Computation of Partial Equilibrium Solidification with Complete Interstitial and Negligible Substitutional Solute Back Diffusion

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1 Materals Transactons, Vol. 43, o. 3 (22 pp. 551 to 559 c 22 The Japan Insttute of Metals omputaton of Partal Equlbrum Soldfcaton wth omplete Intersttal and eglgble Substtutonal Solute Bac Dffuson Qng hen and Bo Sundman Department of Materals Scence and Engneerng, Royal Insttute of Technology, S-1 44, Stocholm, Sweden A smple numercal scheme s presented to smulate partal equlbrum soldfcaton wth complete ntersttal and neglgble substtutonal solute bac dffuson n mult-component and mult-phase systems. Based on ths scheme, a computng tool capable of usng Thermo-alc databases drectly has been developed for the estmaton of soldfcaton behavor of steels and other ntersttal-contanng alloys. Agreements between calculated and expermental as well as DITRA results have been obtaned on the mcrosegregaton, fracton of eutectc, and freezng range of several steels. Ths suggests that the partal equlbrum assumpton and proposed numercal scheme are reasonable and satsfactory, and confrms that the carbon bac dffuson plays a very mportant role n the soldfcaton of steels. (Receved December 12, 21; Accepted January 25, 22 Keywords: numercal smulaton, soldfcaton, thermodynamcs, mcrosegregaton, steel 1. Introducton The exact behavor of a soldfyng system under local equlbrum condtons s controlled by the thermodynamcal propertes and solute dffuson n both lqud and sold phases and can only be predcted by usng software wth full ntegraton of thermodynamcs and netcs, such as DITRA. 1 However, n practce the soldfcaton process can often be approxmated n smple and effcent ways wthout evong tme-consumng netc calculatons. The two extremes are equlbrum (lever-rule and non-equlbrum (Schel-Gullver methods. 2 4 They both assume unform solute dstrbuton n the lqud; and ths s usually reasonable due to the large atomc moblty and presence of convecton n the lqud. For the sold, the lever-rule method assumes that the solute dffuson s also so rapd that unform composton s mantaned, whch s reasonable for ntersttal solutes le and only. For substtutonal solutes n the sold, on the other hand, dffuson s much slower and can often be neglected. In ths case, each nfntely small porton of the sold, once formed, s expected to retan the same composton through the soldfcaton course. Ths s the other basc assumpton of the Schel-Gullver method. The Schel-Gullver smulaton wth varable partton coeffcents calculated from thermodynamc databases often gves a reasonable approxmaton to the normal soldfcaton behavor n Al and -based alloys, 5 but s by no means successful n the nvestgaton of steels where the bac dffuson of ntersttal carbon must be taen nto account. 6 A natural and more realstc approxmaton to the stuaton n steels s expected to be a partal equlbrum one,.e. complete ntersttal but neglgble substtutonal solute bac dffuson n solds. Ths approach has been appled n a crude way by Fredrsson and Hellner 7 to study the nfluence of carbon on the segregaton of chromum n steels through combnng Schel-Gullver s segregaton equaton for r and the lever rule for together wth some phase dagram nformaton. In order to realze the partal equlbrum approach exactly, a numercal scheme wth the ad of a thermodynamc database must be adopted because t s the chemcal potental or actvty rather than composton of the ntersttal that remans unform throughout the sold. In a recent report, an effort n ths drecton has been descrbed brefly by Kozeschn. 8 The present wor ntends to develop a practcal numercal scheme and buld a computng tool for partal equlbrum soldfcaton smulaton. Applcaton examples wll be gven to show that ths approach s an excellent approxmaton to the soldfcaton behavor of steels except for those domnated by the ferrtc prmary phase, where the bac dffuson of substtutonal solute s hardly neglgble. 2. Methodology There s no shortage of analytcal solutons to the problem of soldfcaton wth solute bac dffuson, but t seems that none of them s sutable for our purpose of partal equlbrum smulaton. In ths secton, we wll frst examne the lmtatons of the exstng analytcal models and explan specfcally why they should fal upon the partal equlbrum condton. A numercal method wll then be ntroduced for Schel-Gullver soldfcaton and further modfed for partal equlbrum soldfcaton. 2.1 mtatons of analytcal models Varous analytcal equatons have been proposed to tae nto account the solute bac dffuson n the soldfcaton process over past years. They were usually derved from a bnary soldfcaton case nvolvng only one sold phase and, consequently, can only be utlzed to analyze the prmary-phase soldfcaton stage. For mult-sold-phase soldfcaton stages, whch s normally unavodable n commercal alloys, no analytcal soluton s feasble so far. Even wthn the ablty of the analytcal methods, there s a ptfall wth all of them f the stuaton of partal equlbrum s encountered n multcomponent systems. The frst of those analytcal equatons was developed by Brody and Flemngs: 9 x S /K = x = x {1 (1 2αK f S } (K 1(1 2αK, (1 where x S s the concentraton of component n the sold

2 552 Q. hen and B. Sundman at the lqud/sold nterface, K the partton coeffcent, x the concentraton n the lqud, x the ntal or overall concentraton, and f S the volume fracton of sold formed. The bac-dffuson parameter α s a constant related to the sold dffusvty D S of solute element, secondary dendrte arm spacng λ and local freezng tme t f by the expresson α = 4D S t f /λ 2. (2 Apparently, eq. (1 cannot be appled to descrbe the behavor of nfntely fast-movng elements. As lyne and Kurz 1 ponted out later, the use of eq. (1 should be lmted to very slow-movng elements, otherwse solute atoms are not conserved. lyne and Kurz examned ths lmtaton and proposed a modfed form of the Brody-Flemngs equaton and appled t also to fast-movng elements. They suggested that nstead of usng the parameter α nsde eq. (1 one could use a functon of t Ω(α = α{1 exp( 1/α} (1/2 exp( 1/2α. (3 Other smpler or more complex equatons have been proposed by Ohnaa, 11 Kobayash, 12 and astac and Stefanescu. 13 Asymptotcal analyses of all these modfed Brody-Flemngs equatons 1, 11 or completely new ones, 12, 13 lead successfully to the equlbrum or lever-rule equaton f D S : x S /K = x = x (1 (1 K f S 1, (4 where x S s the concentraton n the sold, and the Schel- Gullver s equaton f D S : x S /K = x = x (1 f S K 1. (5 Ths s approprate n bnary systems because eq. (4 holds f there s only one solute element and ts dffusvty approaches nfnty, and eq. (5 holds f ts dffusvty s close to zero. In mult-component systems, the dffusvtes of the solutes may all be approxmated as nfntely large, or all neglgble, or some nfntely large and some neglgble. For the frst scenaro, the asymptotcal result eq. (4 should be met for each solute element; for the second scenaro, the asymptotcal result eq. (5 should hold for each solute element. For the thrd scenaro, whch has here been called partal equlbrum, applcaton of these analytcal equatons 1 13 results n eq. (5 for some solute elements due to D S and eq. (4 for the others due to D S. However, that cannot be a good descrpton f there s a thermodynamc nteracton between elements of the two nds. From a thermodynamc pont of vew, f a solute has nfntely large atomc moblty, equlbrum shall be attaned for t everywhere n the whole system, that s to say, ts chemcal potental or actvty s unform. For the partal equlbrum condton, the concentraton of each nfntely slow-movng element vares throughout the sold accordng to eq. (5 and at the same tme the unformty of chemcal potental should be mantaned for each nfntely fast-movng element. As a consequence, the concentraton of each nfntely fast-movng element could not be unform n the sold, and thus eq. (4 cannot hold for the element even f D S. The fact that the asymptotcal results can not hold for all the scenaros suggests that those analytcal equatons, whch are approprate for bnary systems, lac thermodynamcal consstency f appled to partal equlbrum stuaton n mult-component systems. Even though one may be tempted to combne eqs. (4 and (5 drectly to treat the partal equlbrum soldfcaton, that approach s thermodynamcally ncorrect and may lead to very dfferent results from that of a real partal equlbrum calculaton. Ths wll be demonstrated n the Results and Dscusson secton. The solute redstrbuton n a soldfcaton process s very senstve to the value of the partton coeffcents used n analytcal models. The partton coeffcent wll seldom be a constant n the temperature-composton space as assumed n all analytcal models. Wth the advance of APHAD (Aculaton of PHAse Dagrams technque, 5, a number of software and thermodynamc database systems, such as Thermo-alc, 17 are now avalable for the calculaton of the phase equlbrum of an alloy system from whch the partton coeffcents can be calculated readly at any temperature and composton. Therefore, when analytcal methods are employed, t s better to use them for numercal calculatons by combnng them wth approprate thermodynamc software and database rather than applyng constant partton coeffcents. 2.2 umercal method The soldfcaton process s drven by the extracton of heat, whether the alloy s bnary or mult-component, whether t nvolves one sold or several solds, and whether t can be approxmated by condtons of equlbrum, non-equlbrum, or partal equlbrum. If we lower the temperature successvely for a small degree T from the lqudus temperature and calculate the phase equlbrum of the gven system at each temperature step wth a thermodynamc database, the equlbrum soldfcaton path and fracton of sold as well as other nterestng propertes can be obtaned readly. For the Schel-Gullver model, owng to the local equlbrum assumpton at the phase nterface, we can use the same technque but eep changng the composton and amount of the lqud on whch a phase equlbrum calculaton s performed. For the partal equlbrum model, the same strategy wll be used but the change of composton and amount of the lqud at each step s dfferent owng to the ntersttal bac dffuson Schel-Gullver model Supposng we have an alloy wth m (m 1 solute elements, an ntal composton of x ( = 1,...,m, and total amount of one mole of atoms ( = 1, ts Schel-Gullver soldfcaton process can be smulated by followng the steps descrbed below: ocate the lqudus temperature T, whch can be obtaned together wth the number of equlbrum sold phase(s, n (n 1, and the composton of each sold phase ( = 1,...,n, x, by a phase equlbrum calculaton on a system of ntal composton and amount wth the fracton of lqud set to unty. At ths temperature, we have f =, f S = f =, and f = 1 f S = 1, where f s the amount or fracton of the sold phase (the amount wll be expressed n moles and s dentcal to fracton, f S the amount of sold, and f the amount of remanng lqud. The composton of the lqud phase, x, s of course equal to

3 omputaton of Partal Equlbrum Soldfcaton wth omplete Intersttal and eglgble Substtutonal Solute Bac Dffuson 553 x. 1 ower the temperature from T to T 1 by a small decrement T and the frst small amount of sold phase, 1 f, s formed. The value of 1 f and the composton of the sold phase and lqud at ths temperature step, 1 x and 1 x, can be obtaned by performng phase equlbrum calculaton on a system of composton x and amount f. The amounts of sold formed thus far can be thought of as: 1 f = f + 1 f and 1 f S = 1 f. The amount of remanng lqud s 1 f = 1 1 f S. 2 ower the temperature from T 1 to T 2 by a small decrement T and another small amount of sold phase, 2 f, s formed from the lqud left from step 1, whch has a composton of 1 x. The value of 2 f and the composton of newly formed sold phase(s and remanng lqud, 2 x and 2 x, can be obtaned from a phase equlbrum calculaton on a system of composton 1 x and amount 1 f. The amounts of sold formed so far are 2 f = 1 f + 2 f and 2 f S = 2 f. The amount of remanng lqud s 2 f = 1 2 f S. Snce there s no dffuson n the sold, there wll be no transport of atoms among newly formed porton of sold and the old ones although the chemcal potental of each solute element s dfferent n dfferent parts of sold. The local equlbrum holds between the newly formed sold phase(s and remanng lqud and there s no redstrbuton of atoms between lqud and sold phase(s. 3 ower the temperature and repeat smlar calculatons untl the remanng amount of lqud s less than a gven threshold. Apparently, the steps after can be generalzed as follows: lower the temperature from T to T +1 ( by a small decrement T and a small amount of sold phase, +1 f wll be formed from the lqud left from step, whch has a composton of x. The value of +1 f and the composton of newly formed sold phase(s and remanng lqud, +1 x and +1 x, can be obtaned from a phase equlbrum calculaton by settng the amount of a system to f and composton to x. The amounts of sold formed thus far are +1 f = f + +1 f and +1 f S = +1 f. The amount of remanng lqud s +1 f = 1 +1 f S. hec f +1 f s smaller than a gven value or not. If not, contnue to the next step and repeat smlar calculatons. Otherwse, termnate the smulaton. The whole computatonal process s schematcally drawn wthn the dashed lne rectangle n Fg. 1. The numercal scheme outlned above s a general one and wll wor n mult-component and mult-phase systems. It has been used 18 and mplemented nto a specal module 19 nsde the Thermo-alc. The fact that ths numercal scheme reduces to the Schel-Gullver equaton for constant partton coeffcents can be llustrated as follows. From any step to + 1, snce t s an equlbrum calculaton, +1 f S, +1 x S, +1 x, x and f are related to each other by the lever-rule. Here subscrpt of S s omtted because only one sold phase s consdered n the analytcal Schel-Gullver method. As a result, we have ( x +1 x S +1 f S / f = ( +1 x whch may be rearranged nto ( +1 x x (1 +1 f S / f, (6 +1 x S +1 f S = ( +1 x x f. (7 Substtuton of f = 1 f S and +1 x S eq. (7 gves ( +1 x = +1 K +1 x nto +1 K +1 x +1 f S = ( +1 x x (1 f S, (8 whch can be rewrtten as +1 x (1 +1 K +1 f S = +1 x (1 f S. (9 If the step s small enough, the step superscrpt can be gnored and we get df S (1 K = dx 1 f S x. (1 Assumng a constant K, we obtan eq. (5 after a smple ntegraton of the above equaton from to f S on the left sde and x to x on the rght sde Partal equlbrum model The numercal algorthm for the partal equlbrum model can be obtaned by smply modfyng the scheme for the Schel-Gullver model presented n the precedng secton. At each teraton step, the composton and amount of the remanng lqud wll be adusted accordng to the partal equlbrum assumpton. For smplcty, carbon wll be the only ntersttal element to be dscussed n the present wor. Acceptng that there are complete ntersttal and neglgble substutonal solute bac dffusons n the sold, we shall allow carbon to move freely among the lqud and all portons of the sold phases formed at the present and prevous temperature steps, so that ts chemcal potental µ becomes equal everywhere throughout the whole system. At the same tme, substtutonal solutes shall be completely frozen n the remanng lqud and each porton of the sold formed, whch means that ther u-fractons (defned as u = = x = /(1 x 14 shall not change durng the bac dffuson of carbon. Therefore, at any step, the followng relaton can thus be appled to each porton of the sold phases formed p µ p u = = µ u = (11 where p denotes the porton formed at the step p (1 p. eq. (11 yelds ndependent relatons f sold phases have formed n all the steps. We need one more equaton to obtan 1 + unnown varables,.e., the new content of carbon n the remanng lqud and the portons of sold phases, ( x and ( p x. By consderng the mass conservaton of the ntersttal element, we get the necessary

4 554 Q. hen and B. Sundman START x, ( 1 T T lq, x S 1, x ( S x T T x x, T x ( x ( ' ', x x, S 1 S T ntersttal element? Y ' ( x 1 ( x p p ( 1 p u u x ' ( p p x ' x p p 1 ( x ' STOP Fg. 1 omputatonal scheme for partal equlbrum soldfcaton. equaton ( x x 1 ( x + ( p x p x 1 ( p x p = (12 where and p are the amount of the remanng lqud and the p-th porton of sold phase before the carbon bac dffuson at the step. Havng obtaned ( x and ( p x, we may easly get the new content of substtutonal solutes n the lqud and sold phase(s, ( x = and ( p x =, as well as the new amount or fracton of the lqud and sold(s, ( and ( p, usng the relatons: ( x = = 1 ( x 1 x x = (13 ( p x = = 1 (p x 1 p x p x = (14 ( = 1 x 1 ( x (15 ( p = 1 p x p 1 ( p x S (16 After ths partal equlbrum adustment to the phase composton and amount, the remanng lqud wll be n a metastable equlbrum state and some sold phase(s would form f an equlbrum calculaton s performed on t at the current temperature step. However, we shall not perform such an equlbrum calculaton, whch wll apparently lead to smaller and smaller adustment untl the new lqud composton s on the lqudus surface at the current temperature. We wll use the metastable lqud drectly at the next temperature step for the local equlbrum calculaton and subsequent partal equlbrum manpulaton. The full computatonal procedure s shown n Fg. 1. ow let s loo bac at the partal equlbrum condton eq. (11. As we sad before, t conssts of ndependent relatons f sold phases have formed n all steps. For commercal alloys, a normal smulaton has hundreds or over one thousand temperature steps, dependng on the freezng range and temperature step. As the smulaton contnues, becomes larger and larger and also ncreases, there wll be a formdable number of equatons to solve. In order to sm-

5 omputaton of Partal Equlbrum Soldfcaton wth omplete Intersttal and eglgble Substtutonal Solute Bac Dffuson 555 plfy the problem we shall approxmate the ntersttal bac dffuson by summng up the amount and calculatng an average composton for each nd of sold phase thus formed, and then even out the ntersttal chemcal potental among the remanng lqud and each sold phase of the average composton. As a consequence, at any step, eqs. (11 and (12 become µ ū = = µ u = (17 ( x x 1 ( x + ( S x x 1 ( x S = (18 where µ s the average chemcal potental of carbon n the sold phase of an average u-fracton of substtutonal solute ū =. x and ( x are the average mole fractons of carbon n the sold phase before and after evenng out the chemcal potental. In ths smplfed approach, we are not tracng the propertes of ndvdual porton soldfed at each temperature step but the average ones for each nd of sold phase thus formed. Therefore, nstead of solvng a frghtenng 1 + dmensonal problem, we are now comfortably dealng wth a 1 + one and wll have a tremendous gan n computatonal tme. However, admttedly, due to ths smplfcaton we wll lose the ablty to follow the change of composton n each ndvdually soldfed part and cannot obtan the actual solute dstrbuton at the termnatng step T,.e., p x (1 p T. evertheless, we may stll use the sold composton at the sold/lqud nterface at each temperature step to represent the fnal solute dstrbuton for substtutonal elements, and, fortunately, ths may gve even better agreements between the calculaton and measurements than the orgnal approach whch solves the problem rgorously by employng eqs. (11 to (16. In fact, the unavodable bac dffuson of substtutonal elements n a real alloy, even though very small, wll perhaps be large enough or more than enough to cancel the composton change of substtutonal solute due to the ntersttal bac dffuson. 3. Results and Dscusson A computng tool, called SHEI (SHEl wth Intersttal arbon, has been developed for smulatng partal equlbrum soldfcaton by usng the numercal method presented n the precedng secton. The prmary goal of devsng ths tool s to tae nto account of bac dffuson of carbon n the soldfcaton of steels, but the usage s not restrcted to carbon and steels. Alloys wth other ntersttal element(s can also be treated. Wth the ad of the TQI (Thermodynamc cal- Ulaton Interface, 2 an applcaton program nterface of the Thermo-alc, we have made t possble for ths computatonal tool to use the thermodynamc data drectly from databases avalable to the Thermo-alc software pacage. Wth use of ths new tool, t wll be as easy to perform a partal equlbrum smulaton as to do a Schel-Gullver calculaton nsde the Thermo-alc system, but the results wll be closer to realty f ntersttal elements are nvolved. A few examples wll now be demonstrated. Ono et al. 21 has recently reported ther expermental results on Fe 1.84%r.95% and Fe 5.34%r.92% alloys soldfed at a coolng rate of.167 K/s. Usng SHEI and Thermo-alc together wth the TFE 22 database, partal equlbrum and Schel-Gullver smulatons were performed for these two steels. In addton, DITRA, a software wth full ntegraton of thermodynamcs and netcs, has also been employed to corroborate the measurements and our smple calculaton results. The TFE 22 thermodynamc database, MOB2 23 moblty database, and expermental parameters,.e., coolng rate and secondary arm spacng, were used n DITRA smulatons. It should be mentoned that n ths study, all partal equlbrum and Schel-Gullver smulatons have been termnated when the fracton of the remanng lqud become less than 1%. Fgure 2 shows the agreement between the calculated and expermental results on the mcrosegregaton of r n these two steels. In Fg. 2(a, t can be seen that the DITRA smu- Mass fracton, r Mass fracton, r Fe-1r-1 (a Schel-Gullver DITRA Partal Equlbrum Experment [21].8 Fracton of sold Fe-5r-1 (b Schel-Gullver DITRA Partal Equlbrum Experment [21].4 Fracton of sold Fg. 2 Mcrosegregaton of r n Fcc phase n steel (a Fe 1.84%r.95% and (b Fe 5.34%r.92%. The dashed lnes serve as a gudelne for the expermental data. 21

6 556 Q. hen and B. Sundman Mass fracton, r Bcc E E Fcc D PE M 7 3 D PE SG SG M E-3 Mass fracton, Fg. 3 qudus surface proecton and soldfcaton paths of Fe 1.84%r.95% and Fe 5.34%r.92% steel. Square and crcle ndcate the start for steel Fe 1.84%r.95% and Fe 5.34%r.92%, respectvely. Symbols wth mar the end ponts accordng to varous smulatons: E Equlbrum, SG Schel-Gullver, D DITRA, PE Partal Equlbrum. The styles of lnes follow those n Fg. 2 except that the dashed lnes represent the results from the analytcal method. laton result at f S = s almost exactly the same as the measurement, and the partal equlbrum value s slghtly lower. Ths dfference s due to the fact that the bac dffuson of r has been completely neglected n the partal equlbrum treatment. Overall, the two smulaton results agree wth each other better than wth the expermental results, whch means that the partal equlbrum assumpton and numercal algorthm are reasonable and satsfactory, but the thermodynamc database used may need mprovement or some of the expermental measurements may nvolve a large uncertanty. By comparng wth the Schel-Gullver smulaton whch neglects bac dffuson completely, we can clearly see from Fg. 2 that the mcrosegregaton of r n the prmary Fcc phase s enhanced due to the bac dffuson of carbon, a fact observed by many nvestgators but never fully explaned. onsderng steel Fe 5.34%r.92% where the effect s most promnent, t s apparent that ths s largely because the formaton of eutectc mxture s postponed (see Fg. 2(b and the enrchment of chromum n the lqud becomes more and more severe towards the end of soldfcaton. The change of the lqud composton n the soldfcaton process s shown n Fg. 3. From ths plot, we see that the soldfcaton path s greatly changed due to the bac dffuson of carbon. As a matter of fact, t s the change of soldfcaton path that causes the delay of eutectc formaton. Because of the change of soldfcaton path, the varaton of partton coeffcent of r for the partal equlbrum soldfcaton s also dfferent from that for the Schel-Gullver smulaton, see Fg. 4. Ths can explan why the mcrosegregaton curve for partal equlbrum has a dfferent curvature from that for Schel-Gullver smulatons n Fg. 2, where we can see that the former one les more and more above the latter one as the soldfcaton contnues. From Fg. 2, we can also see that there s a sharp decrease n r content of the eutectc Fcc phase towards the Partton coeffcent r Fe-5r-1.6 Equlbrum.55 Schel-Gullver DITRA Partal Equlbrum.5 Fracton of sold Fg. 4 Partton coeffcent of r varyng wth the fracton of sold n steel Fe 5.34%.92%. end of soldfcaton. Ths s caused by the formaton of r carbde n the eutectc reacton. In Fg. 3, the results from a drect combnaton of eq. (4 for and eq. (5 for r by usng thermodynamcally calculated partal coeffcents are also shown. It s clear that these results devate severely from those of the real partal equlbrum calculaton and DITRA smulaton. Therefore, one must be extremely careful when beng tempted to use ths nave partal equlbrum approach. The sold fracton vs temperature curves for the Fe 1.84%r.95% and Fe 5.34%r.92% steels are depcted n Fg. 5. The agreement between the smple partal equlbrum smulaton and the more sophstcated DITRA smulaton s apparent. A favorable comparson wth the expermental results on the eutectc fractons and freezng ranges s gven n Table 1. The soldfcaton of the Fe 1.84%r.95% and Fe 5.34%r.92% alloys nvolves only two sold phases, and can be modeled by usng DITRA convenently although t needs more computatonal tme. For mult-sold cases n conventonal commercal steels, DITRA s not applcable, but SHEI can be used wth the same ease. Fgure 6 shows the predcted sold fracton vs temperature curves for the AISI M2 steel. 24 The solute compostons used n the calculatons are.88%,.3%s,.32%mn, 3.9%r,.36%, 4.9%Mo,.1%u,.3%o, 6.1%W, and 1.9%V. Although the formaton of M 7 3 nstead of M 2 s not n conform wth the expermental nformaton, 24 and ndcates a need to mprove the thermodynamc database, t s remarable that the estmated freezng range from the partal equlbrum calculaton s very close to the expermental ones whch was measured at an average coolng rate of 2. and.5 K/s. 24 From the above examples, t may be concluded that the partal equlbrum approach s a bg mprovement to the Schel- Gullver method, and the results obtaned from the partal equlbrum approach s very close to that from DITRA n the soldfcaton of the steels nvestgated. Ths success suggests that the basc partal equlbrum assumpton,.e., com-

7 omputaton of Partal Equlbrum Soldfcaton wth omplete Intersttal and eglgble Substtutonal Solute Bac Dffuson 557 Table 1 alculated and expermental results on the eutectc fracton and freezng range of soldfed Fe 1.84%r.95% and Fe 5.34%r.92% steels. Fe 1.84%r.95% Fe 5.34%r.92% Eutectc fracton Freezng range, K Eutectc fracton Freezng range, K ever-rule Schel-Gullver 11.69% % 287 DITRA 5.5% % 187 Partal equl. 6.74% % 186 Experment % % 213 Temperature, (a Fe-1r Equlbrum Schel-Gullver DITRA Partal Equlbrum 12 Fracton of sold Temperature, q+bcc 2 q+bcc+fcc 3 q+fcc 4 q+fcc+m 6 5 q+fcc+m 6 +M 6 q+fcc+m 6 +M+M 7 3 Equlbrum Schel-Gullver Partal Equlbrum 1 Fracton of sold 4 AISI M2 5 6 Temperature, (b Fe-5r Equlbrum 11 Schel-Gullver DITRA Partal Equlbrum 15 Fracton of sold Fg. 5 Sold fracton vs temperature curve for steels (a Fe 1.84%r.95% and (b Fe 5.34%r.92%. plete ntersttal and neglgble substtutonal bac dffuson, s reasonable for the case studed. As a matter of fact, we can verfy ths pont by mang a rough estmaton of the lyne- Kurz bac-dffuson parameter usng eqs. (2 and (3 although the analytc method s not thermodynamcally consstent all the tme. For the austentc prmary phase, the dffuson coeffcent of carbon s n the order of 1 9 m 2 s 1 and that of Fg. 6 Predcted sold fracton vs temperature curves of AISI M2 steel. The trangles mar the expermental soldus temperatures for the coolng rates of.1,.5, and 2. K/s, respectvely. 24 substtutonal elements of 1 13 m 2 s Knowng that the secondary dendrte arm spacng and local freezng tme are n the order of 1 4 m and 1 3 s, respectvely, we have α = 4 for carbon and α =.4 for substtutonal elements. Examnng the plot of the lyne-kurz functon gven n Fg. 7, we see that Ω(α s very close to the hgher lmtng value.5 for carbon and close to for substutonal elements, whch would then lead to eqs. (4 and (5, respectvely, accordng to the lyne and Kurz equaton. Ths ndcates that the bac dffuson of carbon s almost complete and that of substtutonal elements s neglgble. ow let us mae a smlar estmaton for the ferrtc prmary phase. Usng the same values for the austente except the dffuson coeffcent of substtutonal elements, whch s about 1 11 m 2 s 1 for the ferrte, 23 we have α = 4 for carbon and α = 4 for substtutonal elements. In ths case, we now from Fg. 7 that the bac dffuson of substtutonal elements s no longer neglgble, but rather close to complete. Therefore, n steels domnated by the ferrtc prmary phase, the partal equlbrum approach s not expected to gve very good predctons although t certanly gves better results than the Schel-Gullver method. Maybe t could be mproved by treatng r as a fast-dffusng element n ferrte. However, f all alloyng elements are fast-dffusng n ferrte, then the model would approach the lever-rule model. Smulatons have been carred out for several ferrtc steels

8 558 Q. hen and B. Sundman ( Fg. 7 Varaton curve of the lyne-kurz functon Ω(α wth α. The up end means that the bac dffuson s complete; the low end means that the bac dffuson s neglgble. Temperature, AISI 41S 143 Equlbrum Schel-Gullver 142 DITRA Partal Equlbrum 141 Fracton Sold Fg. 8 Predcted sold fracton vs temperature curve for AISI 41S steel. Three DITRA results have been mared wth the correspondng coolng rate n K/s. The trangles mar the expermental soldus temperatures for the coolng rates of.1,.5, and 2. K/s, respectvely. 24 expermentally nvestgated n Ref. 24. It was found that for these steels the soldfcaton behavor s ndeed closer to the lever-rule than the partal equlbrum model. Here we tae AISI 41S 24 as an example. The results on the sold fracton vs temperature relaton are depcted n Fg. 8. For DITRA calculatons, three expermental coolng rates 24 have been used. From the plot, t can be seen that even under the largest coolng rate, the DITRA curve s closer to that of the leverrule than the partal equlbrum, although the expermental soldus ponts are slghtly lower than the DITRA predctons. Ths s more or less true for the mcrosegregaton rato f the equlbrum chromum concentraton n ferrte at each temperature step s drawn for equlbrum soldfcaton, see Fg. 9, although we now that there should be no mcrosegregaton n the lever-rule model Weght fracton, r Equlbrum Schel-Gullver DITRA Partal Equlbrum AISI 41S E-3 12 Fracton Sold Fg. 9 Predcted mcrosegregaton of r n AISI 41S steel. DITRA results have been mared wth the correspondng coolng rate n K/s. The equlbrum curve shows the change of equlbrum composton n Bcc phase durng the equlbrum soldfcaton process. 4. Summary The soldfcaton model called partal equlbrum smulaton s a reasonable approxmaton for steels except for those domnated by the ferrte as the prmary phase. A smple numercal scheme has been descrbed for the applcaton of the partal equlbrum soldfcaton n mult-component and mult-phase systems. Based on ths scheme, a general computng tool capable of usng Thermo-alc databases drectly has been developed. Agreement between calculated and expermental as well as DITRA results have been obtaned on the mcrosegregaton, fracton of eutectc, and freezng range of several steels. Ths confrms that the partal equlbrum approxmaton s reasonable for the steels nvestgated and suggests that the proposed numercal scheme s satsfactory. However, for steels where prmary precptaton of ferrte predomnates, the lever-rule may gve better results. Accordng to our smulaton, the nfluence of carbon on the mcrosegregaton of chromum n the prmary Fcc phase n the Fe r system s due to the change of soldfcaton path upon the bac dffuson of carbon. Acnowledgements Part of ths wor s fnancally supported by the E-funded VESPISM proect. The authors wsh to than Prof. M. Hllert for constructve suggestons. REFEREES 1 A. Borgenstam, A. Engström,. Höglund, and J. Ågren: J. Phase Equlbra 21 ( G. M. Gullver: J. Inst. Metals 9 ( E. Schel: Z. Metalld., 34 ( M.. Flemngs: Soldfcaton Processng, (McGraw-Hll, ew Yor, Y, Saunders and A. P. Modown: APHAD alculaton of Phase Dagrams: A omprehensve Gude, (Pergamon, ew Yor, Y, M. Hllert,. Höglund and M. Schaln: Metall. Mater. Trans. A 3A

9 omputaton of Partal Equlbrum Soldfcaton wth omplete Intersttal and eglgble Substtutonal Solute Bac Dffuson 559 ( H. Fredrsson and. Hellner: Scand. J. Metall. 3 ( E. Kozeschn: Metall. Mater. Trans. A 31A ( H. D. Brody and M.. Flemngs: Trans. TMS-AIME 236 ( T. W. lyne and W. Kurz: Metall. Trans. A 12A ( I. Ohnaa: Trans. ISIJ 26 ( S. Kobayash: Trans. ISIJ 28 ( astac and D. M. Stefanescu: Metall. Trans. A 24A ( M. Hllert: Phase Equlbrum, Phase Dagrams and Phase Transformatons, (ambrdge Unversty Press, ew Yor, Y, Kaufman and H. Bernsten: omputer alculaton of Phase Dagrams, (Academc Press, ew Yor, Y, APHAD omputer ouplng of Phase Dagrams and Thermochemstry, (Pergamon, ew Yor, Y. 17 B. Sundman, B. Jansson and J.-O. Andersson: APHAD 9 ( B. Sundman and I. Ansara: n The SGTE aseboo Thermodynamcs at Wor, ed. K. Hac (The Insttute of Materals, ondon, UK, 1996, pp Q. hen and B. Sundman: unpublshed wor, B. Sundman and Q. hen, Thermodynamc alculaton Interface (TQ: Programmers Gude and Examples, (Thermo-alc AB, Stocholm, Y. Ono, T. Taech and K. Og: J. Japan Inst. Metals 57 ( TFE Thermo-alc AB steel database, Thermo-alc AB, Stocholm, MOB2 Moblty database, Thermo-alc AB, Stocholm, A Gude to the Soldfcaton of Steels, Jernontoret, Stocholm, 1977.