Calculation of α + γ phase boundaries in Fe-C-X systems from the central atoms model

cqLPH.+JD Vo1.9, No.1, pp. 43-58, 1985 Printed in the USA.

0364-5916/85 $3.00 t .OO (cl 1985 Pergamon Press Ltd.

CALCULATION OF Q+Y PHASE BOUNDARIES IN Fe-C-X SYSTEMS FROM THE CENTRAL ATOMS MODEL

M. Enomoto*, ** and H.I. Aaronson* *Department of Metallurgical Engineering and Materials Science,

Carnegie-Mellon University. Pittsburgh, PA 15213. U.S.A. **Returned to Tsukuba Laboratories, National Research Institute for Metals, 1-2-1 Sengen, Sakura-Mura, Niihari-Gun, lbaraki 305, Japan

ABSTRACT

The Central Atoms Model generalized to multicomponent solutions is placed in a form which permits direct application to the evaluation of a-7 phase equilibria in Fe-C-X alloys. From the activity expressions thus obtained the ortho- and paraequilibrium phase boundaries are calculated for five of these alloys, where X is successively Mn, Ni, Co, Si and MO. The results are generally in good agreement with the calculations made from the Hillert-Staffanson regular solution model and the available experimental data on orthoequilibrium y/(a+y) phase boundaries. The model is then used to calculate the free energy change associated with the proeutectoid ferrite reaction in the same ternary systems.

Introduction

The activity of carbon in the ferrite and especially in the austenite phase of Fe-C alloys has been repeatedly investigated on the basis of various statistical models (t-6). In some instances, the results have been used to determine the activity of iron, and then to calculate various thermodynamic quantities relevant to the proeutectoid ferrite reaction, particularly at temperatures below that of the eutectoid, where the austenite + ferrite region exists only in metastable equilibrium form. Counterpart studies in Fe-C-X alloys have been attempted only through relatively simple models (7,8) because of the mathematical complexities involved. More rigorous statistical models of Fe-C-X alloys, e.g., the quasi-chemical model (9) and the central atoms model (lO,lll, have so far been utilized only to consider the implications of the composition- and temperature-dependence of the activity of carbon in austenite.

The Central Atoms Model, developed almost simultaneously by Lupis and Elliott (10) and by Mathieu, Durand and Bonnier (11). has several important advantages over previous solution models applied to this problem. These include: instead of assuming random mixing, the most probable atomic configuration of the system is sought under the condition that the free energy of the system be a minimum; in seeking the minimum, the overall configuration of atoms in the nearest neighboring shell is considered. By doing so, the difference in the energy of a central atom with the variation of configuration at a constant total number of solute atoms is indirectly taken into account, which apparently was not accomplished by the quasi-chemical model {9); also, unlike most previous models the vibrational contribution to the entropy is included.

Foo and Lupis (12) generalized the Central Atoms Model (CAM) to multicomponent solutions. Substitutional and interstitial solutes are treated in a similar manner, but each is taken to occupy a different sublattice. However, only the activity function for carbon was developed into a form ready for immediate use. In the present paper, the Foo-Lupis generalization wili be briefly summarized, and in the process of so doing equivalent useful relationships will be obtained for the activities of iron and a substitutional solute, X. These three relationships will then be applied to calculation of austenite + ferrite (a + 7) phase equilibria and of free energy changes attending the proeutectoid ferrite reaction in Fe-C-X alloys. Full use will be made in these applications of data on carbon activity in Fe-C-X alloys which has recently become available (12-15) as well as the thermodynamic functions of binary Fe-X systems being compiled by Kaufman and Nesor (16.17). The numerical results thus secured will be compared with those ___-_______-________~~-~~~~~~~~~~~-~~~~~~~~~-~~--~-~~~ Received 28 May 1984

43

obtained from other analyses, but particularly that of the widely used Hillert-Steffanson (18) (H5) regular solution treatment, for which a number of similar applications to Fe-C-X systems has been reported (8).

Derivation of CAM Activity Functions in Fe-C-X Systems

Foo and Lupis (72) wrote the activities, a, of the solvent atom, 1, the substitutional solute, 2, and the interstitial solute, C, as:

In a, = In Y, + r In Yv - In P, - r In PV - py/RT t l-Al

In a2 =InY2+r?nYy+Zh2+2n~2-lnP2-rInP -&Rf Y I WI

In aC a ‘In YC + zrhC + z’h’o - In PC + In P - $RT ” Il-Cl

A brief description and explicit formulae or tables will now be given for each of the quantities in these equations. ratio in the

pi” (where i = 1, 2 or C) is the chemical potential of the i’th pure component. Yi is the atomic sublattice to which the 7th species of atoms belongs. Thus i = 1 and 2 appear in the

substitutional subtattice and i = C and v (whsre v = vacancies) in the interstitial sublattice. The Yi are related to the mole fractions, xi as:

x1 Y, q F

x2

C Y, I l_x

C

Xt YC = -

r( 1 -xc! *

r-t 1 +r)xC

‘v 3 ril-xc1

123

where r is the ratio of the number of interstitial sites to that of substitutional sites. Taking yr as the ratio of the number of i atoms to the number of solvent atoms, 1 or v, in each sublattice:

1” x2

Y, = y2 = 7

133 xC

y, = r - (7 + rIxCJ Y, = 1

Z and L are respectively the nearest-neighbor coordination numbers of substitutional and interstitial sites to a substitutional site. 2’ and z’ are the corresponding coordination numbers to interstitial sites, Their values for fee and bee lattices are shown in Table I.

Table 1. Coordination Numbers

II (bCC) 8 6 2 4 3

y (fee) 12 6 6 I.2 1

A. and A’. are the Lagrangian multipliers of the i’th species. Yheses terms were introduced in the course ‘of calculating the most probable configuration of solute atoms by minimizing the totat free energy of the system through maximizing the grand partition function while maintaining a mass balance for each species. For mathematical convenience, Ai and Arr are expressed as:

7 = exp (Ai - d$,.“‘i 14-A] i

@; = exp Arr t4-81

where a@ ,,‘I) is the difference in the energy state of the Fe central atom when one atom of i’th component Iis present in the nearest neighbor shell and when all the sites in the same shell are occupied by solvent atoms, i.e., Fe atoms in the substitutional lattice.

PHASE BOIJNOARIES IN Fe-C-X SYSTEMS 45

Explicit relationships for wi and w,’ can be obtained by solving the mass balance equation, They are given for Fe-C-X alloys as Eq. 1451 of Ref. (12). For present purposes, however, since the solute concentrations rarely exceed 0.15 at temperatures of usual interest, the use of the approximate expanded form of that equation, retaining terms up to second order in y,, will suffice:

w,orw’= l+ay +b 2 2 2 + acyc 22y2 + b2Cy2yC + %cyE

The coefficients of the y terms, which appear in Eq. [513 this Table, X,, is the interaction coefficient between solutes

.1

A = 22 1 - exp (-Jf(‘) ‘2 + a+, “)I 2

x 2c

= 1 - exp (-b+“’ + a+, “‘j ‘C C

x = cc 1

- exp (-B)(‘)) ‘C

Table 2. Coefficients of

a 2

a C

b 22

b 2c

O2 x 22 0 2x22-1 0

wC x

2c 0 x2c(x2c-1) xic

w; 0 x 2c 0 G

co’ 0 x c cc

0 0

of Ref. (12). are summarized in Table 2. In i and j:

[61

the w Function

b cc

0

0

x2c(x2c-1)

xcc (2Xcc-1)

All other X,, are equal to zero. 6):) and dp’,c’ are the counterparts of a), (‘I for the central atoms I I I

of substitutional solute and carbon, respectively. It has been shown (12) that the reciprocity relationship X = X. holds. In this derivation a linear dependence of the energy of the central atom upon the number o? solbte atoms in the nearest neighboring shell is assumed. Other dependencies have also been employed (19-21).

Returning to Eq. [ 11, P is the normalization factor for the probability of distribution of atoms, i.e., the summation of the probabilities of the most probable configurations in the nearest neighboring shell with respect to j and k, where j and k are the numbers of substitutional and of interstitial atoms in the shell surrounding the central atom i. They are written as:

PI = Y:Y: exd-frb)(l + y2~2)z(1 + ycwcY

p2 q y:y: exd-+fb) I1 + Y,w,(~-~,,)~~~I + ycwc(i-xzc)~~

pC = Y:‘Y:’ exp(-+::I {l+y2W;(l-x2c)~z’ { l+Yc+-xccH”

r71

P ” = Y?‘Y$ (1 + Y2w;)z’ (1 + ycw$

where p (I) is the potential of the i’th atom whet it is entirely surrounded by solvent atoms and is equal to p,/RT”‘& definition.

From Eqs. [41 and 171, the activities (Eq. [ 11) become:

In a, = lnY,-S-A [8-A]

46 M. ENOMOTO and H.I. AARONSON

ln a2 = ln Y, + 22 In m2 + 2z In W; - S - A + In yy

ln aC = lnyc + 22’ln oc + 22 tn urJ + lnirrp)

where:

s = (2 + rZ’)ln Y, + {(z + rz’) - r)ln V ”

A = Z ln(l+yzw.J + 2 ln(1 + ycwc) + Z’r ln(1 + yzw;) + z’r In(l + y,~~)

and

The last two relationships can be confirmed by taking

rim&[ ,,+]x = D and trnno[ ln$]x = o in Eqs. c8”

C 2

If we again retain terms up to 2nd order terms in yi,

In a, = In Y, - S’ - A’

In a2 = In Y, + 2ZX22y2{1 - y,(l - p X,,))

+ 2zx2cY,il - Y&l x2c

- 2) + y2X2,) - S’ - A’ + In y:

ln ac = lnyc

+ 22 ’ X,,y,l 1 -

where

ES-81

E8-C1

[B-AI

19-83

tS-Cl

S’ t (Z + rZ’)Y,(l + $9 + (z + rz’ - r)Yo(l + T)

A’ L (Z + Z’r)Y,(l - G) + (Z + Z' f)Yc(l - $4

+ ZY”2& + (z + Z’r) Y~Y~X~~ + z'ry&.c

Under the assumption of a linear dependence of the energy of the central atom on the number of solute atoms in the nearest neighboring shell, only one parameter is needed to describe the interaction between two atoms. From Eqs. [Sl, a connection can be found to Wagner’s interaction parameter, a weil- established thermodynamic quantity which can readily be determined by experimental measurements (12):

f2 (2) = 2zx*z t 10-A]

5 + x Phase Boundaries in the Fe-C System ---

c IO-B1

f 10-C]

The activities of Fe and C in Fe-C alloys are obtained from Eqs. [8-A[ and [8-C] as:

PHASE BOUNDARIES IN Fe-C-X SYSTEMS 47

lna, = -I ln[l + y,] - z’r In 1 + v,QJ;

[ 1 1 + Y, Ill-Al

In ac = In yc + Wry?) + 22’ In UI(J [11-B]

Similar relationships were derived in ref. (5). The phase boundary compositions can be obtained by solving equations:

4, = P,,~ for i = 1, C.

For In yp and Y in Table 3. to&ther wtt

in austenite, the values reported by Foo and Lupis (5) are used. They are given ‘?I the valu es for ferrite used in this calculation. Two investigations (6,7) have

shown that there are strong attractive interactions between carbon atoms in ferrite. Thus c fc’ and A are negative. However, as a result of problems with either the experimental data or the an&&es use % the value of the interaction parameter could not be accurately defined. be made that et”

The assumption wili accordingly = 0. The error caused by this assumption is expected to be very small because of

the extremely lo%‘carbon concentrations in the ferrite phase.

Table 3. Activity Coefficients and Interaction Coefficients of Carbon Used in This Study

Ferrite(15,24) Austenite (5)

lnyr -5.191 + 12431/T -2.1 + 5300/T

x -----__ CC

1 - exp (-0.10 - 290/T)

Figs. 1 and 2 show calculated y/(.z+y) and a/(a+y) boundaries, respectively, in the Fe-C system. The values of FP - LIP are taken from the tabulation by Orr and Chipman (22). Data on lnyO” from three different sources (15,‘$3,24), for which analytical expressions are summarized by Lobo and Ge$$r (15). were utilized. They yielded significant differences in the a/(a+y) curve but were virtually without effect upon the y/(4+7) phase boundary. Since the value obtained from Dunn and Mciellan (24) (see Table 3) produced the best agreement with the experimental Q/(Q+~) curve, this one was utilized in all further calculations. The phase boundaries calculated on the CAM approach are compared with those obtained from the Hillert- Staffanson model, the quasi-chemical treatment (4) utilized by Shifiet et al (61, and the experimental phase diagram in Figs. 1 and 2. The HS model gives the best fit with the experimental y/(a+y) and a/(,~+~) curves. The reason for superior fitting is not certain, particularly in view of the fundamental improvements contained in the CAM model.

a + x Phase Boundaries in Fe-C-X Systems -- - --

Data on &.

Three parameters, In y:, X2 element. They must be evaluated f

and X are added by the introduction of one substitutional alloying or both

“f’ errrte and austenite.

Considerable data area available on E(‘) , the temperature dependencies of which have been summarized by Kirkaldy et al (13). Although th% parameter has also been evaluated by applying the CAM to published data on the activity of carbon in austenite (121, the temperature ranges over which this was done were usually too small to permit determination of the temperature-dependent expressions given by Kirkaldy et al (13) were employed. Table 4 summarizes these expressions for ct2) and also compares their values at 1OOO’C with those evaluated from the CAM model (12); agree&&t between the two approaches is seen to be quite good.


I 8

0 2.0 4.0 6.0 1 CARBON, at X

\ .

61 J

Figure I. Various calculations of the y/la + yl boundary in the Fe-C system.

,001 , ,

0 0.05 0.10 0.15

CARBON, ot x

Figure 2. Various calculations of the a& + ~1 boundary IR the Fe-C system.


Table 4. Wagner Interaction Parameters for Carbon, cx(“, Used in This Study

Alloying Values of e (2) at 1000DC

Element Ferrite Austenite Calculated k$orted (12)

Mn --_ 1.86 - Y(12) -4.3 -4.3

7600 Ni _-_ -2.2 + - 3.7 4.1

T

2800 co --_ 2.2 2.2

T

Si 1500, 7310

14.8 - - T

4.84 + - 10.6 11.2 T

17870 MO 58.0 - y(T>742'C)' 3.855 - 7 -10.2 -__

172000 -151 + -(T>742'C)

T

ft2’ data have been reported for MO (15) and Si (14) over narrow temperature ranges, as recorded in Table dy For the other alloying elements considered, the assumption is made that 6:: = 0.

Conversion of e2rc) to Xzc is accomplished by means of Eq. [10-B].

Yy and 52 I’) in iron --

Data on the activity of X in Fe-X alloys are relatively scarce. Accordingly, the data needed on ycc and 6 “) have been calculated from the equations for the excess free energy, GeX, in Fe-X systenk develoied by Kaufman and Nesor (16,171 from the relationship:

ex %

r GeX + (1 - X2g 2

and

In yy = +;x1x2=o

'2 f2) = [z]x2_o

[I21

[13-A]

[13-B]

The calculated values are shown in Table 5.

The two important points concerning the GeX expressions of Kaufman and Nesor should be noted. One is that the temperature ranges to which they apply encompass the regions of interest during the present study. The second is that even though these expressions are cast as polynomials of quasi- or sub-regular solution type, they nonetheless represent accurately the data to which they were fitted and hence may be considered for present purposes to be model-independent.

‘Calculrted from the data in the range 635’C 2 T 5 1 OOO’C in Ref. (14).

2 Calculated from rhe data !n ths rmgs 757’C 5 T 5 846’C for the first and 662’C 5 T 5 727’C in Ref. (15).


Table 5a. 12) y:and s2 Used in the Calculations

Ferrite

Alloys ln Yy, (2)

(2.0

493.1 Fe-Mn -0.5636 + -

124.6 -1.893 - -

T T

Fe-Ni 1.596x10-4T - 1.909 7.88x10-3T - 2.410

x10-'T2 + F x~O-~T~ - - 4559

T

Fe-Co 6.022x10-3T - 2.610 -1.356x10-2T + 6.46

x~O-~T~ -- 4444 8180

T x~O-~T~ + -

T

Fe-Si 15601

0.9562 - - 9059

29.09 - - T T

Fe-MO 3875

-1.006 + - 7460

4.026 - - T T

Table 5b. 12) yyand c2 Used in the Calculations

Alloys In Yw 2.Y

Fe-Mn 2270

2.043 - - T

Fe-Ni -4.608~10-~T + 1.965

251.6 x10-'T2 + -

T

Fe-Co 2.507~10-~T - 5.170

279.3 x10‘'T2 - -

T

Fe-Si 16400

0.9562 - - T

Fe-Ho 2982

-1.006 + - T

(2) (2.y

4511 -4.087 + -

T

7.716x10-3T - 3.288

9397 x10+T2 - -

T

3.763~10-~T - 7.750

X10-8T2 + 880.7

T

29.09 - y

5822 4.026 - 7

Results of Calculations and Comparisons with Other Models Two types of U+Y phase boundary are recognized in connection with the proeutectoid ferrite reaction

in Fe-C-X alloys: orthoequilibrium and paraequilibrium (7.26.27). The former represents the situation in which the chemical potential of each of the three species is the same in both phases. Hence the ortho- or complete equilibrium phase boundaries of the O+Y region can be obtained by solving three simultaneous equations:

Pi.0 = Pi.y [ 141

with appropriate relationships substituted for the chemical potentials. In the paraequilibrium case, the ratio of X to Fe atoms is the same in both phases; carbon atoms redistribute so as to permit their chemical potential to become uniquely the same in austenite and ferrite (16.17):

PC.CZ q ky

(P,,a - P,.y) + +2.1, - ‘2.y) = 6

where

[15-A]

[ 15-53


X x2 5 +!E E

X 2.r

1.a X 1.Y

Calculated orthoequilibrium

successively Mn, Ni, Co, Si and

refs. 16 and 17. Two sets of

[I61

phase boundaries for five representative Fe-C-X systems, where X is

MO, are shown in Figs. 3-7. for each element is given in

calculations are shown for eat In the “a” Figures,

isothermal sections at several temperatures include both y/(a+y) and a/(a+y) boundaries. The

paraequilibrium boundaries are included in these figures only for representative temperatures in order to

avoid confusion. The “b” figures show the y/(a+y) boundary and so the paraequilibrium boundary below

which transformation to ferrite can take place without partition of X but not of C, both at a constant

proportion of X; usually this proportion

order that an exact comparison may be

comparison purposes, the experimental

the Hillert-Staffanson regular solution

Kirkaldy et al (27). which apparently are

where available.

is 3 at% though in some systems a little deviation is accepted in

made with the published phase diagram or experimental data. For

data are used wherever available, and also calculations based on

model. The predictions from a sim#er method developed by

based upon temperature-independent t 2 values, are also included

CAM-calculated y/(a+y) boundaries for the orthoequiiibrium case agree particularly well with

experiment when X is Co and MO; in the remaining systems agreement is generally reasonable. No

systematic differences are found between the predictions of the CAM model and those of the other two

tested. A combination of the relatively dilute solutions of both austenite and ferrite studied and the

different sources of basic thermodvnamic data, individuallv orocessed, is orobablv resoonsible. From the

viewpoint of the present authors, however, Figs. 3-7 are taken as supporting the utility’ of the CAM in the

context of Fe-C-X systems and encouraging the application of this model to higher order systems.

CARBON, 01 X

Figure 3. (a) Isothermal and (b) isoconcentration sections of Fe-rich region of the Fe-C-Mn phase diagram.

52 M. ENO~TO and tf.1. ~RONSaN

Fr-C-Ni (orthof

- CAM ---_ “.3

-- Shmn~ti&(ze

0 0.5 1.0 1.5 2.0 2.5

CARBON, at%

+ \

3.0

J.,tXNlCKEL Boo-

- CAM

Figure 4. (a) Isothermal and Ib) isoconcentration sections of Fe-rich region of the Fe-C-Ni phase diagram.

6.0 I I , I

Fe-c-co lorthol

-CAM

1.0 2.0 5.0 4.0 5.0 6.0 CARSON. at x

-CAM

A ExPtl. ,501

I.0 20 3.0 4.0 5.0

CARBON. al X

Figure 5. (a) Isothermal and (b) isoconcentration sections of Fe-rich region of the Fe-C-Co phase diagram.


Figure 6. (a 1 I!

4.

B

0

sothermal and (b) isoconcentr /

Fe-C-t& lortho) - CAM

I Ioo.a*c --- HS

va+y -I- Collated Expll. (~61

1.0 2.0 3.0 4.0 50 CARBON. ot X

0 IO 20 1.0 40 50

CARBON. 01 X

Figure 7. (a) isothermal and (b) isoconcentration sections of Fe-rich region of the Fe-C-MO phase diagram.

atic 2n sections

$0 20 30 40 CARBON. ot X

of Fe-rich region of the Fe-C-Si phase diagram.

Free Enerw Changes Associated with the Proeutectoid Ferrite Reaction

A major effect which alloying elements exert upon the nucleation and growth kinetics of proeutectoid ferrite formation takes place through their influence upon the free energy change attending this reaction. These have been previously examined through a simpler model. They are reconsidered here on the basis of the CAM and the HS models.

The driving force for the nucleation and growth of proeutectoid ferrite, i.e., the difference in the free energies of the parent and the precipitate phases at the composition of the latter, is conveniently expressed in vector notation as:

where e = fr,, ,t . pcl and ,X austenite at equih rrum and that &

= (X,, X , X6). Ez? ,xY” and X designate compositions in ferrite and of the a ulk alloy. reZpectivel~.y Under orthoequilibrium conditions (Eq.

[141), this relationship can be rewritten in a form more convenient for purposes of calculation:

AG = (,#=’ - ,+&j) l ,e”y [ISI


It should be noted that an equation identical in form is obtained under the condition of paraequilibrium (Eqs. 1151). The total free energy change is obtained by replacing ZzYwith & in Eq. [181.

The free energy change calculated from Eq. [ 181 for both ortho- and paraequilibrium in Fe-O.5 at% C- 3 at% X alloys as a function of temperature for the five Fe-C-X systems under consideration is shown in Figures 8-12. Equivalent plots are provided in these Figures for calculations based upon the HS model. And a CAM-based calculation is included in each figure for an Fe-O.5 at% C alloy. The displacement of both Fe-C-X curves relative to that for Fe-C is qualitatively similar to those found earlier (7). except in the case of Fe-C-MO, where both the CAM and the HS models do not make AG less negative than in the comparable Fe-C alloy at any temperature investigated.

Inspection of Figures 8-12 indicates that when X is a ferrite-stabilizing element (Si, MO) the difference in AG between ortho- and paraequilibrium is usually appreciably smaller than when X is an austenite-stabilizer (Mn, Ni). When X is an austenite stabilizer, the paraequilibrium a + 7 region is a triangle in an isothermal section of the Fe-C-X diagram, one of whose apexes lies on the Fe-X side of the diagram. If X is a ferrite stabilizer, this apex can be regarded as being much displaced toward the pure X corner of the isothermal section. This change in the shape of the a + r region results in a marked reduction in the difference between the paths of the ortho- and paraequilibrium boundaries of the o + 7 region. When the bulk concentration of alloying element is relatively small, e.g., Alloy No. 1 in Figure 13, the tielines of ortho- and paraequilibrium are nearly coincident in both austenite and ferrite stabilizer Fe-C- X systems. However, as shown in Figure 13a. the carbon contents corresponding to the two equilibria exhibit appreciably larger differences when X is an austenite stabilizer. This explains immediately the larger difference in AG between ortho- and paraequilibrium when X is an austenite stabilizer. As the concentration of alloying element increases, e.g., Alloy 2, the slopes of the ortho- and paraequilibrium tielines deviate increasingly; however, because of the greater similarity of the paths followed by the ortho- and paraequilibrium phase boundaries when X is a ferrite stabilizer, higher bulk concentrations of X (Alloy 3) are needed to produce the same difference in composition between austenite and ferrite. A more slowly changing slope of the tielines when X is a ferrite stabilizer contributes further to this tendency.

h Fe-0.5at%C-3at%Mn

-1.0 -CAM

_-__“S

TEMPERATURE,‘C

Figure 8. Free energy change of proeutectoid ferrite reaction in an Fe-O.5 at%C-3 at%Mn alloy.

PHASE ROUNDARIES IN Fe-C-X SYSTEMS 55

TEMPERATURE.‘C

Figure 9. Free energy change of proeutectoid ferrite reaction in an Fe-O.5 at%C-3 at%Ni alloy.

c Fe - 0 5 at % C - 3 at % CO -1.0

-CAM

0 E

:

F 0 * -0

“a

TEMPERATURE,OC

Figure IO. Free energy change of proeutectoid ferrite reaction in an Fe-O.5 at%C-3 at%Co alloy.


I I I I

Fe-0.5at%C-3at%Si

- CAM

--- HS

700 800 900 TEMPERATURE ,*C

Figure 11. Free energy change of proeutectoid ferrite reaction in an Fe-O.5 at%C-3 at%Si alloy.

I Fe-O.5 at%C-3at XMa I

200 I

700 800 900 mO0 TEMPERATURE ,OC

Figure 12. Free energy change of proeutectoid ferrite reaction in an Fe-O.5 at%C-3 at%Mo alloy.


(a)

Figure 13. Schematic diagram of typical appearance of Fe-rich corner of Fe-C-X phase diagram when: (a) X is an austenite stabilizer, and (b) X is a ferrite stabilizer.

Summary

The formalism of the “Central Atoms Model” (CAM) (10-12) has been placed in a form permitting direct application to calculation of both orthoequilibrium and paraequilibrium relationships between austenite and ferrite in Fe-C-X alloys. The constants needed to use this model were evaluated from published data for five of these alloy systems, where X is successively Mn, Ni, Co, Si and MO. The CAM was then used to calculate both sets of phase boundaries and also the free energy change associated with the nucleation and growth of proeutectoid ferrite. Equivalent calculations were also made by means of the Hillert-Staffanson (18) regular solution model. Good agreement was generally found between these two models and the available experimental data on orthoequilibrium (r/(4 + 7) phase boundaries. The main interest in making these tests and applications of the CAM model is that it may be particularly well suited for extension to quaternary and higher order Fe-C base alloys; research on this extension is currently in progress (31).

Acknowledqements

The authors wish to thank especially Dr. C.H.P. Lupis for his advice which initiated the present calculations and reviewing the manuscript. They also express their appreciation to NSF for sponsoring this work under Grant No. DMR-El-16905 from the Division of Materials Research.

References

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3. L.S. Darken and R.P. Smith: J. Amer. Chem. Sot.. 1946, Vol. 68, p. 1172.

4. R.B. McLellan and W.W. Dunn: J. Phys. Chem. Solids, 1969, Vol. 30, p. 2631.

5. E.-H. Foo and C.H.P. Lupis: Proceedings ICSTIS, Suppl. Trans. ISIJ. 1971, Vol. 11. p. 404.

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Calculation of α + γ phase boundaries in Fe-C-X systems from the central atoms model