A sub-regular solution model used to predict the component activities of quaternary systems

Calphad Vol. 21, No. 3, pp. 301-309, 1997 Q 1998 Published by Elsevier Science Ltd

Pergamon

Printed in Great Britain. All rights reserved 0384-5916198 $19.00 + 0.00

PII SO364-5916(97) A SUB-REGULAR

00031-X

SOLUTION MODEL USED TO PREDICT THE COMPONENT QUATEBNARY

ACTIVITIES

OF

SYSTEMS

Zhang Xiao Bing, Jiang Guo Chang, Tang Kai, Xu Jian Lun, Ding Wei Zhong, Xu Kuang Di Shanghai Enhanced Lab of Ferrometallurgy,

Shanghai University

Shanghai 200072, P.R.China

ABSTRACT A sub-regular solution model used to predict the component activities in a homogeneous quatemary system was developed in Shanghai Enhanced Lab of Ferrometallurgy. SELF-SReM4.

The component

activities were described with the polynomials

in which a group of Aju parameters conditions.

involves.

It was evaluated

region of a

It was designated as

of the content variables,

along with the reliable boundary

The deduction of the model and the procedure of the evaluation of A,

parameters were

introduced in detail in this paper.

In 1992, a sub-regular solution model used to predict the component activities in part of the liquid region of ternary systems, as C-Fe-X (X = Mn, Si, Cr, Ni) was published by the authors( 1). Hereafter, the model is designated as SELF-SReM3.

SELF refers Shanghai Enhanced Lab of Ferrometallurgy.

Notwithstanding,

it is more significant

activities of a quatemary production

of MnFe

to acquire a model that could be employed

to predict the component

system. The incentives arise from the following considerations. and MnSi,

the component

activities

of C-Fe-Si-Mn

and

In order to improve the

CaO-MnO-Al,O,-SiO,

are

indispensable. For efficiently running the smelting reduction process of the master alloy of stainless steel, it is vital to get the component activities of C-Fe-Cr-Ni. Those of C-Fe-Cr-P and C-Fe-Mn-P are much important data that is needed to guide the dephosphorization technology of Fe-Cr, Fe-Mn alloys. So it was decided to develop SELFSReM4 through expanding and modifying SELF-SReM3.

SELF-SREM4 _____~ In homogeneous region of a quatemary system, Y, Z, U was denoted as the content variables. This procedure was used for easily running. Y=l Z=l

-x, -(X*/Y)

U=l

-(X,NZ)

Original version received

(1) (2) (3) on 18 September

1996, Revised version on 26 August 301

1%~

302

Z.X. BING eta/.

X denotes the molar fraction of the components. system were shown as TABLE 1.

The Values of Y, Z, U on the boundaries

of the quatemary

Tl element 1 2 3 4

Y

Z

U

0 1 0 1 1 111

0 0

sub-binary 12 13 14 23 24 34

Y

Z 0 1 1

1 1 1

U 0 1 0 1

sub-temaxy 123 124 134 234

Y

Z

U 0 1

1 1

1

Take G,‘” to denote the excess partial molar free energy of component

1, and G’” to denote the excess integral

molar free energy of mixing of the quatemary system. According to the high order sub-regular solution model, G,‘” and G’“could be expressed as polynomials of the composition(2,3). j’

it’

I

G,'"=~~~

Ajk,Y'Z*Uf= G,ex(Y,Z,U) 0 0 0 j’ k’ I G’“= ccc $Pjk,Y’zs’= G’“(Y,Z,U) 0 0 0

(4)

(5)

In these polynomials, the (A,,) and (ek,) are the parameters to be evaluated. And thej’, the model. From Gibbs-Duhem Eg.,

k’, I’ are the order of

G,-= G’” -[Y(acc”/aY)]

(6)

therefore, qkf=

Ajk,

/(l-j)

(7)

(Alk,) must be zero, because (qk,) cannot be infinite even if j = 1. Namely, A,U = 0. On the other hand, when the system degenerates to pure component 1, the following regularity G,“(O,Z,U) = W(O,Z,U) = 0

(8)

should emerge. So, A,,, = P,,~,= 0. If Z = 0, as TABLE 1 it is the sub-binary system of 12. The description

of Gler(Y,O,U) and GeX(Y,O,U) by means

of (A,,,) and (pj,,) is due equivalent to that based on (cz~),~,and ( #jj)12, that is the parameters binary itself. Thus from Eg. (4) and (5),

G,‘“(Y,O,U)=~&4,,,YW’=~(n,),,Y’ 0 0 0 j’ I’ G’“(Y,O,U) = ccQjo,YiT’=h( 0 0 0

belonging

to 12 sub-

(9)

l#j),,Y’

The following conditions are due necessary to make these two polynomials zeroifj22and 121;Thesecondis(~,,,,,)=O,ifj~l,Z>l.

(10) to be correct. The first is (Ai,,) should be

COMPONENT According to Gibbs-Duhem

ACTIVITIES

OF QUATERNARY

303

SYSTEMS

Eg., and Eliminating the parameters which have been proved to be zero, then get

G,‘“=~~~,~,z~‘{(l-k)+(k-l)/z}+~~~Aj~,Y’z*V’{l+~-k)/[~(~-j)]+(k-~/[~z(~~)]} z 0 0 0 0

(13)

G,“=~~y,,,,z’U’{(l-k)+(k-l),z+NZUj+~~~Aj~,Y’z~’{~++(i-k)/[Y(~-j)]

2 0 0 ~(k-“~I,Yz(l-j)]+NIYzU/(l-j)l)

,,z~l+~~~Aj~~z~‘,(i~~

G”=Aip 0 0 Besides,

(14) (15)

z 0 0

some other parameters

degenerate simultaneously

,“,“,(GYY)=(

are also should be zero. When the three sub-binary

to pure component

systems

of 12, 13, 14

1, the following limit(4) should be achieved.

1-Z)G~(O,O,O)+Z( 1-U)G,‘“(O, 1,O)+ZUG,‘“(O, 1,l)

(16)

Combining with Eqs of (11) to (16), get

(17)

Thus, P,~, should be zero if either k >_2 or 12 2. 2 , GzCx( l,O,O) = 0 resulting from Eq. (12), so

For pure component j’

PlCtl=-

(18)

CAjOOl(l_j) 2

By use of the similar procedures

for pure component

3 and 4, get G,‘“( 1,l ,O) = 0 as well as GF( 1, 1,l) = 0, thus from

Eqs. (13) and (14) get I’ k' Pm+ VII0 =-CCA&l-j)

(19)

2 0

Finally, the SELF-SReM4 I’

k’

1

G,‘“=~~~ 2

can be shown as follows,

”

A,,YjzS

(21)

0

(22) z i’

2

0

0

k’

G~=-~~A,,+~~~A,,YiZ~‘{l+(l-k)l[Y(l~)]+(k-~/[YZ(l~)]} 2 0 ” z 0

(23)

304

2.X. BING eta/. GI’“=-~~~Ai~,I(l-j)+~~~A,~,Y’Z~‘(l+(l-~)/[Y(l~)]+(~-~/[YZ(l~)]+//[Yzu(l~)]} 2 0 0 2 0 0 j’ k’ I’ i’ k’ I j’ k’ I G’“=- CAi,,YI(l-J’) -~~A,,YZI(1_j)-~,~,~,Aj~YZU/(1-j)+~~~Ai~,Y’Z~’/(1-j) 2 z 1 2 I 1 2 0 0

It is clear, that only one group of (Ajk,)parameters is needed in SELF-SReM4.0 and G’” for a homogeneous

(24) (25)

to predict G,‘” .

GF ,

GqeX. G:

region of a quatemary system.

1. To evaluate (A,,) parameters based on the sub-binaries The G,‘” and G’” are able to be expressed as the polynomials six relations between a parameters

of the sub-binaries

of a parameters in the sub-binaries,

and (A,,,) parameters

some of (Aik,) can be evaluated from the known properties of the sub-binaries,

of the quatemary. as TABLE 2.

(a,) u=A,,o (a,)

as the following

On the other word,

(26)

f,=tAjko

(27)

(a,),,=~A,,(l-k)l(l-j) (6=&t, (I-WU_j)

(29)

2

2

0

i’

AZ’

(30)

(a,)34=CCA;k,(l-k)/(l-j) 2

(31)

I

It may be quite clear, that to evaluate the entire Ajk, parameters properties

are possible

model. If the quatemary

system behaves like a sub-regular

larger than 2. In this case, the entire A,, parameters based on the sub-binary thermodynamic

thermodynamic

of the regular solution

solution, then it was claimed thatj’, k’, 1’ all should be

of the quatemary

system was not possible to be evaluated only

properties.

2. To evaluated the (A,,,) parameters based on sub-temaries In the sub-temaries, the G,‘” and G’” are also expressed between

just along the sub-binary

only if thej’, k’, I’ all are equal to 2. This is the right characteristics

/3 parameters of the sub-temaries

as the polynomials

of p parameters.

Four relations

and A,k,parameters of the quatemary were obtained. Some other A, could

be evaluated from the boundary conditions polynomials of b parameters in TABLE 3.

of the sub-temaries.

Glex of sub-temaries

could be expressed

(P,kL =40

(32)

(Pi&=~AIW

(33)

=t‘+,, 0

(Pi1h34 (pl,)*34=$Ajk,

2

(34) (35)

as

COMPONENT

ACTIVITIES

TABL ,E 2. Evaluation of LZ pararr j-binary

known properties

SYSTEMS

305

.ers through the properties of sub-binary systems the equations used for the evaluation of a parameters

12 13 14

at least one of (G,‘“),,

23 24

at least one of (G,‘“),,, i = 3 or 4 either scattered information or systematic information could be used.

i=2,3or4

either scattered information or systematic information could be used.

34

at least one of (G,‘“),,, (G,‘“),,, (G’“),,, either scattered information or systematic information could be used.

(G,“)z,=&at),i z

{Z”+kZ’l[Z(l-k)]}

(G”)S,=&),i

(Z’-Z)l(l-k)

W),,=~(a,),, U’ (G&=&a,),, 2

{U’+KJ’l[U(l-r)]-1/(1-o}

(G,“),,=&,)j,

(u’-U)/(l-l)

TABLE 3. Evaluation of /? parameters ;ub-binary

known properties i = 3, 4

123, 124

(Gle&,

134

(G,‘“),,,

through the properties of sub-ternary systems

the equations used for of the evaluation of p paramwrers i’ k’ (G,‘“),,i=CC(Ba)I*iY’Z

2 0

234

Nevertheless,

OF QUATERNARY

i’ I

(G,‘“),,,=CC(a,>,,,yiU’ * 0

(Gr%

(Gz%=

the p parameters

i&P,),,.,

Z’U’

can also be evaluated from different known properties belonging

and or the ternary system itself. TABLE 4 indicates some of the examples. As 123 sub-ternary,

to sub-binaries

if some scattered or

systematic information of ( cz i )12,(a j )13,(a t )23as well as (G,‘“),,, are known, the (/?jk),23 parameter could be evaluated by means of the equation shown in TABLE 4. The others could be calculated from eq. (36) to eq.(38) When k’ 2 k 2 2, (p&

= -*+2X&

3

(36)

Whenj’>j?2, (Pi,),*) = (a,),, -(a,),, (P/O)123= (a, )I2

- ~(&)I,, 2

(37) (38)

306

Z.X. BING eta/. TABLE 4. The evaluation of known properties (a,),,, (a jh3. (aA. (G,‘“),,,

123 ~_.~_p to ~_different ____ ___..._-__ known ..-- nrnnertien ~__I-__---~~~ _ in ~~~ ~~~sub-temarv _.~ ~~~~~ .._, according ~~~~~ ,B narameters I

the equations used for the evaluation of p parameters P,~),*~[r

kt(

+Y’l(~-j)~(z*-Z)=G,“),,,-o~ a,J12Y’2

3 2 i’ Z=y( a,),,Y’+Y2$(

a~)2,-(Z~-Z)l(l-k)

(a,),,, (a,),,, (GICX),,, (a,),>, (akL.

i’ &( z $(

p,,t,2,YfZ+~=&

(a,),,,

$(

p,,),,,Y’(m)+f~(

p,,,,,,~(Z’j)=(G;.,,,,cl-Z)~( 2

z

3 2

(G,“),,,

2

(ak12,, G’“),,,

j’

o,),lY’ - Zk( 2

p,,),,,[y’+Y2/(l_j)lZ*=(G,“),,,-~( qm+& 2

3 2

If j’, k’ all are 4, the entirety of (/3,&,

2

arMW-4

P,3,,,~y’+~*~~~~~l~~*-~~=

(G,CX),2,-~( a,),,Y’+Y’ t( a,)&*-l)/(lk) 2 z i’ c( p,,),z,YZ+$i( B,~),*,rz’=(G,“),,,-~( 2 * 3 2

(a,),,. (G,‘“),,,

a,),,Y’

namely the A,,

a,),*Y

is 15. If only (G,‘“),,, was known, then the total 15 p

parameters have to be evaluated directly from the first Eq. in TABLE 3. However, it should be noticed that the more the parameters

evaluated by means of one Eq., the more the difficulty in the evaluation procedure,

accuracy of the evaluation

results. On the other hand, TABLE 4 provides more known properties

the binaries and that of ternary itself so fewer parameters ternary could be treated with the similar procedures

and the less the including that of

need to he evaluated from one equation. The other sub-

as that of ternary 123.

3. To evaluated the A, parameters based on the information If the sub-ternary

parameters,

as (Pi&

(PJrz4,

of quatemary itself or some sub-binary (P,Jrj4, (P&,

parameters,

as (a k )24,

(a ,)34r have been evaluated, together with the G,‘” of several points (Y,Z,U) belonging to the homogeneous of the quatemary

were known, some A,

parameters

could be obtained base on the equations in TABLE

region

5. The rest

could be calculated by use of equations that were shown in TABLE 6. If j’, k’, I’ all are 4, then every ternary gets 15 p parameters, and the sum of (A,,[) parameters is 63. Generally, an A, may appear in these h parameters and in the others simultaneously. If no information of the quaternary itself had been introduced, Besides,

these 63 A,, parameters could not be separated one by one.

there are the four relations

between

these

p parameters

of different

sub-temaries

TABLE 7. So, these four equations could he used to check the reliability of the boundary conditions for selecting the boundary conditions.

as exhibited

in

and as the tool

1. The fundamental idea of SELF-SReM4 For the solution of component activities of a polynary system, the first thing is to acquire the activity of the “first component” throughout the investigated homogeneous region. According to SELF-SReM4, for a ferroalloy, C was

COMPONENT selected as the first component

ACTIVITIES

OF QUATERNARY

due to that many of the accurate thermodynamic

ferroalloys had been proposed based on careful examinations

TABLE 5. The evaluation of (A,)

(P,3124, (P,,)1341

(PA34

parameters

according to different

I’ k’ I ~~~Aa,

[Y’+Y’I(l-j)](Z’-Z)(U’-U)=G,“-kk(

G,=,

(Pj11134r

cp,k,,,,,

Cak)24

G,‘“, (P,L

known properties

P&Y’Z’

- fi[(p,,),,,-

P,I)IzI~~* - $$[(&h.-

I'

k'

$$Ak,Y*(z’-z)(IJ-~)+$$$~~~, [U’+Y’I(1_j)lZ’(U’-U)=G,.‘-~~( P,A,Y’Z’I4

2

(P,k),,,> (4h4

2 0

i’ I (Pjk)1231rzw _CCC P,i)134‘rz(u’-u) * 2 j' k' , i’ I

I

CC< G,‘“, (P,Jm.

in

2 I 3 2 z (p,,),,,Iy’zTJ - 22 ( p,,h,,Y’z(v’-u) + ik ( Pk,L4Y2[(z’-m(~-mw’-u) 2 2 2 2

G,‘“, (P,~,,~,, ~~IA,,,Y’(Z’-Z)(U’-U)=G,“-~~( (P,h,,,> (P,,L

properties of C dissolving

carried out by some investigators.

the equations used for the evaluation of (4k,) parameters

known properties G,‘”> (P,dm

307

SYSTEMS

J’

k'

2

2

I

$+kjy2(zk-z)(u'-u) j'

k'

ak)24y2 ll(Zk-ZY(l-k)lU

Pjh)lll~(z”-z)u+~(

pj,)l)dYjZU’+CC(

2

I’ +

I’

I

y,F,y, A,,, [Y'Z' +Y2U( l-j)](U’-U) = 3

j'

12 k'

G,‘=xX( p,&,Y’Z’ - ~~[(p,*),~~-(p,,),~Ipz~+~( a,)u=W”-U)/(I-~ 2 2 I * 0 J’ k’ I A Y’Z*(U’-U) = G,“-kt( p&YjZ’ - it[ (pp),,,-(p,,),,,]y’Z”U FFF jkf z I 2 0

(P&,

TABLE 6. The evaluation of the rest (A,,) parameters conditions

the equations for evaluation of the rest (A,,..) parameters

k’>k>2, I’ll?2

A&&++$+&

j’ rjr2, 1’2122

A,,, = (p,/)w

j’ 2j > 2, k’>k>2

A,,= (P,3,~I-(P,3,2,-~AI(/

-2

Aj,, 2

Z.X. BING et a/.

308

TABLE 7. The relations between conditions

the relations between the p parameters (PjO),J,=

j’lj12,

In SELF-SReM4, throughout evaluation

Pjk>l*3

of GleX and G’” versus the composition

the liquid region and on its boundaries. procedure

k( "

the polynomials

evaluated under the restriction

p parameters of different sub-temaric

The (A,)

of the boundary conditions

had been completed,

parameters

(properties)

the component

of the system

were established

included in the polynomials

should be

which have to be precise and enough. If the

activity prediction

by means of this model is a quite

simple step. 2. The order of SELF-SReM4 of GleX and G’” versus the composition

The polynomials considered

as real equations describing

degree was mainly determined SELF-SReM4

established

in SELF-SReM4

model should not be

the system, but a kind of approaches quite closing to the reality. The closing

by the selected order of the polynomials.

One of the reasons illustrating the merit of

superior than regular solution model is it owns a higher order.

3. The boundary conditions (properties) According to either regular solution model, or the geometric model that was recently suggested component

activities

binaries. For C-Fe-Si, region. Certainly,

of a ternary system were determined

it does not behave as regular solution and its sub-binary

this system is difficult

boundary conditions

only by the thermodynamic

of the sub-binary

itself was utilized simultaneously

properties

by Chou(S), the of the three sub-

C-Si owns a very narrow liquid

to predict with regular solution model. On the contrary,

but also the some information

in SELF-SReM4

not only the

that belongs to the ternary and the polynary

to evaluate (,4jU) parameters.

In mathematics,

this is not a simple

numerical insert procedure, but a fitting procedure under mutual restriction resulting from multiple aspects. This is the another reason to elucidate the good result of SELF-SReM4. In SELF-SReM4 model, generally the used boundary properties are scattered information, can systematically predict the component activities in a homogeneous region.

however,

the model

4. The sources of the possible errors In principal, the error of using SELF-SReM4 to predict component activities may result from two sources. the closing degree of the model to the reality. Second is the accuracy of the evaluation of (A,) parameters. this accuracy was farther determined by two factors. One is the boundary conditions used, whether they are and reliable or not. The another is the chose order. As just mentioned, the higher the order, the better the degree, if there is no difficulty in mathematics treatment. Nevertheless, the higher the order, the more parameters had to be evaluated. So, the higher the order, the more difficulty the precise evaluation of parameters.

First is In fact, enough closing the AjU the A,

COMPONENT

ACTIVITIES

OF QUATERNARY

SYSTEMS

It was found that the result for [C] could be better but those for other components order was chose. As mentioned,

(A,,) parameters

309

were even worse, if too high an

are evaluated in several steps and get various effects on G,‘“. Too

high an order results easily in that the restrictions

coming from different aspects to be less balanced. Therefore,

it is

a must to select an optimum order for a given system. 5. The flexibility of SELF-SReM4 1). SELF-SReM4

could be used successfully

not only for alloys but also for slags.

2). It does not necessary to use all of the boundary conditions. the application of SELF-SReM4. 3). Inserting the conditions

shown in TABLE 1, the SELF-SReM4

or sub-binaries.

Thus, if the information

the sub-system.

Furthermore,

turn to research S-Fe-X,-X,.

Some of them in shortage cannot make a barrier for

This just is its flexibility. of its sub-system

if the thermodynamic Measuring

would be degenerated

is not available, SELF-SReM4

properties of a Fe-X,-X,

the activity of S in the quatemary,

to either its sub-temaries could be used to predict

are not easy to measure, then it is better then it is able to.use SELF-SReM4

for

predicting Fe-X,-X,.

CONCLUSION 1). SELF-SReM4

is a high order sub-regular

Lab of Ferrometallurgy.

The component

slag could be successfully

predicted with SELF-SReM4

paper elucidates the SELF-SReM4 2). According simultaneously

region of a quaternary

system either alloy or

through the evaluation of one group of A,,, parameters.

The

and illustrates the procedure evaluating the Ajti parameters.

to the known boundary

only some approaches

solution model developed in SELF referring the Shanghai Enhanced

activities in a homogeneous

were elucidated

conditions, as examples.

there are different approaches to evaluate A,,. In this paper The more the conditions, the less the Ajk, should be evaluated

from one equation, and the better the accuracy of the results.

3). In general, an A,, may be involved in different a parameters and b parameters

simultaneously.

So, this situation

could be used to check the results of evaluation, or to check the reliability of some boundary conditions. 4). In SELF-SReM4, All the used information

the standard of the used boundary conditions

should be taken as the relevant pure material.

should be of the same temperature.

5). The A,, parameters were evaluated step by step. At first, Ajm might be evaluated from binary 12 . Secondly A,M might be evaluated from ternary 123. Lastly the remained A, should be evaluated based on the other information. On the other hand, if A,,JA,, = lo-“‘, the contribution of two terms of (A,,Y’) and (A,, ZkU’) in the polynomials may get a magnitude difference of (k+Z-m). Therefore, if the boundary condition concerning with a component is more and quite reliable, in principal this component

should be put ahead in the series.

REFERENCES 1. Jiang Guo Chang, Zhang Xiao Bing, Xu Kuang Di, Acta Metallurgical Sinica (Eng. ed.) 5B( 1992),476--482 2. A.D.Pelton, S.M.Flengas, Canadian J. Chemistry, 47(1969),2283--2292 3. Wang Zhi Chang,, Chinese Sciences (Chinese ed.), 8A(1986), 862--873. 4.L.S.Darken, J. Am. Chem. SoC.72(1950),2909--2915. 5. Kuo-Chih Chou, CALPHAD X X III, Madison, Wisconsin, USA. ,( 1994).