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The Model of Super Order Sub-Regular Melts.

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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 (gcjiang@online.sh.cn).
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 [6] 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 [7] 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 [8].
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 [9], 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. A review on thermodynamic solution model, The 8th Japan-China
Symposium on Science & Technology oflron & Steel, ISIJ, Japan, 182-1 86.
Zhang, X.B., Jiang, GC., and Xu, K.D. 2000. Evaluation of component activity in molten
MnO-Si02-AI203-CaO system with model SELF-SReM4, J. Imn Steel Res. Inf.,7( l), 6-8.
Miettinen, J. 1998. Approximate thermodynamic solution phase data for steels, CALPHAD, 22(2),
275-300.
Grifing, N.R., Forgeng, W.D., and Healy. GW. 1962. C-Cr-Fe liquidus surface, Trans. Metal. SOC.
AIME, 224(2), 148-159.
40 7
Guo-Chang Jiang, Shao-Bo Zheng and Kuang-Di Xu
8.
Zhou, GZ. 1978. The activity on the boundary of two-phase field in ternary systems. Science in China,
8(3), 312-324.
9.
408
Pelton, A.D., Bale, M.R. 1986. A modified interaction parameter formalism for non-dilute solutions,
M e i d . Trans., 17A,1211-1215.
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