Growth of irregular eutectics and the AlSi system

Acta metall, mater. Vol. 39, No. 4, pp. 469~480, 1991 Printed in Great Britain. All rights reserved

0956-7151/91 $3.00+ 0.00 Copyright © 1991 Pergamon Press plc

GROWTH OF IRREGULAR EUTECTICS A N D THE A1-Si SYSTEM P. M A G N I N t , J . T. M A S O N and R. T R I V E D I Ames Laboratory, USDOE and Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011, U.S.A. (Received 18 December 1989; in revised form 24 August 1990) Abstract--The directional solidification of the lamellar A1-Si eutectic alloy is studied experimentally. It is shown that the average undercoolings and spacings lie on the theoretical AT-2-V~ curve, but not at the extremum point. The operating factor, ~b, describing this departure from the extremum conditions is experimentally shown to be independent of the growth velocity. The minimum spacings observed in the irregular microstructure formed are shown to be higher than the extremum value. The results are compared with those previously obtained in the Fe--C and Fe--Fe3C alloys, the latter being shown to be an irregular eutectic despite the regularity of the microstructure formed. The theory of irregular eutectic growth is reexamined, based on the new model previously developed. A solution is obtained in the case of a non-isothermal s/1 interface, which is identical to the isothermal solution if a weighted average undercooling is used. The effect of a kinetic undercooling term is studied, although it can usually be neglected even for irregular eutectics. It is shown that irregular eutectics grow with a nearly isothermal s/1 interface, the different spacings observed in the microstructure being characterized by different growth rates. These rates oscillate locally around an average value, and significant departures from the steady-state conditions occur, The operating range is described by its average value (~b) and its extent (~/). It is shown experimentally that the latter is always very close to the value of the average spacing, independent of the q~-value. Rrsumr---La solidification de l'eutectique lamellaire A1-Si est 6tudire exprrimentalement. On montre que les surfusions et espacements interlamellaires moyens se trouvent sur la courbe throrique A T - 2 - V , mais pas ~ l'extrrmum. Le facteur oprrationnel ~b drcrivant cette diffrrence par rapport au point extrrmum est obtenu exprrimentalement comme une constante, indrpendante de la vitesse de solidification. On montre que l'espacement minimum mesur6 dans la microstructure irrrgulirre est suprrieur ~ la valeur correspondant au point extrrmum. Ces rrsultats sont comparrs ~ ceux obtenus prrckdemment dans les alliages Fe~2 et Fe-Fe3C, ce dernier se comportant comme un eutectique irrrgulier bien qu'il conduise ~i une microstructure rrgulirre. La throrie de la croissance eutectique irr6gulirre est rr-rxaminre sur la base d'un nouveau modrle drvelopp6 prrcrdemment. Une solution est obtenue dans le cas d'une interface non-isotherme. Cette solution est identique ~t celle correspondant au cas isotherme si l'on utilise une surfusion "moyenne". L'effet d'une surfusion cinrtique est 6tudir, bien qu'il puisse grnrralement ~tre nrglig6 m~me dans le cas des eutectiques irrrguliers. On montre que la croissance des eutectiques irrrguliers conduit ~i une interface presque isotherme, les diffrrents espacements interlammaires observrs dans la microstructure 6tant caractrrisrs par diffrrentes vitesses de croissance. Cette dernirre oscille localement autour d'une valeur moyenne, conduisant ~ des conditions non-stationnaires. Le domaine d'espacements interlamellaires observrs est reprrsent6 par sa valeur moyenne (q~) et son 6tendue (~/). On montre exprrimentalement que cette dernirre est toujours de l'ordre de grandeur de l'espacement moyen, indrpendamment de la valeur de ~.

Zusammenfassung--Gerichtet erstarrte eutektischer A1-Si Lamellen werden untersucht. Die Autoren zeigen, dass die Unterkiihlungen und die mittleren Lamellenabst~inde auf der theoretischen A T - 2 - V Kurve liegen, aber nicht am Extrempunkt. Der experimentell erhaltene Faktor ~, der dieses Abweichen vom Extremum beschreibt, h/ingt nicht vonder Wachstumsgeschwindigkeit ab. Die gemessenen Minimalabst~inde der unregelm~issigen Mikrostruktur sind gr6sser als die entsprechenden Werte am Extrempunkt. Sic werden mit friiher erzielten Resultaten der Legierungen Fe-C und Fe-Fe3C vergiichen; letztere verh/ilt sich wie ein unregelm/issiges Eutektikum, bildet aber eine regelm~issige Mikrostructur. Die Theorie fiber unregelm~issiges eutektisches Wachstum wird an Hand eines neuen Modells, das kfirzlich entwickelt wurde, fiberpr/ift. Ffir den Fall einer nicht-isotherrnen s/1 Grenzfl/iche wird eine Lrsung erhalten. Sie is mit einem isothermen Fall identisch, wenn eine "mittlere" Unterk/ihlung verwendet wird. Die Auswirkung einer kinetischen Unterkfihlung wird untersucht, obwohl sic normalerweise ffir regelm~issige und unregelm~issige Eutektika vernachl/issigt werden kann. Es wird gezeigt, dass unregelm~issige Eutektika mit einer fast isothermen Grenzfl~iche wachsen und dass die beobachteten unterschiedlichen Lamellenabst~inde der Mikrostruktur dutch verschiedene Wachstumsgeschwindigkeiten charakterisiert werden. Letztere oscillieren lokal um einem Mittelwert und ffihren zu nichstationn~iren Bedingungen. Der Bereich der beobachteten Lamellenabst~inde wird dutch seinen Mittelwert (~b) und durch seine Ausdehnung (r/) dargestellt. Experimentell wird gezeigt, dass letzterer immer in der Gr6ssenordnung des mittlerem Abstandes liegt, der unabhfingig yon ~b ist. tPresent address: Laboratoire de Mrtallurgie Physique, Ecole Polytechnique Frdrrale de Lausanne, CH-Ecublens, CH-1015 Lausanne, Switzerland. 469

470

MAGNIN

et al.:

GROWTH OF IRREGULAR EUTECTICS

INTRODUCTION The theory of eutectic growth has been studied by a number of authors and it is now rather well understood for the growth of regular eutectics [1, 2]. According to these models, which are based on the diffusion in the liquid and on the thermodynamics of interfaces, the interface undercooling, AT, is related to the solidification rate, V, and the lamellar spacing, 2. Consequently, one still has to determine the operating range, that is the spacing(s) actually selected during the eutectic growth under a given growth rate condition. For a given growth rate, the interface undercooling goes through a minimum as a function of lamellar spacing, and it has been proposed that the growth occurs at this minimum (spacing 2~x) which corresponds also to the maximum growth velocity for a given undercooling. Experimental studies, on the other hand, indicate that the actual average spacing of irregular eutectic is larger than 2e~, so that the average eutectic spacing can be characterized by 2 = ~ 2~x [3, 4], where tp is an operating parameter reflecting the spacing adjustment mechanism that is different from the minimum undercooling principle. For regular eutectics, the competition mechanism between the growing phases causes the growth to occur very close to the extremum spacing [5], and the q~-value is then close to unity. Most of the eutectic alloys of practical interest (Fe-C, AI-Si) are irregular eutectics. They are characterized by the presence of a faceted phase, whose growth kinetics are strongly affected by planar defect mechanisms and therefore they are very anisotropic. The growth of each lamellae of this phase occurs preferentially in a direction determined by its local crystallographic orientation, thus preventing the spacing adjustment mechanism to drive the growth toward the minimum undercooling point. As a result, the solidification of irregular eutectics occurs within a whole range of spacings growing simultaneously, according to the following mechanism (Fig. 1) [6]: when two lamellae converge, the growth of one simply stops when 2 becomes smaller than a critical spacing, 2 ~ , below which the surface energy effect prevents further growth. Conversely, the solidification of diverging lamellae occurs until another critical spacing, 2br, is reached, whereupon the growth becomes unstable and one of the lamellae branches into two diverging lamellae, thereby reducing the spacing. Growth of irregular eutectic has been studied theoretically by several authors [6-8], emphasizing two modifications of the regular eutectic growth theory: • The isothermal s/1 interfce assumption made in the regular model was assumed to be no longer valid for irregular growth [7] and was subsequently relaxed, leading to a numerical [6] as well as an analytical [8] solution. The latter was, however, shown to be negligibly

Fig. 1. Growth mechanism of irregular eutectics, as proposed by Fisher and Kurz [6]. different from that for the regular eutectic growth. • A steady state growth with an operating point significantly different from the minimum undercooling value was considered (Fig. 2). The minimum spacing, 2 n , was assumed to be the extremum value, 2e~, while the maximum spacing, 2br, was determined by a stability analysis [6] or a morphological criterion [8]. The average spacing, which is close to the arithmetic average between 2~n and 2br, was described by the quantity ~ = ~b2~x, using a tk-value much greater than 1. The ~b-value could be a slight function of the thermal gradient [6] as well as of the velocity [8] at extremely low growth rates.

/ 4

hmin

I

0

I

I

I

i

I

i

I

2

3

4

5

6

lamellar spacing [~.rn]

Fig. 2. Operating range irregular eutectics, according to Jones and Kurz [4].

MAGNIN et al.: GROWTH OF IRREGULAR EUTECTICS These theoretical studies were, however, based on the Jackson Hunt model. One of the goals of this paper is to reexamine the growth of irregular eutectics with the new approach presented by Magnin and Trivedi [2], which does not imply the assumption of an isothermal s/1 interface. In addition, it will be shown that irregular eutectic growth is fundamentally a non-steady-state process, which does not follow the growth behavior shown in Fig. 2. The operating range must, therefore, be entirely reexamined. In order to check the validity of the proposed theory, good measurements of the spacings and undercoolings as a function of the growth rate are needed for some irregular eutectic alloys. Unfortunately, due to the irregularity of the microstructure formed and the wide variety of morphologies which can occur, reliable data are difficult to obtain from the literature. Careful measurements have been carfled out in a previous study with the Fe--C eutectic system [9]. In this paper, the other industrially important irregular eutectic alloy (A1-Si) will be studied experimentally, emphasizing the spacings and undercoolings measurement in the lamellar microstrueture. The experimental results will be presented first and compared with those found in the literature for AI-Si, as well as for F e - C (reinterpreted with the revised physical constants established in Ref. [2]) in order to get more general information about the irregular eutectic growth mechanism. The theoretical model of irregular eutectic will then be examined with the help of these observations. EXPERIMENTAL STUDIES

Experimental The samples were prepared from pure A1 (99.999%) and Si (99.998%). Alloys of eutectic composition (12.6 wt% Si) were obtained by melting these elements under vacuum in a graphite crucible. The samples were cast (at about 620°C) into rods of 5 mm diameter in graphite coated alumina tubes, and then machined to about 4.7 mm diameter. The directional solidification experiments were carried out in a vertical Bridgman-type resistance heated furnace, described in [10]. The samples were contained in an alumina tube of 5/8 mm diameter (I.D./O.D.) in the center of the furnace. The lower part of the tube was placed into a water chamber in order to ensure a well defined thermal gradient. The maximum temperature in the furnace was 750°C, and the thermal gradient at the s/1 interface was 8 K/mm. At the beginning of the experiment, the samples, which were approximately 150 mm long, were heated until their upper parts (about 100 mm) were melted. A thermocouple (chromel-alumel) contained in an alumina tube of 1.5 mm O.D. was then placed about 50 mm into the liquid. After a 20 min holding time in order to stabilize the thermal conditions, the furnace and water chamber were raised at constant velocity with a screw driven by a computer controlled

471

precision stepping motor. When the s/1 interface had just crossed the thermocouple junction, the sample was quenched by raising the water chamber at a very high velocity (25 mm/s). A study was carried out in order to ensure that the steady-state (i.e. solidification rate = furnace velocity) was reached. Only the center part of the sample (about one third of the total length) was used, in order to avoid the heat flux changes associated with the end effects. All experiments were done under argon atmosphere in order to prevent any oxidation. The output from the thermocouple was continuously recorded. The thermal gradient and the quenching temperature could then be measured directly on the temperature-time curve. Note that measuring the quenching temperature on the T(t) curve allows to take account of the delay (typically 2 s) necessary to change the heat fluxes and to record them on the thermocouple. On the other hand, the distance between the quenched interface and the thermocouple junction could be measured very accurately (_+25/~m) in metallographic observations of longitudinal sections of the samples. The interface temperature could then be obtained from the above measurements. This technique is much more sensitive and reliable then the one generally used, based on a change in the slope of the T(t) curve when the s/1 interface crosses the thermocouple junction. The thermocouple was not strongly cemented to its alumina tube and was easily recovered and reused in all experiments, thus avoiding the need of a precise and difficult calibration. The thermocouple was simply calibrated in an indirect way by setting the interface temperature extrapolated to zero growth rate equal to the eutectic temperature. The lamellar spacings were measured by metallographic observation of transverse sections of the sample, about 10 mm behind the quenched interface to ensure being in the steady-state region. The average spacing was obtained from many measurements (typically over 150 lamellar spacings) in various representative areas of the sample. These measurements were carried out perpendicular to the general direction of the lamellae in very carefully selected regions where the latter is more or less kept constant over some area within the grains. However, these regions were not limited to those where the lamellae are strictly parallel (as often seen in the literature), since the latter are not representative of the overall microstructure. The minimum spacing was determined with about 20 measurements of the distance at which two converging lamellae stop growing, as well as of the minimum distance observed between two parallel lamellae (both values giving the same result). A good criterion for the determination of the maximum spacing could not be found. The latter was therefore not measured directly. However, it can be shown that the average spacing is very close to the arithmetic average between the minimum and maximum spacings. The maximum spacing can then be

MAGNIN et al.: GROWTH OF IRREGULAR EUTECTICS

472

Table 1. Summary of the experimental results obtained with the lamellar A1-Si alloy (G = 8 K/mm) V

AT

~,

~m/s) (K) (,um) 0.1 - - 108 2 0.3 24.2 10 0.9 10.7 50 2,3 4.8 100 3.0 3.3 200 4.5 2.5 350 5.4 1.8 500 6.9 1.57 10 8.0 100 2.6

J'min

'~p

(#m) (#m) 33 13.3 9.7 5.9 3.9 2.7 1.8 1.3 1.03 0.95 "~ G = 16Kmm J

58O Te=577.2

e

~" 575 8 - 570

,,Iv , i , 51 i , i i .v[~rn/s] 2 I0 0 I00 200 350 500 Fig. 3. Measured s/1 interface temperature as a function of the growth velocity. The solid line corresponds to the predictions of the theoretical model [equation (3)], using the physical constants given in Table 2 (D = 4.3.10 -9 m2/s, ,k = 3.2). determined from the previous measurements by the relation 2br = 2~[ -- )~in.

Results The results of the growth temperature measurements are given in Table 1 and in Fig. 3. The theoretical proportionality relationship between the growth undercooling and the square root of the solidification rate is shown to be very well verified, giving a A T / x / ~ constant of 3 0 6 K m - m s m. The very high accuracy of these results demonstrates the advantage of the experimental method used, which IO0 ~ E

Fig. 5. Silicon patterns observed in the slowly solidified microstructures (2 Itm/s). Transverse section, as polished.

allows both very precise measurements and the use of the same thermocouple in order to avoid calibration errors. The results of the lamellar spacing measurements are given in Fig. 4. Both the average and minimum spacings are shown to follow very well the theoretical relationship, giving 2 z V constants of 1158.10-is m3/s and 350.10-1Sm3/s, respectively. A t low solidification rates, the silicon is observed to occasionally form complex patterns (Fig. 5) characterized by a spacing, 2p, slightly lower than the minimum spacing observed in the rest of the microstructure. When the solidification rate is as low as 0.1/~m/s, the microstructure consists mostly of twin lamellae, as shown in Fig. 6. A few experiments were carried out with a higher thermal gradient (16 K / m m ) produced by increasing the maximum furnace temperature to 950°C. The microstructure consisted of shorter lamellae and exhibited about 25% lower spacings than those obtained at the lower thermal gradient. The thermocouple used was unfortunately broken at the beginning of these experiments, and the growth undercoolings were therefore not measured.

t/}

Comparison with the literature q

0.10.1

,I

I0]

L I00

,. I000

Growth velocity [p.m/s]

Fig. 4. Measured average (~] and minimum (2~.) spacings as a function of the growth velocity. The solid lines correspond to the theoretical predictions [equation (2)], with ,~-values of 3.2 (~[), 1.76 (2~i,) and 1 (2ex), respectively. x : spacings, 2p, of the patterns shown in Fig. 5.

The experimental results obtained in this work are difficult to compare with those found in the literature, since the latter exhibit considerable disagreements and are often interpreted according to growth laws different from the 22V = c o n s t a n t and AT/x ~ = constant relationships, emphasizing a thermal gradient influence which is actually probably only an indirect effect. The experimental ~2 V constant of 1158" 10 -18 m3/s obtained for the average spacings can be considered

MAGNIN et al.: GROWTH OF IRREGULAR EUTECTICS

473

reported very high undercooling values ( A T / v / V = 1000 and 1220Km-1/2s 1/2, respectively) which are probably due to an experimental problem, as already suggested by Elliot and Glenister [13]. Extremely high undercoolings have been reported by Hogan and Song [11] (AT/x/'-V = 1680 K m -1/2 s 1/2) and Glenister and Elliot [15] ( A T / v / V = 1738 K m - m S1~2) for the fibrous microstructure, obtained by alloying with strontium. No attempt was made to confirm or deny this result, which will be discussed in the kinetic undercooling section of the theoretical part of this work.

Comparison with existing theoretical models The average spacings and undercoolings measured for different growth velocities will now be compared with the theoretical model. The growth rate and the interface undercooling are related by the following equation [1, 2]

A T = KIAV + K2/2 Fig. 6. Twin lamellae microstructure produced at extremely low growth rates (0.1 #m/s). Transverse section, as polished. to be in good agreement with those given for comparable thermal gradients by Hogan and Song [l l] (770 "10 -18 ma/s) and Toloui and Hellawell [12] (800-1000"10-1Sma/s), if one notices that those measurements were carried out only in the regions were the lamellae are strictly parallel, leading to values slightly lower than the true average spacing of the microstructure, as measured in this work. It can be shown that the 10-20% difference between the spacings obtained can be attributed to the different measurement methods used. The A T / ~ - V constant obtained here (306 K m -~/2 s 1/2) is slightly lower than the one reported by Hogan and Song (464 K m - m sin). The measurement technique used in the latter paper (analysis of the changes in the slope of a T(t) curve) is believed to be more difficult to calibrate and less reliable than the method used in the present work. The experimental results are, however, in disagreement with those of Elliot and Glenister [13], who found 1.5-2 x higher spacings following a relationship close to ~[3V = constant, and twice larger undercoolings, and A T / v / V = constant law becoming a AT/V°25=constant relationship at growth rates slower than about 80#m/s. These differences are probably related to a different growth morphology of the silicon phase due to the very low thermal gradient used in their work (0.8 K/ram). The microstructures and the low growth rate-undercooling law given there seem to indicate the growth of primary silicon particles rather than a coupled eutectic solidification mechanism. In addition, it is not sure that the very low thermal gradient used was sufficient to allow the occurrence of a well controlled directional solidification. For a thermal gradient of 8 K/mm, Toloui and Hellawell [12] and Steen and Hellawell [14] have

(1)

with inc. P Kl--D f~h

[F~ sin 0~

Fp sin 0,\

where m i is the liquidus slope of the phase i (i = e, fl), including the sign (i.e. m~<0), ~--Im~lmp/ (Iraqi + rap) is a weighted liquidus slope, f~ andfp are the volume fractions of the phases, P is a function of the volume fractions, approximately given by 0.335 (f~f~)l.6S, C* is a weighted eutectic tie-line length [21, D is the diffusion coefficient in the liquid, and Fi and 0~ are the Gibbs-Thomson coefficients and contact angles imposed by the surface tension balance at the three-phase junction. The above result is for the non-optimized case. If the operating point of eutectic growth is assumed to occur at a spacing 2 = ¢2ex, then the eutectic growth can be described by the following relationships [4]

22 V = ¢2Kz/K ~ A r / x / ~ = (c~ + 1/cb)x//-~K2 ,~AT = (dp2 + 1)K2.

(2) (3) (4)

The experimental results on average spacings and undercoolings, measured at different velocities, are compared in Fig. 7 with the theoretical non-optimized relationship [equation (1)], using the physical constants given in Table 2. As discussed in Ref. [2], these values were established very carefully and should be reliable. The value used in Fig. 7 for the diffusion coefficient, D = 4.6.10-gm2/s, is the one measured directly by Petrescu [16], with the density correction discussed in [2]. The theoretical curves drawn in Fig. 7 are then obtained completely from independent measurements and should be accurate.

MAGNIN et al.: GROWTH OF IRREGULAR EUTECTICS

474

' I0 -

500~m/s

200~m/s =~ _._ "J~,~"~

~'~ ! m .~

i

0.i 3.I

IO~.m/s

I

I

I0 I00 lomellur spocing [H.rn] Fig. 7. Comparison between the theoretical non-optimized A T - 2 - V relationship [equation (l)] and the experimental average spacings and undercoolings measured at different growth velocities. The theoretical curves (solid lines) are obtained completely independently from the experimental results, with the D value measured in [16]. Table 2. Physical properties of the lamellar Al-Si eutectic system, according to [2] ~IAJ) #(si) c~" (wt%) 87.7 m (K/wt%) -7.5 17.5 f (--) 0.873 0.127 F (10 7mK) 1.96 1.7 0 (o) 30 65

Comparison with the experimental results obtained here then leads to two conclusions: • the growth of A1-Si eutectic does not occur at the extremum spacing, 2ex, *the average spacings and undercoolings obtained correspond to a point lying on the theoretical A T - 2 - V relationship, to the right of the extremum spacing. These conclusions are obtained independently of any experimental errors or unknown parameter. This demonstrates that the operating point factor, ~, represents a physical mechanism (analogous to the a* parameter for dendrites) and not simply a fitting parameter added in order to correct the predictions of the theoretical model. In addition, Fig. 8 shows

that the ~b-value is found experimentally to be independent of the growth velocity. As discussed in Ref. [2], the experimental ~2 V and AT/x//ff constants also allow one to calculate the value of the diffusion coefficient if the system parameters other than the operating point factor are known. From our experimental results, the best fit is obtained for D = 4.3" 10 -9 m2/s with ~b = 3.2. Furthermore, since this value lies within the experimental error of the direct measurement, it will be used in the rest of this work. The experimental results are compared in Figs 3 and 4 to the theoretical relationships [equations (3) and (2)]. Although the measured minimum spacings, 2~i,, are shown to follow the 2~nV = constant relationship, Fig. 4 shows that they are definitely larger than the extremum spacings, 2ex, in contrast to what was previously believed. Note that a similar result is obtained from the lamellar spacings studies in gray cast iron [9, 17] when the revised physical constants determined in [2] are used, as shown in Fig. 9. At extremely low velocity (0.1 mm/s), The microstructure consists mainly of twin lamellae separated by a distance corresponding to the extremum spacing (Fig. 6). Note that, although the lamellae are much more parallel and form a more regular structure than at higher growth rates, the ~[2V constant obtained with the average spacing (and thus the operating point factor, q~), is unchanged. This result was also observed in cast iron [18]. As shown in [8], the influence of the thermal gradient on the eutectic growth is negligible as far as the microstructure remains perfectly lamellar. However, the gradient has been found to affect the growth morphology. It has been shown in the case of cast iron that a morphological change (in Fe--C, a degeneration of the graphite lameUae with increasing velocities at high solidification rates) leads to a modification of the spacings [9, 17]. The thermal gradient can then influence the spacings (i.e. the operating factor) only by an indirect effect, thus explaining the disagreements found in the literature. The 25% reduction in the lamellar spacings observed

~100o

o ,

I0.01 [

,,

O. I

I

I

I

I0 I00 Growth VeLocity [~am/s]

I •

I000

Fig. 8. Experimental operating point factor, ~b obtained from the spacings measured at different growth velocities.

0.1

I

/

I0

.

I00

Growth velocity [/~m/s] Fig. 9. Average (:;[) and minimum (2~i.) spacings measured purely lamellar Fe~C eutectic microstructure [9, 17], and comparison with the theoretical relationships (solid lines), using the physical constants given in Ref. [2].

MAGNIN et al.: GROWTH OF IRREGULAR EUTECTICS

I0 E

Eo,

~

"'k3v

=

const.

0.1

0'0110

I

I

I

102 I03 104 Growth velocity [/zm/s]

I m

105

Fig. 10. Average spacings measured at low (directional solidification [9]) and high (laser surface treatment [18]) growth rates in Fe-Fe 3C, and comparison with the theoretical relationship (solid lines) as well as with the 23V = constant relationship (dashed line) often found in the literature. in this work for a twofold increase in the gradient corresponds to the G -1/3 dependence reported by Toloui and HellaweU [12]. This relationship is, however, valid only in a relatively narrow range of thermal gradients (as far as a good correspondence between G and the morphology of the microstructure formed can be established) and has been considered as overestimating the influence of G by several authors [11, 13]. As predicted by theory, no representative spacing below the extremum value, 2ex, could be found in this study, even though special morphologies have been observed (Figs 5 and 6). This result is valid even in the case of the Fe-Fe3 C alloy (white cast iron). The spacings merasured in Fe-Fe3 C at low (directional solidification [9]) and high (laser surface treatment [19]) growth rates are compared in Fig. 10 with the theoretical relationship, using the physical constants determined in [2]. One can see that, despite the regularity of the microstructure formed, this system is actually an irregular eutectic growing with an average ~b-value of 1.8, as also indicated by the faceted behavior of the cementite phase [2]. However, instead of having a range of spacings growing simultaneously, the system selects only one value which oscillates around an average 22V = constant relationship, staying always above the 2ex value. Because of these departures, a ~3V = constant relationship has often been reported in the literature for this alloy. IRREGULAR EUTECTIC GROWTH THEORY

The theory of regular eutectic growth has been reviewed in other papers of this series [2, 5]. In order to treat the problem of the growth of irregular eutectics, two aspects need to be reconsidered: • the validity of some of the assumptions made in order to establish the relationship between the solidification rate, the growth undercooling and lamellar spacing (equation [1]) has to AM 39/~-D

475

be re-examined. The hypothesis which are most likely to be affected by the specific conditions produced by the irregular growth are: the isothermal s/l interface, the simplified geometry assumed for the solute balance, the negligible kinetic undercooling, and the steady-state growth. .The operating range of irregular eutectics is controlled by a very different mechanism than the nearly extremum growth of regular eutectic alloys, and must therefore be totally reconsidered. These different theoretical aspects will be examined in this section. Non-isothermal s/I/interface

The solidification of irregular eutectics occurs with a wide range of spacing growing simultaneously. Because of this, the s/1 interface is much less fiat and isothermal than in the case of regular eutectics, and the validity of these assumptions must therefore be examined carefully. Assuming for now that the kinetic undercooling is negligible (which will be discussed in a subsequent section), each point of the s/1 interface must be in thermodynamic equilibrium. This condition can be written T[x, z = I(~)] = Te - ATe(x) - ATr (x)

(5)

where T[x, z = I(x)] is the temperature imposed by the growth conditions at the s/1 interface, described by the relation z = I(x), and ATe(x) and ATr(x) are the local solute and curvature undercoolings, whose values are given in Ref. [2]. As shown in the latter paper, the surface tension balance at the three phasejunction imposes fixed contact angles. This additional condition is fulfilled by an adjustment of the boundary layer composition, and can be taken into account by equalizing the equilibrium contact angles, 0~ and 0a, to those calculated from the integration of the interface curvature given by equation (5). Although the calculations were carried out in Ref. [2] for the case of an isothermal s/1 interface, the latter condition was introduced only as a simplification which can now be relaxed. Supposing that the thermal gradient in the liquid, G, is in the z-direction (perpendicular to the s/1 interface) and is nearly constant around the interface, the local growth undercooling, AT(x) = T¢ - T[x, z = I(x)], can be expressed as AT(x) = AT ° - GI(x)

(6)

where A T ° = T o - T(x, z = 0) is the growth undercooling at the level of the three-phase junction. Substituting equation (6) into the curvature expression (equation (14) of Ref. [2]) and integrating it as described in [2] leads to AT° = K l g V + K:/2 + F

(7)

476

MAGNIN et aL: GROWTH OF IRREGULAR EUTECTICS

with

F = FnG ~ - ~ I + - ~ ) where 7 ~ and fa are the average positions of the s/l interface of ct and fl phases, respectively. Equation (7) differs from the solution obtained with an isothermal interface only in the corrective factor, F. Note that this factor depends only on the average positions, 7 ~ and r a, and not directly on the actual interface shape, which is then not a very sensitive parameter. Therefore, although an exact solution could easily be obtained numerically, one can replace the actual s/l interface shape by an approximate solution without seriously affecting equation (7). As proposed by Fisher and Kurz [6], the s/l interface shape can be approximated in each half-lamella by a cubic curve given by the contact angles, the symmetry condition and the s/l interface position at the center of the phases, calculated from equations (5) and (6). Substituting this solution into equation (7) leads to an expression for the corrective factor, F, identical to that given in equation (13) of Ref. [8]. The coupling condition proposed in the latter paper is then equivalent to the s/1 interface equilibrium condition, and the solution proposed there is therefore valid. As shown in Ref. [8], the correction introduced for a non-isothermal interface is negligible, except for very large spacings obtained at extremely low growth rates (below 0.1 #m/s), in which the leading part of the s/1 interface becomes slightly less undercooled (in directional solidification, i.e. under positive thermal gradient) than predicted by the isothermal model. However, even in the latter case and without making any assumption for the s/l interface shape, expressing the local growth undercooling as AT(x) = AT' - G(I(x) - 7')

(8)

where ~-~i is the average undercoolings of the phase i (i = ~,/~), and following the procedure described in Ref. [2] leads to A"~P= m a A ~ + Im~IA--Ta= K 12V + K2/A. Irn~l + m~

(9)

The isothermal solution is then also the exact solution for non-isothermal interfaces if the weighted average undercooling, A"/', is used. Note that, except for extremely low growth rates, the temperature variations along the s/1 interface are negligible when compared to A"~P(see the operating range section). A"/" can then be considered simply as the growth undercooling.

Solute balance for the real interface shape In order to solve the diffusion problem, the solute balance boundary condition was established in Ref. [2] for a flat s/1 interface. The exact solution is very difficult to obtain because the real interface shape, and thus the boundary condition, depends in a complex way on the diffusion field. An analytical

solution can be found under certain conditions [20], but its use is not very practical and the assumptions made are very restrictive. A more general solution was obtained by Series et al. [21] with an electrical analogue experimental technique. Their numerical results show that the flat s/1 interface assumption has a negligible effect on the growth undercooling calculations. This conclusion can be explained intuitively by the following remark. Even when the s/l interface is not flat, as in the case of irregular eutectic growth, the amplitude of the function describing its shape, l(x), is at most of the order of magnitude of a few lamellar spacings, as can be seen in quenched interfaces (larger variations are sometimes reported in the literature, but they are related to the growth of primary phases and not to a eutectic s/l interface). At usual growth rates, the peclet number, p = 2V/2D, is very small. The amplitude of the interface shape is then always much smaller than the diffusion distance D/V. Therefore, the real interface shape cannot affect the general diffusion field, and only the short range composition variations close to the s/1 interface are modified. Since the latter are produced by a local solute rejection at the interface, they can be thought to simply follow the real interface shape, without any further modifications. A very good approximation of the interface composition is then given by the C(x, z = 0) value obtained with the flat interface boundary condition (and not, as already mentioned in [6], by C[x, z = I(x)], which doesn't take into account the influence of the actual interface shape on the short range composition variations). In addition, it will be shown later that the irregular eutectic growth implies local departures from the steady-state conditions. In this case, such departures will affect the local composition more than the variations of the interface shape, which can then be neglected.

Kinetic undercooling The assumption of negligible kinetic undercooling, ATk, is known to be well satisfied for most regular eutectic systems. However, in the case of irregular eutectics, one of the phases is faceted and could exhibit a significant kinetic undercooling. It has been proposed that this effect could modify the growth mechanism [22]. This conclusion was, however, introduced mostly in order to correct the predictions of a theoretical model assuming the growth at the extremum conditions, which is now known not to be valid for irregular eutectics. The kinetic undercooling is not believed to affect seriously the growth of most irregular eutectic for several reasons: o First, the average values of undercoolings and spacings measured in very irregular AI-Si and F e - ~ systems have been shown above to be predicted perfectly without considering a kinetic undercooling. These results show that

MAGNIN et al.: GROWTH OF IRREGULAR EUTECTICS the kinetic undercooling cannot have a very large effect on the growth mechanism of these systems. • Second, most faceted phases found in metallic systems exhibit growth difficulties with high kinetic undercooling in specific crystallographic directions only. Since their growth occurs only in favorable directions, the resulting kinetic undercooling is therefore not likely to be much higher than that in the case of non-faceted phases. e Finally, the growth of irregular eutectics occurs far from the extremum conditions, i.e. with much higher total interface undercooling than for regular growth so that the ATk value for a faceted phase remains negligible when compared to AT. The kinetic effect could become important if impurities are present in the system. For example, when Sr or Na atoms are added to produce a fibrous A1-Si microstructure, they were shown to be selectively adsorped on specific cryostallographic sites on the silicon particles [23]. If this mechanism prevents the growth in the favorable directions observed for the lamellar structure (which is consistent with the rod morphology), a high kinetic undercooling of the silicon phase can result [24]. The kinetic undercooling influences the eutectic growth as follows: as shown in [2], the volume fractions of the phases regulate the composition of the boundary layer in order to ensure similar growth undercoolings of both phases. If the undercooling of the faceted phase is raised by a high ATk value, this mechanism modifies the boundary layer composition in order to reduce the chemical undercooling of this phase. The influence of the kinetic undercooling is then distributed over the whole s/1 interface. Introducing a ATk term in equation (5), the treatment described in Ref. [2] leads to

AT=KI2V+K2/2 Jr m~-A-T~+Im~I-AT~ (10) Im~l + rnp and

6C=6c + ~

Im~lJ

where A---T[and ~ are the average kinetic undercoolings of ~ and fl phases (the one corresponding to the non-faceted phase being negligible), 6C is the difference between the average s/1 interface composition and the eutectic composition, and JC ° is the corresponding value calculated without kinetic undercooling (equation (21) in Ref. [2]). The kinetic undercooling represents a local mechanism at the s/1 interface and is, therefore, independent of the spacing. The extremum spacing, 2ex, obtained from equation (10) is then identical to that given by equation (1). Consequently, as long as the operating point factor, ~b, is unchanged, the effect of the kinetic

477

Table 3. Spacings (~[exp)and growth undercoolings (AT exp) measured by Hogan and Song [11] in the fibrous A1-Si microstructure, and comparison with the predictions of the theoretical model with negligible kinetic undercooling (~[th, ATth). The kinetic undercooling for the Si phase (ATSki) that would be necessary to explain the differences between the calculated and experimental undercoolings is obtained from equation (10) V ~[cxp ~th A T exp A T th A Tksl (/~m/s) (#m) (#m) (K) (K) (K) 10 20 40 50 80 100

4.7 3.6 2.9 2.6 2.1 1.8

5.5 3.9 2.7 2.4 1.9 1.7

5.50 7.47 10.24 12.06 15.01 16.69

0.68 0.96 1.36 1.52 1.92 2.15

14.9 20.1 27.5 32.6 40.5 45.0

undercooling of the faceted phase upon the solidification process is to raise the average undercooling by a quantity given by the last term of equation (10), without changing spacing. The experimental results obtained by Hogan and Song [11] for the fibrous A1-Si microstructure are compared in Table 3 to the predictions of equation (10), using the solution established for the rod geometry [2] and assuming as a first approximation that the diffusion coefficient value and the physical constants given in Table 2 for the lamellar A1-Si are still valid for the fibrous eutectic. However, since the growth of the silicon phase is in this case nearly equally difficult in every direction, the growth mechanism described in Fig. 1 is less likely to occur. The microstructure formed then becomes much more regular and the alloy solidifies with a lower ~b-value. As can be seen in Table 3, good agreement is obtained for the spacings when a ~b-value of 2.2 is used. The kinetic undercoolings calculated for the Si phase are, however, extremely high and proportional to the square root of the solidification velocity, rather than the usual linear or square dependence on V for rough or stepped interfaces, respectively. Even if the growth of the rod AI-Si eutectic is really affected by a kinetic undercooling effect (i.e. if the experimental undercooling values are reliable and if a small addition of Sr, typically 0.05 wt%, cannot significantly change the physical constants), this effect would be due to a very unusual growth mechanism. Therefore, the kinetic undercooling can be neglected in most irregular eutectic alloys, and need to be considered only in very special cases.

Steady state growth We now consider the case of two lamellae growing in a converging way, for which the spacing will change from •br to 2=, over a distance of the order of a few average spacings. The corresponding undercooling variations given at constant growth velocity by equation (1) can easily be as high as several degrees or more, as can be seen in Fig. 2. With a typical thermal gradient of 10K/mm, this would correspond to a distance of at least one hundred

MAGNIN et al.: GROWTH OF IRREGULAR EUTECTICS

478

spacings, in contradiction with the assumed geometry. As the converging lamellae are growing the s/1 interface temperature is imposed by the thermal conditions and the interface undercooling is therefore much more constant than the fast decreasing one predicted at constant velocity by equation (1). As a result, the local solidification rate is increased up to the value corresponding to the imposed s/1 interface temperature. For diverging lamellae, the opposite effect is observed. During the growth of irregular eutectics, the local solidification rate is therefore not constant, but oscillates continuously around the average value imposed by the experimental conditions. Significant departures from the steady state can then occur locally. The change in the boundary layer composition, AC, due to a variation of the growth velocity, A V, can be estimated according to the simplification proposed by Jackson and Hunt for the equations developed by Smith et al. [25]. For the solidification at an average growth rate, V, of a liquid of exact eutectic composition, Co, the average composition in the boundary layer is Ce + 6C and obtains IAC[~<

1+ CJ

P l"

(11)

As shown above, the actual variations of the s/l interface temperature are one or two order of magnitude lower than those predicted at constant velocity by equation (1). As a first approximation, the growth undercooling can then be assumed to be a constant, AT, and the local growth velocity is given as a function of the spacing by V=

AT - K2/2

(12)

It can be seen from the experimental results that 2=, is of the order of the half of the average spacing, 2, implying that 2b~~ 1.5 ~ since it can be shown that ~ ~ (2~n + ~,br)/2. Introducing these values into equation (12), one can show that the corresponding change in growth velocity, A V, is of the order of the average velocity, F. Since 3C is negligible when compared to C~ [2], one finally gets IACI ~< 16C/21.

03)

The changes in the boundary layer composition due to the non steady-state conditions are then small, at least for the solidification of a liquid of eutectic composition. In addition, these variations can be compensated by an adjustment of the volume fractions of the phases, as shown in Ref. [2]. However, the eutectic growth mechanism is extremely sensitive to the composition at the s/1 interface. If the variations in the boundary layer composition are too fast, the thermodynamic equilibrium condition for the s/1 interface [equation (5)] can no longer be fulfilled. As shown in [2], this leads nearly immediately to cut off the growth of one phase. Therefore, the effect of the departure from the steady-state conditions has to be

considered for the determination of the operating range. Equation (12) can be integrated in order to calculate the average velocity obtained when two lamellae are growing at a given undercooling, AT, with a spacing varying from 2~n to ~'br (or conversely). This velocity, V, can be used in equation (1) to determine the corresponding average spacing. Provided that the faceted phases are growing straight, it is found that this spacing is within + 1% of the average spacing derived from the microstructure, ~ = (2~an+ 2br)/2. Despite the departure from the steady state conditions, equation (1) is then still valid for the average AT, ], and V" values, as verified experimentally in Fig. 7.

Operating range As explained by the mechanism described in Fig. 1, the main difference between the regular and irregular eutectic growth is that, in the latter case, the growth direction of each lamella of the faceted phase is determined mostly by its own crystallographic orientation, independently of the heat flow direction. The microstructure produced is then irregular, thus leading to a whole range of spacings growing simultaneously. The problem which has to be solved is to determine the two limits, 2min and 2br, within which the growth occurs. Traditionally, 2~n was considered to be equal to 2ex, while 2br was determined by a stability or morphological criterion [6, 8]. However, the experimental results and the mechanism depicted in Fig. 11 shows that 2~n is larger than 2ex. One must therefore obtain a criterion for both 2~n and 2hr. The problem is further complicated by the fact that the growth does not occur at the steady-state. Actually, the non steady-state conditions could even constitute the fundamental reason for the determination of ,~n and ~-br, the growth of one phase being cut off by the departure from the s/1 interface composition needed in order to ensure thermodynamic equilibrium. In addition, equation (1) was established for the growth of a microstructure with a uniform spacing. In reality, different spacings are growing simultaneously. The growth conditions are then not only a function of the single spacing between the two lamellae considered, but depend on all spacings growing in the neighborhood. For the irregular eutectic the opening range can be described by the two dimensionless parameters, ¢ and q, defined as

¢ = 2¢---~ rt=

(14)

)~br -- •mila

The ~b-value has been shown experimentally to be very well defined and independent of the growth velocity. When the solidification rate is reduced to

MAGNIN

GROWTH OF IRREGULAR EUTECTICS

et al.:

(a) 300

~200 3o lO0 _~

l

I

I

2

I

)

5 4 5 Iornellor spocing

I

6

Co)

I

V=ITO~oop.m/s

5 4

~__~~/~rn

z~3

i2 ~

Is Xb¢

o

,

spacings, 2m~n and 2br , are not determined by two different phenomenon, but rather by a global mechanism responsible for the extent of the operating range (t/) as well as for the average operating point (~b). Interestingly, the ~b-value obtained in the Fe-Fe3 C alloy is the minimum value compatible with the 2n~n~ ~/2 relationship, 2r~n being in the latter case equal to 2ex. Although it could be closely related with a departure from the steady-state compositions, the exact mechanism responsible for the determination of the operating range is still not clear. In particular, the influence of the initial composition of the liquid (and possibly of the thermal gradient) need further experimental and theoretical investigation. Unfortunately, it seems that only non steady-state calculations in a non-periodic microstructure (i.e. with different spacings growing simultaneously) could lead to a complete description of the operating range of irregular eutectic growth. CONCLUSIONS

I I

479

I

3

I

;

lamellospaci r ng[p.m]

6

Fig. 11. Operating range of irregular eutectics with local departures from the average solidification rate, shown (a) on the V-). curve, (b) on the AT-). curve. Numerical values are for the A1-Si eutectic solidifying at an average velocity of 100 #m/s. extremely low values, the growth of the faceted phase has enough time to occur even along the unfavorable crystallographic orientations, and the lamellae can then be aligned parallel to the heat flow direction. The microstructure obtained in these conditions in the At-Si alloy (Fig. 6) shows that these parallel lamellae tend to grow at the extremum spacing, according to the regular growth mechanism. However, this occurs only in separate groups of two lamellae, and the average spacing of the microstructure is still much higher than 2ex. This observation constitutes very strong support that the mechanism responsible for the determination of the ~b-value is fundamentally independent of the growth velocity.. The r/ factor describes the extent of the eutectic range. It can be seen in Table 4 that the experimental t/-values are very close to 1 for the three alloys considered here, although the corresponding tkvalues range from 1.8 to 5.4. This suggest that the two Table 4. Experimentaloperatingrangeof differentirregulareutectic alloys Fe-C [9, 17] A1-Si Fe-Fe3C[9, 19] •2rV (fl nl3/s) 1213 2435 859 ~:V (#m3/s) 460 1158 410 2~inV (#m3/s) 65 350 125 (--) 5.4 3.2 1.8 n (--) 1.25 0.90 0.90

The directional solidification of the lamellar A1-Si eutectic is studied experimentally, and the results are compared with those previously obtained in the F e - C and Fe-Fe3C alloys, reinterpreted with the revised physical constants established in Ref [2]. In particular, it is shown that the Fe-Fe3C system is actually an irregular eutectic, despite the regularity of the microstructure formed. The spacing and undercooling values measured in the A1-Si alloy show independently of any experimental errors or any adjustable parameter that the average ~ and AT values lie on the theoretical A T - 2 - V curve, but not at the extremum spacing, 2ox. This provides an experimental justification for the use of the operating point factor, ~b, which is further experimentally shown to be independent of the growth velocity. The minimum spacings observed in the irregular microstructure formed are shown to be higher than the extremum value, in contrast to what was previously believed. Comparison with the results obtained in the F e - C and Fe-FeaC eutectics shows that the ~'min value is always of the order of the half of the average spacing, independent of the -value. The theory of irregular eutectic growth is reexamined, based on the model developed in [2]. A solution is obtained in the case of a non-isothermal s/1 interface. It is shown that the latter is, however, negligibly different from the isothermal solution, which is still exactly valid if a weighted average undercooling of the s/1 interface is used. Although the kinetic undercooling can usually be neglected even in the case of irregular eutectics, a solution including a ATk term is developed, showing that this leads to increase the average undercooling of the s/1 interface and to modify the boundary layer

480

MAGNIN et al.: GROWTH OF IRREGULAR EUTECTICS

composition, without changing the spacing as long as the operating point is not affected. It is shown that, although they are higher than for regular eutectics, the temperature variations along the s/1 interface of irregular eutectics are still very small. The different spacings observed in the microstructure are then characterized by different growth rates and not as previously believed by a variable growth undercooling. The solidification occurs with a velocity continuously oscillating around an average value, and significant departures from the steadystate conditions can occur locally. The mechanism responsible for the determination of the operating range is unknown, and can probably be fully explained only by considering departures from the steady-state conditions in a non-periodic microstructure. However, the operating range can be described by its average value (q~) and its extent (r/). The average spacings, undercoolings and growth velocities are shown to satisfy the A T - 2 - V relationship, despite the local variations of the solidification rate. It is shown experimentally that the extent o f the operating range is always very close to the value of the average spacing, independent of the q~-value. This suggest that the two spacings, 2m~, and 2b~, are determined by the same mechanism. Acknowledgements--This work was carried out at Ames Laboratory which is operated for the U.S. Department of Energy by Iowa State University under contract no. W-7405-ENG-82. This work was supported by the Office of Basic Energy Sciences, Division of Materials Sciences. One of the authors (PM) wishes to thanks the "Fonds National Suisse de la Recherche Scientifique", Bern, for financial support.

REFERENCES 1. K. A. Jackson and J. D. Hunt, Trans. Am. Inst. Mitt. Engrs 236, 1129 (1966). 2. P. Magnin and R. Trivedi, Acta metall, mater. 39, 453 (1991). 3. R. Trivedi and W. Kurz, in Solidification Processing o f Eutectic Alloys (edited b~'D. M. Stefanescu, G. J. Abbaschian and R. J. Bayttzick), p. 3. Metall. Soc., Warrendale, Pa (1988). 4. H. Jones and W. Kurz, Z. Metallk. 72, 792 (1981). 5. P. Magnin, Acta metall, mater. To be published. 6. D. J. Fisher and W. Kurz, Acta metall. 28, 777 (1980). 7. T, Sato and Y. Sayama, J. Cryst. Growth 22, 259 (1974). 8. P. Magnin and W. Kurz, Acta metall. 35, 1119 (1987). 9. P. Magnin and W. Kurz, Metall. Trans. 19A, 1955 (1988). 10. J. T. Mason, An Apparatus for Directional Solidification, IS-4817, UC-37. Available from National Technical Information Service, U.S. Dept. of Commerce, Springfield, Va (1982). 11. L.M. Hogan and H. Song, Metall. Trans. 18A, 707 (1987). 12. B. Toloui and A. Hellawell, Acta metall. 24, 565 (1976). 13. R. Elliott and S. M. D. Glenister, Acta metall. 28, 1489 (1980). 14. H. A. H. Steen and A. Hellawe11, Acta metall. 20, 363 (1972). 15. S. M. D. Glenister and R. Elliott, Metall. Sci., p. 181 (1981). 16. M. Petrescu, Z. Metallk. 61, 14 (1970). 17. H. Jones and W. Kurz, Metall. Trans. llA, 1265 (1980). 18. P. Magnin, P h . D . thesis, Swiss Federal Inst. Tech., Lausanne (1985).

19. M. Rappaz, M. Gremaud, R. Dekumbis and W. Kurz, in Laser Treatment o f Materials (edited by B. L. Mordike), DGM, p. 45 Oberusel, F.R.G. (1987). 20. K. Bratkus, unpublished work, referred by W. Kurz and R. Trivedi, Acta metall, mater, 38, 1 (1990). 21. R. W. Series, J. D. Hunt and K. A. Jackson, J. Cryst. Growth 40, 222 (1977). 22. G. Lesoult, Ph. D. thesis, CNRS A03795, Inst. Nat. Polytechnique de Lorraine, Nancy (1976). 23. H. Sens, Ph. D. thesis, Inst. Nat. Polytechnique, Grenoble (1988). 24. H. Sens, N. Eustathopoulos, D. Camel and J. J. Favier, Metall. Trans. To be published. 25. V. G. Smith, W, A. Tiller and J. W. Ritter, Can. J. Phys. 33, 723 (1955).

APPENDIX Nomenclature Ce = eutectic composition (wt% or at%) C* = corrected tie line length [2] (wt% or at%) D = interdiffusion coefficient in the liquid (m2/s) F = corrective factor in equation (7) (K) G = thermal gradient in the liquid (K/m) I(x) = s/1 interface shape (m) [~, [~ = average position of ~ and [7 s/1 interfaces (m) K l = constant of the 2 V term in equation (1) (Ks/m 2) K2 = constant of the 1/2 term in equation (1) (m K) P = function of the volume fraction (--) T = temperature field (K) T, = eutectic temperature (K) V = growth rate (m/s) f~, f~ = volume fractions (--) = weighted liquidus slope (~l = ]m~]m#/ (Iraqi + m,)] (K/wt% or K/at.%) m~, m~ = liquidus slopes (m~ < 0) (K/wt% or K/at.%) p = Peclet number (p = 2V/2D) (--) x = coordinate in the s/l interface, origin: center of ~-phase (m) z = coordinate in the growth direction, moving with the s/1 interface (m) F~, F~ = Gibbs-Thomson coefficients (m K) AC = changes in 6C due to the non steady-state conditions (wt% or at.%) AT = undercooling (ATe: chemical, ATr: curvature, A"~ ATk: kinetic) (K) = weighted undercooling [equation (9)] (K) A T ° = undercooling at the three-phase junction (K) AT ~, AT p = average undercooling of ~ and fl phases (K) AV = changes in V due to the non steady-state conditions (m/s) 8C = average composition at the s/l interface minus Ce (wt% or at.%) 8C ° = 6C value without kinetic undercooling (wt% or at.%) #/=extent of the operating range (l# =(2br-~ , ) / ~ ) (--) 0~, 0# = contact angles at the three-phase junction (°) 2 = lamellar spacing (m) 7{= average lamellar spacing (m) 2b~ = maximum (branching) spacing (in) Aex= spacing at the minimum undercooling point (extremum) (m) 2min = minimum spacing (In) 2p = spacing of the low velocities patterns (Fig. 5) (m) 4b = ratio of the mean to extremum spacing (--) Subscript = A-rich phase /7 = B-rich phase i = solid phase (~ and/7) 1 = liquid phase Notation: X = average value of the variable X