 K. Component, mole A B Fe N, kj/mol Fe Cr, kj/mol N Cr, kj/mol Excess measured by Sato et al. 9) and Kobatake et al. 9) were plotted n Fg. 3 and compared wth the calculated values by Seetharaman model, showng that the predcted values can reproduce the permental data qute well. The predcted vscostes for N Cr alloys are gven out as shown Excess Excess Fg. 2. Comparson between the permental data and calculated vscostes for Fe N bnary alloys wth dfferent models at 873 K. 27 ISIJ 298

4 n Fg. 4, llustratng that Cr has smlar effect n Fe Cr and N Cr alloys. 3. Evaluaton for the Vscosty of Ternary Alloy 3.. Excess Actvaton Energy Evaluaton of Bnary Alloy The cess actvaton energy can be pressed by the lnear relatonshp as follows: = x... (6) Where, s the cess actvaton energy of bnary alloy. Accordng to the calculated actvaton energes and the actvaton energes of pure metals, the cess actvaton Fg. 3. Comparson between the calculated vscostes and permental values of Fe Cr bnary alloys at 873 K. energes of three sub-bnary systems n the Fe N Cr ternary alloy can be calculated as shown n Fg. 5, whch are ft usng polynomal equaton descrbed by Eq. (7), where x A and x B are the molar contents of the components, A and B, respectvely, and A k are the coeffcents related to the sub-bnary A-B system and ther values are lsted n Table 4. n = A ( x x ) k= k A B k... (7) 3.2. Methods to Calculate the Excess Actvaton Energes of Ternary Alloys In order to descrbe the cess vscosty of ternary alloy from the measured vscostes of sub-bnary systems, three possble methods have been consdered here. The tradtonal geometrc models can be dvded nto three types: symmetrc (Kohler 24) model), asymmetrc (Toop 25) model) and general soluton model (Chou 26) model). Geometrc models have been used tensvely to predct the thermodynamc propertes of ternary and mult-component systems from sub-bnary systems. 36 4) Hu et al. 4) frst tended geometrc models to predct the surface tensons of the Ag Au Cu ternary alloy wthout comparson between calculated and permental results. Recently, geometrcal models were successfully appled to predct the surface tenson of the N 3 S 2 FeS Cu 2 S matte system and the calculated results were n good agreement wth the permental values. 42) Surface tensons of the Sn Ga In ternary alloy were calculated from the surface tensons of the Sn Ga, Ga In and In Sn sub-bnary systems by usng geometrc models (the Kohler model, Toop model and Chou model) by Yan et al. 43) wth cellent reproducblty. Fg. 4. Predcted vscostes of N Cr alloys by mxng Gbbs energy at 873 K. Fg. 5. Varaton of cess actvaton energes of three sub-bnary A-B alloys wth X B at 873 K. Table 4. Coeffcents of the polynomal for cess actvaton energes of bnary alloys at 873 K. System A A A 2 A 3 A 4 A 5 A 6 Fe N Fe Cr N Cr ISIJ

5 Here, geometrc models are used to predct the actvaton energy nstead of Gbbs free energy. As mentoned above, the meltng pont of Cr s qute hgher than 873 K, so t s mpossble to obtan the so-vscosty curves of the ternary alloy n the whole range. The lqud projecton of Fe N Cr ternary alloy as a functon of temperature by FactSage was shown n Fg. 6. So n ths work, the so-vscosty curves wll be cut off by the 873 K sothermal lne n the ternary Fe N Cr fgures Kohler Model In 96, Kohler 24) proposed a symmetrc geometrc model based on the hypothess of the smlarty of the thermodynamcs propertes of the three components wthn the ternary alloy, whch can be descrbed as the followng equaton: = ( X + X ) ( ) + ( X + X ) ( ) X X X2 X ) X2 X3 + ( X + X ) (... (8) Where, and j are the cess actvaton energy of the ternary system and A-B the bnary system, respectvely; X s the mole fracton of component n the ternary system Toop Model As a classc asymmetrc geometrc model, the Toop model 25) has been wdely used and can be descrbed as follows: X2 X X X X X3 = 2 (, ) + + X + X 3 ( X, X ) ( X2 + X3) 2 X2 X3 23, X2 + X3 X2 + X 3... (9) In ths model, t s of vtal mportance to determne the asymmetrc component. Qao et al. 39) compared some crtera for judgng the symmetry of the ternary system, and Fg. 6. Lqud projecton of Fe N Cr ternary alloy wth temperature by thermodynamcs calculatons. ponted out that f the cess thermodynamc propertes of the three sub-bnary systems are smlar to each other, then the ternary system s symmetrc. Otherwse, f the devatons of the bnary systems A-B and A-C from the deal soluton are smlar, but dffer markedly from that of the bnary system B-C, then the A-B-C ternary system s asymmetrc. In the asymmetrc system, the common component A n two sub-bnary systems wth thermodynamc smlartes should be chosen as the thermodynamc asymmetrc component. Here, ths crteron was tended to our cess actvaton energy calculaton by usng geometrc models. It s better to pck up N as the asymmetrc component for Toop model, snce the Fe N and N Cr two bnary systems are much more smlar shown n Fg General Soluton Model----Chou Model Recently, Chou 26) proposed a general soluton model to predct the thermodynamc propertes of a ternary system from the three boundary system. The most attractve characterstc of ths new model s bypassng the tradtonal methods and subsumng symmetrc and asymmetrc models n a more general approach. Here, the Chou model was also tended to the actvaton energy predcton of the ternary system from the three boundary systems, as shown n Eq. (): E XX 2 E XX 2 3 E XX 3 η = η2 + η23 + η3 E X2 ( ) X2( 2) X223 ( ) + X3( 23) X33 ( ) + X3 ( )... () Where X (j) s the mole fracton of component n the -j bnary system, whch can be calculated from the followng equatons: X2 ( ) = X + ξ2x3 X223 ( ) = X2 + ξ23x... () X = X + ξ X 33 ( ) Where ξ j s the smlarty coeffcent, whch can be pressed as: λ ξ2 = λ + λ2 λ2 ξ23 =... (2) λ2 + λ3 λ ξ3 = λ + λ3 Where λ s the devaton sum of squares, whch can be calculated as follows: dx E E λ = ( η η ) E E λ = ( η η ) dx E E λ3 = ( η3 η32 ) 2 dx 3... (3) It should be ponted out that the smlarty coeffcents mentoned n the Chou model are consdered to study 27 ISIJ 3

6 the applcablty of three models n order to analyze the accuracy of the predcted values. If the thrd component s smlar to the second one, λ =, thus ξ 2 =, and f the thrd one s smlar to the frst one, λ =, thus ξ 2 =. Therefore, t can be judged from the smlarty coeffcent that the thrd component s more smlar to the component one or two. It has been proved 26,44 47) that the Chou s model s a general geometrcal model, all tradtonal geometrc models, ncludng the Kohler model and Toop model, are one of forms of the general geometrcal model under the dfferent partcular stuaton Evaluaton for Actvaton Energy of Ternary Alloy The actvaton energy of ternary alloy can be calculated by Eq. (5) wth the cess actvaton energy and actvaton energes of pure metals. 3 = x +... (4) Smlar to the vscosty calculaton of bnary alloy, the vscosty of ternary alloy can be calculated by combng Eqs. (), (3), (4) and (5). the vscosty. Interestng to fnd that there s a regon that the vscosty of the ternary alloy s even lower than that of N, and the reason may be attrbuted nto the concentraton change n the bndng energy between the atoms determned by the geometrc and compostonal short-range order, whch can also be found n the Fe Cr bnary alloys Iso-vscosty Curve by Toop Model Fgure 8 shows the so-vscosty curves of the Fe N Cr ternary alloy evaluated by the Toop model. Smooth sovscosty curves are also observed, showng that there s no remarkable nteracton force between the components. Smlar nfluences can be found for Fe, N and Cr n Kohler model. But the effect of Cr to decrease the vscosty at the Fe-rch corner s weakened whle that at the N-rch corner s strengthened. Comparng wth the calculated results by the Kohler model, the most apparent pont s that the low vscosty regon (lower than 3.5 mpa s) s enlarged, whch s not reasonable and may cause a major devaton. 4. Results and Dscusson 4.. Iso-vscosty Curves of Ternary Alloy 4... Iso-vscosty Curves by Kohler Model The so-vscosty curves of the Fe N Cr ternary alloy predcted by the Kohler model were plotted n Fg. 7. Smooth so-vscosty curves are observed, showng that there s no remarkable nteracton force between the components. N can lower the vscosty of the ternary alloy at the whole composton range. Cr can result n the decrease of the vscosty at the N-rch corner and Fe-rch corner. But when Cr ceeds 2 n mole%, the addton of Cr wll cause the rse of the vscosty. When the N s fxed and above 4 n mole%, the addton of Fe can ncrease the vscosty. Once the N s fxed and below 4%, Fe can promote the ncrease of the vscosty when Fe s below 4%, then the contnung ncrease of Fe wll result n the decrease of Fg. 8. Iso-vscosty curves of Fe N Cr alloys calculated by Toop model at 873 K (mpa s). Fg. 7. Iso-vscosty curves of Fe N Cr alloys calculated by Kohler model at 873 K (mpa s). Fg. 9. Iso-vscosty curves of Fe N Cr alloys calculated by Chou model at 873 K (mpa s) ISIJ

7 4..3. Iso-vscosty by Chou Model The so-vscosty curves of the Fe N Cr ternary alloy evaluated by the Chou model are plotted n Fg. 9. Smooth so-vscosty curves are also observed, and smlar nfluences can be found for Fe, N and Cr. Comparng wth the calculated results by the Kohler model and the Toop model, all the so-vscosty curves shft away from the Fe Cr lne, ndcatng that the evaluated values by the Chou model are the bggest and have a much more reasonable low vscosty regon, whch can be thought as the mprovement dfferent from other two models Reproducblty of Three Geometrc Models Expermental data of Fe N Cr ternary alloys measured by Sato et al. 2,2) were also marked as red dot n Fgs The effects of N and Cr on vscostes of Fe N Cr alloys are consstent wth the measured results by Sato et al. 2) The reproducblty s evaluated as the average error defned as Eq. (5), and the average errors between the measured results and predcted values by Kohler, Toop and Chou model are below 5%, ndcatng that evaluated vscostes of Fe N Cr ternary alloys by three models reach a good agreement wth the permental data. Averageerror = N σcalc σexpe %... (5) N σ 5. Conclusons In ths work, the vscostes of lqud Fe N Cr subbnary alloys and ternary alloys at 873 K have been evaluated usng mxng Gbbs free energes and geometrc models by consderng the cess actvaton energes from the subbnary systems. The man conclusons are as follows: () The actvaton energes of pure Fe, N and Cr at 873 K are gven ther proposed values are , and J/mol, respectvely. The calculated vscostes of Fe N and Fe Cr alloys can reproduce the permental values and the vscostes of N Cr alloys are predcted n ths work. Cr has a two-sded effect n Fe Cr and N Cr alloys. (2) The so-vscosty curves of Fe N Cr alloy are calculated by three geometrc models (Kohler, Toop and Chou models) from sub-bnary systems. N can decrease the vscosty n the whole range, whle Fe has a two-sded effect. Cr can result n the decrease of the vscosty at the N-rch corner and Fe-rch corner. But when Cr ceeds 2 n mole%, the addton of Cr wll cause the rse of the vscosty. The evaluated values by the Chou model are the bggest and have a much more reasonable low vscosty regon. (3) The average errors between the measured results and predcted values by Kohler, Toop and Chou model are below 5%, ndcatng that evaluated vscostes of Fe N Cr ternary alloys by three models (Kohler, Toop and Chou) reach a good agreement wth the permental data. Expe Acknowledgements The authors are especally grateful to the Natonal Natural Scence Foundaton of Chna (NSFC) (Grant No. 5234) and Chongqng Unversty Graduate Students Scentfc Research Innovaton Project (No. CYB426). Thanks for the support from Chnese Scholarshp Councl (CSC No ). REFERENCES ) J. A. Rddck and W. B. Bunger: Organc Solvents, 3rd ed., Wley- Interscence, New York, (97), 32. 2) D. R. Lde: CRC Handbook of Chemstry and Physcs, 82nd ed., CRC Press, Boca Raton, FL, (22), 2. 3) J. T. Schrodt and R. M. Akel: J. Chem. Eng. Data, 34 (989), 8. 4) Y. Sato, K. Sugsawa, D. Aok and T. Yamamura: Meas. Sc. Technol., 6 (25), ) D. P. Dobson, W. A. Crchton, L. Vocadlo, A. P. Jones, Y. Wang, T. Uchda, M. Rvers, S. Sutton and J. P. Brodholt: Am. Mneral., 85 (2), ) R. Brooks, A. Dnsdale and P. Quested: Meas. Sc. Technol., 6 (25), ) L. Battezzat and A. Greer: Acta Metall., 37 (989), 79. 8) D. Dysthe, A. Fuchs, B. Rousseau and M. Durandeau: J. Chem. Phys., (999), 46. 9) H. Kobatake and J. Brllo: J. Mater. Sc., 48 (23), 688. ) D. Zvkovc and D. Manasjevc: Calphad, 29 (25), 32. ) X. Wang, H. Bao and W. L: Metall. Mater. Trans. A, 33 (22), 32. 2) Y. S. Yang and C. Y. Tsao: J. Mater. Sc., 32 (997), ) J. Chevaler, P. Petrno and B. Y. Gaston: Chem. Eng. Sc., 43 (988), 33. 4) W. Cao, A. Fredenslund and P. Rasmussen: Ind. Eng. Chem. Res., 3 (992), ) Y. Q, T. Cagn, Y. Kmura and W. A. Goddard I: J. Comput.-Aded Mater. Des., 8 (2), ) E. M. Goslng, I. R. Mcdonald and K. Snger: Mol. Phys., 26 (973), ) W. T. Ashurst and W. Hoover: Phys. Rev. A, (975), ) Y. G. Zhang and G. J. Guo: Phys. rth Planet. Inter., 22 (2), ) Y. Sato: CAMP-ISIJ, 6 (23), 28. 2) Y. Sato: CAMP-ISIJ, 6 (23), 7. 2) Y. Sato: CAMP-ISIJ, 9 (26), ) S. Seetharaman and D. Schen: Metall. Mater. Trans. B, 25 (994), ) S. Seetharaman, D. Schen and F. Z. J: Metall. Mater. Trans. B, 3 (2), 5. 24) F. Kohler: Monatsh. Chem., 9 (96), ) G. Toop: Trans. Metall. Soc. AIME, 233 (965), ) K. C. Chou: Calphad, 9 (995), ) S. Glasstone, K. J. Ladler and H. Eyrng: Surf. Rev. Lett., 7 (96) ) D. M. Kundrat, M. Chochol and J. F. Ellott: Metall. Trans., 5B (984), ) E. Schurmann and J. Brauckmann: Arch. Esenhuttenwes., 48 (977), 3. 3) B. J. Lee: Calphad, 7 (993), 25. 3) A. F. Crawley: Int. Metall. Rev., 9 (974), ) R. Novakovc: J. Phys.: Condens. Matter, 23 (2), ) M. Hra: ISIJ Int., 33 (993), ) T. Ida, M. Ueda and Z. Morta: Tetsu-to-Hagané, 62 (976), ) M. Hra: Tetsu-to-Hagané, 78 (992), ) L. V. Kamaeva, I. V. Sterkhova and V. I. Lad yanov: Inorg. Mater., 48 (22), ) K. C. Chou and Y. A. Chang: Phys. Chem. Chem. Phys., 93 (989), ) I. Ansara, C. Bernard, L. Kaufman and P. Spencer: Calphad, 2 (978),. 39) M. Hllert: Calphad, 4 (98),. 4) Z. Y. Qao, X. R. Xng, M. Peng and A. Mkula: J. Phase Equlb., 7 (996), 52. 4) M. Peng, Z. Y. Qao and A. Mkula: Calphad, 22 (998), ) J. H. Hu, S. L. Chen, W. J. Ca and K. C. Chou: J. Unv. Sc. Technol. Bejng, 2 (99), ) L. J. Yan, Z. M. Cao, Y. A. Xe and Z. Y. Qao: Calphad, 24 (2), ) L. J. Yan, S. Zheng, G. Dng, G. Xu and Z. Y. Qao: Calphad, 3 (27), 2. 45) K. C. Chou, W. C. L, F. S. L and M. H. He: Calphad, 2 (996), ) G. H. Zhang and K. C. Chou: J. Solut. Chem., 39 (2), 2. 47) K. C. Chou and S. K. We: Metall. Mater. Trans. B, 28 (997), ISIJ 32