Dev. Chem. Eng. Mineral Process. 14(3/4), pp. 397-408, 2006. The Model of Super Order Sub-Regular Melts Guo-Chang Jiang*, Shao-Bo Zheng and Kuang-Di Xu School of Material Science and Engineering, Shanghai University, Shanghai 200072, P. R. China The SReM model associates with super order sub-regular melts. It is intended to describe systematically the component activities in the entire region of a homogeneous phase, either liquid or solid, in a multiple alloy or slag. A reliable evaluation of component activities can be provided for engineering problems. In this paper the model characteristics, its formalism and a simplified version are introduced The thermodynamic restrictions utilized to check the evaluated parameters were suggested by the SReM model. According to this self-consistent approach, the evaluated parameters of this model are reliable. The Model The model for quaternary systems based on the work of Pelton and Wang [ 1,2] is available [3-51, including its applications. The model of SReM(5) for quintary systems, with the recurrence relation, is given as follows. ...(1 .l) a=2 b=O c=O d=O ...(1.2) ...( 1.3) C4 =[a-I------ a-b b-c YI YlY2 c-d 1- 1 ...(1.4) YlY2Y3 a-1 * Author for correspondence (firstname.lastname@example.org). 397 Guo-Chang Jiang, Shao-Bo Zheng and Kuang-Di Xu 45 --[a-l_-_---a-b Yl b-c YlY2 c - d -____ d 1 1YlY2Y3 YlY2Y3Y4 a-1 ...(1.5) the concentration coordinates, Y / ~ Y ~ . Y are ~ , Ydefined ~ in Table 1; xi denotes the mole fraction of component i ( i = 1,2,3,4,5); a, b, c, d represent the order. Furthermore, the parameters equal to zero in the SReM(5) model can also be exhibited according to the recurrence approach by: (1) if a? 2 and c l 1 and d? 1 , thenAdCFO ; (2) if a? 2 and b? 0 and d? 1 , then A,b#O. The derivation of a SReM(n) model for a system containing n components is very similar to the procedure for the SReM(5) model. The Correlation among Parameters in a Multi-Component System In a ternary system, the associations between ternary parameters and its sub-binary parameters resulted from the recurrence relation are shown in Table 2. In a quaternary system, the quaternary parameters relate with its sub-ternary parameters as given in Table 3. Correspondingly, the relation among parameters of 4 ternary systems in a quaternary system is illustrated in Table 4. Similarly, Tables 5 and 6 apply to a quintary system. YI 1 - x, y2 X 1-2 Y, Y3 Y4 l-- x3 1 -- x4 YlY2 YIYZY, Table 2. The relationship of ternary parameters vs. binary parameters. 398 The Model of Super Order Sub-Regular Melts Table 3. The relationship of quaternary parameters vs. ternary parameters. Model Relationship Table 4. The correspondency among ternary parameters in a quaternary system. Table 5. The relationship of quintary parameters vs. quaternary parameters. Quaternary system 1234 y4=0 Relationship AiLf3d' = AabcO d' 1235 y4= 1 Aibf"' = 1Aahcd d=O c' 1245 Y3= 1 AoDed A:::') = C=U c b* 1345 y2= 1 A::?) = YI= 1 A::?) Aahcd b=O a* 2345 = b-1 C Aobcda-1 o=2 399 Guo-ChangJiang, Shao-Bo Zheng and Kuang-Di Xu The known information of a ternary and its three sub-binaries are collected from various experiment studies with different levels of error. However, when the three binaries make up the ternary, the required information must obey the restriction listed in Table 2. If, under the restriction, the parameter fitting procedure cannot obtain a reliable result, it suggests that some parameters contain significant errors as well as computational problems. Therefore, the restriction provides the ability to continuously check the consistency of the parameters, and ensure their reliability. The relationships in Table 4 can be adopted to compare the reliability of parameters of different sub-ternaries if they make up the quaternary system. Table 6 can be similarly utilized. The self-consistency check of the parameters is characteristic of a SReM model, and is a distinguishingaspect from other thermodynamic models. Table 7. The SReM3 model for transferring parameters to SGTE model. The Conversion of Parameters from SReM3 to SGTE Model The simplest way to arrange the parameters in a SGTE model to be self-consistently checked is to convert A parameters of SReM3 model to L parameters of SGTE model. In SReM3 model, if a* = 5 , a*+b* 400 S 6 and A25 * 0 then 15 A parameters (as shown The Model of Super Order Sub-Regular Melts in Table 7) are involved. If the binary parameters AS2), AA'3)and,4:23) displayed by Table 7 all have been available and reliable, then remain 8 Aob in total should be fitted according to the ternary information. In this case, the SReM3 model can be rewritten as follows: .**(2) Cerdenotes the ternary information at the point of (yl,y2). Based on the information of n points, a matrix as Equation (3) can be deduced for the evaluation of parameters A:?'. c" 1, and w are known values. Suppose the known information is not at several points of (yl, y2) but the value of c y o r Gyor G'", then a similar approach can be used. If all binary parameters are available and reliable, only 3 parameters (A32, A33, in Table 5 should be fitted together with Equation (3). The others as Azl, A22, A23, A31 and A41 could be determined according to Table 3. The 15 Aabparameters evaluated in this way is a self-consistent group. Together with Equation (9,these parameters can be converted to L parameters of SGTE model, and make the new SGTE model to be self-consistent. However, the conversion indicates that a SReM model can be also used in phase prediction. The SGTE model  is given by: A42) G" = x,x2L,, + x,x3Ll,+ xZx,L,, + xIx2x3L,,, ...(4.1) Using yI and y2 instead of the independent variable xl, x2 and x3 in accordance with Table I , then comparison with the SReM(3) model yields: 1 42.3 = -32 ...(5.1) 401 Guo-Chang Jiang, Shao-30 Zheng and Kuang-Di Xu = L12.1 1 21 (5 +- 1 1 2 2 +3 ...(5.2) 20 32 1 = - (- ...(5.3) 22 +32 ...(5.4) EL,,= A 2 0 + -21A 3 0 + -31A 4 , + - 41A 5 0 . , .(5.5) ...(5.6) ...(5.7) ...(5.8) L23.2 = ...(5.9) 20 1 7 + L23.1 = - - 1 '232 8 L,, = A,, +A2, - 3A2, --A,, 36 8 C L13= 402 ...(5.10) 22 8 2A24--A25 +-A3, I +-A4, 1 2 8 1 1 1 +-A30 +-A,l+-A,Z 36 2 2 2 - - A1 3 3 +-A40+-A41+-A42 1 1 1 - - A45 0 - - A 5 41 4 3 3 3 32 32 +A,, + A22 + A23 -3A2, ...(5.11) 3 --A25 ...(5.15) The Model of Super Order Sub-Regular Melts The Design of Component Sequence in SReM Model In the SReM model the sequence of components is significant. The first component is chosen according to the concept of similarity. In a ternary system of 123, if the binaries of 12/13 are mostly similar, and the binaries of either 21/23 or 31/32 are less similar, then component “1” should be the first one. For a ferro-metal, C is usually selected as the first component. In the case of a quaternary alloy composed of C, Si, Mn and Fe, two sequences can be adopted. One is C-Fe-Mn-Si, because a lot of reliable information of C-Fe and C-Fe-X are available. An alternative is C-Mn-Fe-Si, if it is the intention to study a high [%Mn] region. When the de-Al and de-Ca processes for SiFe refining were studied, the selected sequence was Si-Al-Ca-Fe. For quaternary MnO-based slag, the sequence used was Mn0-SiO2-AI2O3-Ca0. The Component Activities in C-Fe-Gr Alloy Figure 1 is the phase diagram of C-Fe-Cr ternary alloy at 1873 K  which includes a non-stoichiometric compound, (Fe,Cr)7C3,and the phase line a’ p’ parallel to that of C-Fe binary is of this compound. In fact, so far one don’t know the precise position of point p’ under 1873 K. Using $I’ to denote a point on the line a’ p’, the compositions xFc*and xCr*of every $I’ value are not the same, however the ratio of (xFe*+xcr*)/xc* = (0.710.3). The molten metal at point a is in equilibrium with ( F ~ , C I - )at ~ Cpoint ~ a’. If representing the alloy components [C], [Fe], [Cr] with [MI, then their activities should obey the relationships: ...(6.1) ...(6.2) K , , , ] is the equilibrium constant of the formation reaction of Cr7C3at 1873 K. On Figure 1, a region containing metal, (Fe,Cr),C3 and Cr3C2can be seen. These three phases are in equilibrium at points p and p’ as given by: ...(6.3) ...(6.4) K,,, is the equilibrium constant of the formation reaction of Cr3C2at 1873 K. The non-stoichiometric compound (Fe,Cr),C3is a solid solution at 1873 K, and it is usually simplified as a line because its phase region is very narrow. This supposition allows the application of the following SReM(3) model to describe (Fe,Cr)7C3.The Bob parameters are used in this model to avoid confusion, and by definition: y l = i-xc = 0.7. 403 Guo-Chang Jiang, Shao-Bo Zheng and Kuang-Di Xu ...(7.I ) ...(7.2) ...(7.3) The combination of Equations (6) and (7) describes the equilibrium between molten alloy and carbides. The equilibrium of melted and solid alloys on the curve covering the Cr region can be described using the approach suggested by Chou . These provide the complete information of the C-Fe-Cr system. On this basis, the component activities, etc., in the entire homogeneous phase region can be evaluated precisely, if (xFe*)p"and (xcr*)p. can be determined by experiment. CHROMIUM, WCICHT PLRCCNT Figure 1. Eeguilibrium between the liquid phase and non-stoichiometric compound. 404 The Model of Super Order Sub-Regular Melts The Simplified SReM(9-) Model to Describe a Multi-System Containing many Dilute Components Consider an electrical Si steel as an example in order to illustrate the simplified model. It is widely known that the component activities in a low-alloy steel are usually calculated by means of the interaction coefficient method developed by Wagner and Lupis. However, the electrical Si steel does not belong to the category of low-alloy steel if [Si] = 3.0% or 0.058 mole. The interaction coefficient approach can be correct only if E,' = E; , so U ,= & is required by the criterion. Table 8 shows that electrical UP Y," Si steel does not obey the criterion, thus a simplified SReM(9) model was developed to describe it, which includes the 9 elements: Si, Al, 0, S, Mn,C, N, P and Fe. The composition coordinates in the model are defined in Table 9. Table 8. The ratio of activity coeflcients for various PASiI. Composition of alloys y: 0.005C - 0.001Si rnol 0.57 0.0013 0.599 0.00138 0.57 0.0013 1.042 0.00285 0.00SC- 0.058Si mol y:, Yr Ys Yr - YS Y!, Y: 1.05 1.062 1.83 2.19 The SReM(9jmodel can be easily obtained by a similar approach to the SReM(5)model according to the recurrence relation. However for the electrical Si steel, the components are all dilute, except for Fe and Si. Therefore, the SReM(9) model can be simplified by taking a* = 4 and b* = c* = d* = e* =f = g* = h* = I . Furthermore, when a 2 2, if one of b, c, d, e,f; g, h is zero then AobcdLfg,, = Oo Therefore, the simplified SReM(93 model contains 24 parameters in total, which are assumed to have the following relationship : The modified interaction coefficients resulted from Equation (9), which was the approach of Pelton , are adopted as the known information of the region closest to Si-Fe binary in the evaluation of other parameters apart from AZmO, A x , ~ w ~and A~oooooo. 405 Guo-ChangJiang Shao-Bo Zheng and Kuang-Di Xu ...(9) The parameters of the simplified SReM(9-) model for electrical Si steel were evaluated and are given in Table 10. Then the corresponding excess free energy can be calculated according to the set of Aubcdefgh. Table 10. The parametersfor electrical Si steel. Concluding Remarks on the Characteristics of the SReM Model The SReM model describes super order sub-regular melts, and is based on diverse information usually obtained from experiments. The SReM model provides integrated knowledge. In this model, the truncated polynomes are used and only one set of A ~ ~ parameters ~ ~ , . . are involved. These parameters represent the excess free energy of the associated clusters. Therefore, the SReM model is a theoretical model assisting the research studies aimed at linking micro structure and macro properties. Differing from the empirical model, the A parameters in the SReM model are divided into several levels. The parameters of a sub-system can be universally used in various multi-systems involving the sub-system. For example, in all Fe-based alloys containing carbon the parameter of A F - ~is~unique. The SReM model unifies the activity calculation of dilute melts and concentrated melts. If k"' denotes the n order Wagner-Lupis interaction coefficient, then Auhc corresponds to ,$")/RT when a* + b* + C* = n. Using this approach, the interaction coefficients for the Mn, Si, Cr, etc., based alloys are available. 406 The Model of Super Order SubRegular Melts The SReM model aims at a precise thermodynamic description, and reliability is achieved as follows. It is claimed to utilize accurate data including information from phase diagrams for the parameter fitting procedure. In a poly-system, information of either the saturated or the compound separated curves and surfaces are frequently more reliable. In the SReM model, it is guaranteed that the more precise information has more effect on the calculated results in two ways. First, the parameter fitting procedure always starts from binaries, then ternaries, and so on. Since, usually more binary information is available and they are more precise. Second, the correct component sequence is determined, for example, in the C-Fe-Cr ternary alloy. The ucof C-Fe binary is most reliable, therefore select C as the first component and then the precise values of Ado can be obtained, which have significant effect on the model results. During parameter fitting in the SReM model, the errors arising from the use of truncated polynomes are fixed in the evaluated parameters. As the error of ,4F+Fe used for the homogeneous quaternary regions of C-Cr-Fe-Ni and C-Cr-Fe-P, could not increase in the SReM4 model. Finally, it is claimed to constrain the evaluated parameters by the restrictions shown in Tables 2 , 4 , 6 . A simplified SReM model was developed, it aims at describing a multi-system comprising more than 2 concentrated components and other dilute components. In this simplified model, the modified interaction coefficient approach suggested by Pelton is considered appropriate in offering the information of the area closing to a Fe-X binary. By converting to the SGTE model, the SReM model can also be used for prediction of phase diagrams. Acknowledgments Dr. X.B. Zhang, and Dr. K. Tang made significant contributions to the earlier development of the SReM model. This research work had financial support from the National Natural Sciences Foundation of China under Grant No. 598740 16, 59832080. References 1. 2. 3. 4. 5. 6. 7. Pelton, A.D., and Flengas, S.N. 1969. An analytical method of calculating thermodynamic data in ternary systems, Can. J. Chem., 47,2283-2292. Wang, Z.C. 1986. An analytical method of calculating thermodynamic properties in ternary systems. Science in China (Serial A), 16(8), 862-873. Zhang, X.B., Jiang, GC., Tang, K., et al. 1997. A sub-regular solution model used to predict the component activities of quaternary systems, CALPHAD, 21(3), 301-309. Jiang, GC., and Xu, K.D. 1998. 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