Eutectic growth: A modification of the Jackson and Hunt theory

Acta metall, mater. Vol. 39, No. 4, pp. 453~t67, 1991 Printefl in Great Britain. All fights reserved

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

EUTECTIC GROWTH: A MODIFICATION OF THE JACKSON AND HUNT THEORY P. MAGNIN~ and R. TRIVEDI 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 Jackson and Hunt theory of eutectic growth is modified in order to get a better understanding of the physical mechanisms driving the solidification process. The density difference between the eutectic phases is taken into account when calculating the diffusion field, and the isothermal interface coupling condition is replaced by an equilibrium criterion for the three-phase junction. Although the problem can be solved in the general case, a simpler solution is derived for the case of nearly isothermal interface, which differs from that of Jackson and Hunt only by a corrective factor reflecting the density differences between the phases, the correction being quite large in the case of cast iron. The volume fractions of the eutectic phases are shown to be adjusted by the movement of the three-phase junctions. The boundary layer composition, which depends on this mechanism, is shown to be responsible for the well-known occurrence of austenite halos surrounding the primary graphites in cast iron. The curvature undercooling is written as the ratio of the volumetric interface energy contained in the eutectic microstructure to an average volumetric entropy of melting, which allows a quantitative evaluation of the effect of an anistropy in the ~/fl interface energy. Reliable values of the relevant physical constants are provided for several important eutectic alloys.

R~sum6---Le mod61e th6orique de croissance eutectique propos~ par Jackson et Hunt est modifi~ dans le but de permettre une meilleure compr6hension des m6canismes contr61ant la croissance. La difference de densit6 pouvant exister entre les phases eutectiques est prise en compte dans le calcul du champ de diffusion, et l'hypoth~se d'une interface isotherme est remplac6e par une condition d6crivant l'6quilibre m~canique des jonctions ~-fl-1. Bien que le probl6me puisse de cette mani~re ~tre r~solu dans le cas g~n~ral, une solution simple est propos6e pour le cas off l'interface peut ~tre consid6r6e comme quasi-isotherme. La solution obtenue ne diff~re de celle de Jackson et Hunt que par un facteur correctif refl&ant la difference de densit6 entre les phases, cette correction 6tant tr~s importante dans le cas des fontes grises. On montre que les fractions volumiques des phases sont ajust6es par le mouvement des jonctions ct-/~-l, qui r6gulent ~galement la composition de la couche limite. Ce m6canisme permet d'expliquer le ph6nom6ne bien connu de la formation de halos d'aust6nite autour des graphites primaires dans les fontes. On montre que le terme de surfusion de courbure peut s'~crire comme le rapport entre l'6nergie de surface contenue dans une unit6 de volume de la microstructure form6e et une entropie de fusion volumique moyenne. On peut de cette mani6re 6tudier quantitativement l'effet d'une anisotropie de l'6nergie de surface entre les phases solides. Un ensemble de valeurs fiables est d6termin6 pour les constantes physiques de plusieurs alliages eutectiques importants.

Zusammenfassuag--Jacksons und Hunts Theorie fiber das eutektische Wachstum wird abge/indert, um die physikalischen Mechanismen, die die Erstarrung kontrollieren, besser zu verstehen. Die unterschiedliche Dichte der eutektischen Phasen wird beim Berechnen der Diffusionszone ber/icksichtigt, und die Annahme einer isothermen Phasengrenze wird durch ein Kriterium ersetzt, dass das mechanische Gleichgewicht am Tripelpxinkt beschreibt. Obwohl das problem in allgemeinen auf diese Weise gel6st werden kann, wird eine einfache L6sung abgeleitet unter Annahme, dass die Phasengrenze quasi-isotherm ist. Sic unterscheidet sich im Vergleich zu Jackson und Hunt nur durch einen Korrekturfaktor, der die unterschiedlichen Dichten der Phasen widerspiegelt. F/ir graues Gusseisen ist diese Korrektur sehr gross. Durch die Verschiebung des Tripelpunktes wird der Volumenanteil der eutektischen Phasen angepasst und die Zusammensetzung der Grenzschickt gesteuert. Dieser Mechanismus erkl/irt das Entstehen der bekannten Austenit-Halos am Prim~irgraphit in Gusseisen. Die Kriimmungsunterkfihlung kann als Volumenverh/iltnis der in der eutektischen Mikrostruktur enthaltenen Grenzfl/ichenenergie und der Schmelzentropie geschrieben werden. Damit l~isst sich quantitativ abschhtzen, wie gross der Einfluss der Anisotropie der ct//~ Grenzflfichenenergie ist. Verl~issliche Werte der physikalischen Konstanten von mehreren wichtigen eutektischen Legierung werden bestimmt.

~'Present address: Laboratoire de M6taUurgie Physique, Ecole Polytechnique F6d6rale de Lausanne, CH-Ecublens, CH-1015 Lausanne, Switzerland. 453

454

MAGNIN and TRIVEDI: EUTECTIC GROWTH INTRODUCTION

The basis of steady-state eutectic growth theory has been established more than twenty years ago by Jackson and Hunt [1] (JH). Since then, several authors have proposed different modifications of this model in order to extend its validity to a wide range of specific cases, such as irregular growth [2, 3], rapid solidification processes [4], and non steady-state growth [5]. As mentioned by Series et al. [6], the JH analysis (and therefore all models based on it) is valid only as long as the densities of the phases are equal. Due to the large composition difference between the eutectic phases, this condition may often not be fulfilled. In addition, no mechanism is provided for the adjustment of the volume fractions of the phases, which have to be considered as material constants. As a result, the calculation of the boundary layer becomes erroneous, and the corresponding term must therefore be eliminated with an additional criterion (such as an isothermal interface). Although the solution obtained is still valid for most eutectic systems, this is not true for the very important case of cast iron (Fe-C). Moreover, some of the physical aspects of eutectic growth are lost in this treatment. The aim of this paper is to re-examine the eutectic growth theory in order to get a better understanding of the physical mechanisms involved in the solidification process. Specifically, the density differences between the phases will be taken into account when calculating the diffusion field. The JH coupling condition (equality between the average growth undercoolings of ~ and t-phases) will be replaced by an equilibrium criterion for the three-phase junction, thus leading to a mechanism of adjustment of the volume fractions of the phases. Although the problem is formulated for the general case, a simpler solution will be established in this paper for the case of an isothermal s/l interface, which coincides with the JH result except for a corrective factor that reflects the density differences between the phases. The latter is shown to be quite large in the case of Fe-C. The effect of the boundary layer composition on the eutectic growth mechanism will be demonstrated, which will also explain the well-known phenomenon of halo formation in cast iron. The different contributions to the total growth undercooling will be separated, and it will be shown that the curvature undercooling represents the energy required for the formation of the interfaces between the two solid phases in the eutectic microstructure. Finally, the validity of the assumptions made in this paper will be examined carefully. Most eutectic growth models contain two stages: first, the solution of the diffusion equationt is obtSince eutectic growth is controlled mostly by the diffusion of solute, the heat equation does not need to be solved and the model is then valid for directional as well as for equiaxed solidification. The complete solution including the heat diffusion has been developed by Nash [7].

_vT

x

Fig. 1. Simplified geometry assumed for the lamellar eutectic microstructure. tained for a planar interface and the interface undercooling is corrected to include the effect of surface energies. This "general non-optimized equation" gives a relationship between the solidification rate, the interface undercooling and the spacing. Thus, at a given solidification rate, thermodynamic/transport considerations allow the growth of any spacing if no restrictions are applied to the interface undercooling. Next, the actual spacing(s) selected by the system ("operating point" or "operating range") are determined by an additional criterion coming either from empirical considerations [8, 9], from a study of the mechanisms of spacing adjustment [1, 10], or from a stability analysis [11, 12]. In this paper, only the non-optimized part of the problem will be considered and the operating point will be examined in a subsequent publication [13]. For comparison purposes, the operating point will be simply described here by an empirical criterion based on experimental results in several alloy systems. DIFFUSION FIELD Steady-state diffusion equation

In order to get an analytical solution, the diffusion field will be obtained for the case of a lamellar eutcctic, using the idealized geometry shown in Fig. 1 (the case of rod eutectic is considered in Appendix 1). The problem in this case is independent of the y-coordinate, and therefore only two-dimensional. The steady-state solute field in front of the planar solid/liquid interface moving at a constant velocity, V, in the z-direction is then given in a frame of reference moving with the interface by the equation 72C ~2C V ~C gxZ ~-~z2 + ~ z = 0

(1)

where C is the solute concentration and D is the solute coefficient in the liquid. The simplified geometry used leads to the following boundary conditions: periodicity:

C(x + 2) = C ( x )

symmetry:

OC/Ox = 0

far-field:

for x = 0 and x = 2/2

C (z ~ oo ) = Ce + A C~o

where 2 is the lamellar spacing, Ce is the eutectic composition and ACoo is the difference between the

MAGNIN and TRIVEDI: EUTECTIC GROWTH initial composition of the liquid and C~. The general solution of equation (1) is then: C = C~ + AC~o + B o e x p ( - V z

+

)

B. exp( - og.z)cos n---2-x

(2)

455

where v, and va are density factors, whose values will be determined in the next section. One verifies that the condition (5) satisfies the necessary symmetry with respect to the composition (i.e. it is left unchanged by a reversion of the phase diagram, with C replaced by ( 1 - C) and ~ and fl-phases interchanged).

n~l

with co, =

p + x/(2nn)2 +p2 27[ 2 ~ n-f

Composition field when p <<27[

Applying the boundary condition (5) to the general solution [equation (2)] leads to:

where B0 and B~ are Fourier coefficients which have to be determined by the conservation of solute at the s/l interface. At small growth rates, the Peclet number, p = 2V/2D, is very small so that the simplified expression of m, will be used in this paper. Solute balance at the s/l interface

z=O = V ( C l -

2 B. = ~ - ~

V

, Co sin(nT[f~)

with

Neglecting the solute diffusion in the solid phases, the solute balance at the s/1 interface leads to

- D T~zC

So =Lv,(co + ac - c,) - f a v A G - c , - ac)

Ci)

(3)

where C l and Ci are the compositions of the liquid and solid (i = c¢ or fl) phases at the s/1 interface. The boundary condition (3) complicates the problem, because the C l and C~ values are function of the solution of equation (1). A simplification must therefore be made if an analytical expression of the diffusion field is required. At low undercoolings, the liquid in equilibrium with the solid phsaes will have composition that is very close the eutectic composition so that we may write: C~ = C~ + 6C, where 6C is an average corrective factor (it will be shown later that, although 6C is very small, it cannot be neglected). The solute balance is then given explicitly by _ O OC

C* = v~(C~ + 6C - C~) + va(C a - C~ - c$C)

(6)

where f~ and fa are the volume fractions, and C~ = (Co + 6C) k~ and C a = 1 - (1 - Co - 6C) k a are the compositions of ct and fl phases, respectively. The expression for the B0 term may be written differently by considering the overall mass balance constraint. For steady-state growth with a negligible diffusion in the solid, the average composition of the solid microstructure must be equal to the initial composition of the liquid, i.e.

LP=C" + f~PaCa

(7)

C¢+ AC~

where p~ and Pa are the densites of ct and fl-phases, and ~ =f~p~ +fapa is the average density of the solid microstructure. Substituting equation (7) into equation (6) leads to

~Z z= 0

fV(C~ + 6C)(1 - k~)

a-phase

'~(- V(1 - C¢ - 6C)(1 - ka)

fl-phase

B o = - A C ~ + 6C(f~v~ +farts) (4)

where k~ and k~ are the solute distribution coefficients of ct and fl phases. The above equations are valid only if the densities of the phases are equal. As long as the diffusion field is considered only in the liquid phase [equations (1) and (2)], this condition is fulfilled since the density of the liquid can be assumed to be a constant. However, the boundary condition (4) involves the solid phases, whose densities may be very different in some eutectic systems (more than a factor 3 for Fe-C). The compositions in equation (4) should therefore be modified in order to take into account these density differences. One can then write

-oK ~Z z=O

SVv~(Ce + 6C)(1 - k~)

or-phase

( -- Vva(1 -- C ¢ - tSC)(1 - ka)

fl-phase

"x

(5)

+ [C~(f~v~ +fava -- l)

+LC~(p~/~ - v~) +fpCa(pp/~

-

va)].

(8)

According to equation (2), the average composition at the s/1 interface (z = 0) is given by Ce + AC® + B0. From the definition of 6C, the latter is also equal to Ce + 6C. This equality provides then an equation for the v factors. Although there is an infinite number of combinations of v~ and vp values which fulfill the above condition, one can easily see that there is only one solution which respects the symmetry of the problem, leading to vi = Pi/P (i = ct, fl). The effect of the density factors is then to bring all compositions back to the average density, ~, which is very close to the density of the liquid. Substituting these values into equation (8) shows that the value of the large bracket is zero, while the multiplicative factor of 6C is equal to unity.

456

MAGNIN

c.°

c,

and TRIVEDI:

cJ

B

Fig. 2. Eutectic p h a s e d i a g r a m , defining some o f the physical

constants used. Substituting the values of vi in equation (6), the C* term is obtained as: C~=

pa(Ce - C °) -~- pfl(C°fl - Ce)

+ p,(1 - k,) - p#(1 - k#) 6C

(9)

# where C o and C~ are the solubility limits at the eutectic temperature, defined in Fig. 2. Note that, in addition to the fact that 6C is very small, its multi° plicative factor is also small (and even equal to zero when p, = p# and k, = k#). The last term of equation (9) can then be neglected, and the Fourier coefficients of the composition field can finally be written

EUTECTIC

GROWTH

The 6C value is determined by the growth conditions, and will be calculated later in this paper. When the densities of the phases are equal, the C*-value i~ identical to the CO constant of the JH treatment. For some of the most common eutectic systems, Table 1 shows that both values are generally very similar. However, this is not true for the case of F e ~ (cast iron), for which not only the density difference is quite large, but also the Fe42 phase diagram is not symmetrical (in contrast to the A1-A12Cu phase diagram, for which the density difference between the phases does nearly not affect the CO value). The above treatment is not a strict mathematical solution of the very complex problem of density changes during the solidification process, since it does not include liquid motions at the s/1 interface which can affect the effective diffusion coefficient (see Appendix 2). However, it provides a self-consistent solution of the diffusion field which incorporates such density changes.

EUTECTIC GROWTH Growth undercooling

Bo = - A C ~

+ 6C

2 V B. = (nn) z ~ C~' sin(nnf~) with C~ - P=(Ce - Co) + P#(C°~ - Ce)

(lO)

During the solidification process, the growth undercooling, AT, is given by the difference between the eutectic temperature, Te, and the actual temperature of the s/1 interface. Note that the growth undercooling is defined here with respect to To, and not to the liquidus temperature corresponding to the local interface composition. The growth undercooling is made up of three contributions:

Table 1. Physical constants for some important euteetic systems. See Appendix 2 for details Fe-C [17, 20, 33-35]

A1-AI2Cu [37-40]

Sn-Pb [35, 37, 41~13]

577.2 -7.5 15.7 12.6 1.64 99.98

548.2 -4.6 3.8 32.7 5.65 52.5

183.0 -0.83 2.43 38.1 2.5 81.0

2.5 2.15 0.873 0.127 87.7 0.89

2.5 4.0 0.54 0.46 46.0 0.98

7.3 10.3 0.63 0.37 83.4 1.06

3.09 2.75 0.676 0.324 10.6 0.96

1.96 1.7 30 65 4.3 3.2

2.4 0.55 65 55 2.8 1

0.79 0.48 65 35 1.1 1

0.8 1.14 60 55 1.24 1

(°C) (K/wt%) (K/wt%) (wt%) (wt%) (wt%)

p= p# f= f~ C~

C*/Co

(g/cm 3) (g/cm3) (--) (--) (wt%) (--)

7.4 2.11 0.926 0.074 31.1 0.32

F= Fp 0= 0# Deft" ~b

(10-7 m K) (10 -7 m K) (°) (°) (10 -9 mZ/s) (--)

1.9 3.7 25 85 1.25 5.4

K! K2

(109 Ks/m 2) (10 -6 mK)

150.5 2.358

22V

(10 -18 m3/s) Equation (18)

460 457

410 404

1158 1155

103 102

AT/~

(Km -w2 s w2) Equation (19)

3300 3327

100-240 159

306 310

23-128 93

(wt%)

0.22

1147.1 - 135 60 4.30 2.11 6.69

AI-Si [14, 35-37]

Te rn, m# Ce Co C~

6C (V = 100 #m/s)

1154.5 - 135 470 4.26 2.08 99.9

Fe-Fe 3C [17, 32-35]

7.4 7.2 0.515 0.485 4.58 1.00 1.9 2.4 50 55 4.7 1.8 6.03 0.752

--0.006

8.30 0.936

0.33

4.62 0.472

-0.07

5.93 0.207 35.0 35.0 39-129 70 --0.17

CBr4~C2C16 [10, 31] 83.0 - 1.48 2.16 8.4 5.08 16.18

0.937 0.356 380 380 -37 031

0.1~ -0.1 ~

has been shown that the kinetic undercooling can usually also be neglected [3, 14]. However, theoretical models including ATk have been developed [14, 15].

• =/~ mtermce

~

Note that the chemical and curvature undercoolings (and thus the total growth undercooling for negligible ATk) represent the difference between the eutectic temperature and the actual equilibrium temperature of the s/1 interface. They are not a departure from the local equilibrium conditions.

f=05

Fig. 3. F(x) function [equation (11)] describing the composition profile at the s/1 interface, for three different values of the volume fraction. (1) The chemical (or solute) undercooling, ATe, which reflects the fact that the composition in the liquid at the s/1 interface locally slightly departs from the eutectic composition, leading to a proportional departure of the equilibrium temperature from the eutectic temperature. From equations (2) and (10), the solute undercooling at the s/l interface is given by ATe(x) = mi(C~ - C(z=0)) C*

=-mi(rC+-~F(x)2V)

i =ct,[J

with _~ 1 / 2~z \ F ( x ) = ~ ,-:~_,2sin(nnf=)cosln-7-x]

,=ltmz)

\

,t

/

(ll)

where mi are the liquidus slopes given by the phase diagram (including the sign, i.e. m, < 0). The function F(x) is plotted in Fig. 3. (2) The curvature undercooling, ATr, represents the effect of the s/l interface curvature on the equilibrium temperature between the solid and liquid phases, and is given by ATe(x) = F~x(x)

i = o~,fl

(12)

where Fi are the Gibbs-Thomson coefficients and x(x) is the local interface curvature. (3) The kinetic undercooling, ATk, represents the difference between the chemical potentials of the liquid and solid phases at the interface, i.e. the driving force of the solidification process. For most metallic eutectic systems solidified at low velocities, this term is much smaller than ATe and AT~ and will therefore be neglected in this model. For irregular eutectics, it T(x,zl

z=Z(x).. ~ t ~

457

MAGNIN and TRIVEDI: EUTECTIC GROWTH

C(x,z)

,

0# ~ /

I " I ° ' Fig. 4. Real s/1 interface shape, defined by the thermodynamic equilibrium with both temperature and composition fields.

Interface thermodynamic equilibrium Until this point, the s/1 interface was assumed to be flat, according to the simplified geometry shown in Fig. 1. We now consider the real interface shape, described by the relation z = I(x) (Fig. 4). Since the eutectic s/1 interface is macroscopically flat when compared to the diffusion distance, D/V, the interface composition is still given by C(x, z = 0) in equation (2) (and not by C[x, z = I(x)], which has been shown to be a poor approximation for the s/1 interface composition [2, 14]). For negligible kinetic undercooling, the thermodynamic equilibrium requires that, at each point of the s/l interface, the temperature, T[x, z =I(x)], imposed by the growth conditions must be equal to the local equilibrium temperature between the solid and liquid phases, i.e.

T[x, z = I(x)] = T~ - ATc(x ) -- ATr(x).

(13)

Substituting equations (11) and (12) into equation (13) then allows one to calculate the interface curvature, giving

x(x)=~[AT[x,z=I(x)] /

... C*

\-]

(i = . , f l )

(14,

Where AT[x, z = I(x)] = Tc - T[x, z = I(x)]. Note that the curvature, x(x), satisfies the following relationship fo: x(x) dx = fo:

[1 + d2I/dx2 (dI/dx)2] 3/2dx

( di y

= - s i n tan -l \ d x ]0"

(15)

Integration of equation (14) (separately for ct and t-phases since dI/dx is discontinuous at x =f=2/2) with the boundary conditions dI/dx = 0 at x = 0 (~-phase) and at x = 4/2 (t-phase) allows one to calculate the interface slope dI/dx. At the three-phase junction, however, the value of the slope is constrained by the necessity of balancing the surface tensions. As it will be shown later, this condition is fulfilled by an adjustment of the s/1 interface composition which allows one to calculate the 6C value. Equalizing the interface slopes obtained at the threephase junction from equation (14) to the contact

458

MAGNIN and TRIVEDI: EUTECTIC GROWTH

angles, 0, and 0,, imposed by the surface tensions balance gives "

l/di sin

0, = - s i n t a n

-

t ~ x)x=A,t/2

1 C*P = ~ [ ~ ( A T ' + m , 6 C ) 2 + m ~ D 2 EV]

•

_

sin 08 = sin t a n -

•

.

i/d/"~+ /-- /

\ dx ]x=f,;q2

1 h C*P =-~I2(AT~+mts6C)2-ma-~ 21V]

with

P = .~=1~

1

spacing

[sin(nnf')lZ

[1 - 0.205 e x p ( - 24f=fe)] x 0.3251(fja) L63 (16) where AT ~is the average undercooling of the phase i. The analytical expression for P, given here,is accurate (better than -I-0.1% for f t> 0.046, where f is the smaller off~ and fB) and is more convenient to use instead of the exact series solution. Note that the term in the large bracket of the approximation is negligibly different from 1 for f > 0.2• The above model can be solved for the general case. However, since the eutectic s/l interface is macroscopically flat compared to the heat diffusion distance, the temperature variations along the s/l interface are much smaller than the variations of the solute and curvature undercoolings. In order to get a simple solution, the average undercoolings of c~ and /7 phases will be approximated by the same value, A T, and the general solution will be derived in a companion paper [14], where it will be shown that both solutions lead to negligibly different results. For an isothermal interface, the growth undercooling, AT, can be obtained from equations (16) as

AT = Ki2V + K212 r~C~ P K1--D f~f~ f F , sin 0,

Fig. 5. General relationship between the solidification rate, the growth undercooling and the spacing [29]. The extremum valley (¢ = I) corresponds to the minimum undercooling at constant solidification rate or the maximum growth rate for constant undercooling. As a result, a new interpretation of several aspects of eutectic growth will be presented•

Operating point Before further discussion, a complete solution of the eutectic growth is needed. Equation (17) shows that the competition between the solute and curvature undercoolings does not lead to a fixed relationship between the spacing and the growth velocity. One then still have to determine the operating point (or range), that is the spacing(s) actually selected during the eutectic growth. It is not the purpose of this paper to discuss the operating point, which will be examined in a subsequent paper [13]. However, at this point, we will simply assume that the growth occurs at a spacing 2 = q~2ex [16], where 2ex is the spacing corresponding to the minimum of the AT - 2 curve for a given velocity (or to the maximum growth velocity for a given undercooling, see Fig. 5), and ~b is a constant reflecting the spacing adjustment mechanism. In this case, eutectic growth can be described by the relationships [17].

22V = dk2K2/KI ATI~IrV = (¢ + 1/¢) Kv/KT~K ~

with

K2=2rht~+

i OXed yf

~ sin 0p'~

maf, ,]

07)

where nq =lm, lmp/(lm:l+ma) is the half of the harmonic average of Im, I and m s. Except for the v-corrective factor in the definition of C~', equation (17) is identical to the Jackson and Hunt solution for an isothermal interface. However, the condition of isothermal interface was introduced here only as a mathematical simplification (which will be relaxed in Ref. [14]), and is not a basic hypothesis needed in order to eliminate the/70 term, as in the JH treatment.

2AT = (q$z+ 1)Kz.

(18)

(19) (20)

DISCUSSION

Volumefractions adjustment The thermodynamic equilibrium of the s/1 interface is determined by equation (13). Thus, the solidification process determines the composition field, whereas the temperature field is imposed by the macroscopic heat fluxes. The position of the s/1 interface is then adjusted in order to be compatible

MAGNIN and TRIVEDI: EUTECTIC GROWTH

459

Boundary layer

The boundary layer composition determined by the mechanism described above can be calculated from equation (16). Using the relations (18-20), one obtains J

aC = (f~/ Irnd - f~/rn e) A T

+ 2(F e sin(0p)/m e - F, sin(0~)/[ rn~ 1)/2 or

Fig. 6. Mechanism of the volume fraction adjustment. When the contact angles at the three-phase junctions are different of 0~ and 0p (bottom), the resultant force applied to the off-equilibrium three-phase junctions leads the volume fractions to their equilibrium value (top). with both fields, i.e. the difference between the s/l equilibrium temperature corresponding to the local composition and the imposed temperature field must be compensated by the undercooling due to the local curvature of the interface. In order to understand how the three-phase junction affects the s/1 interface shape, let us consider the Fig. 6. In the case represented here, the growth conditions are such that the thermodynamic equilibrium of the s/1 interface described in the previous paragraph imposes contact angles, 0~, and 0~, different from those leading to the mechanical equilibrium between the surface tensions. A resultant force is then applied to the three-phase junctions, under the action of which the volume fraction of one phase (~ in Fig. 6) will be lowered. As the volume fraction of co-phase (which is responsible for the solute rejection) decreases, the average composition at the s/1 interface is also lowered, thus diminishing the solute undercooling of ~-phase and raising that of r-phase. Since the temperature field is nearly not affected by the interface shape, these variations of the solute undercooling must be compensated by an opposite change in the curvature undercooling. Noting that the latter is given in average by 2Fi sin(Oi)/fi2 (i = ~, fl), this leads to raise 0~, and lower 0~. As shown in Fig. 6, the volume fraction adjustment driven by the three-phase junction movement regulates the boundary layer composition in order to bring the contact angles to their equilibrium values. Once the correct contact angles have been established, the volume fractions return to their steady-state values in order to prevent further changes in the boundary layer composition. The solidification of a euteetic microstructure is then globally equivalent to the growth of a single-phase with a solute distribution coefficient slightly different than one. Note that the resultant force occurring at the three-phase junctions when the contact angles are off-equilibrium is very large. The volume fractions are therefore nearly immediately adjusted, even during the transient variations in the boundary layer composition induced by a sudden change in the growth rate, as observed by Jackson and Hunt. AM 3 9 / ~

6C

= AT//j¢~mp -_fe[ m~[ th \ [ m , l + m e (F e sin(0 e)/m e -- F~ sin(P,)/] m~[)/(~b2 + 1)'~ -t

F~ sin(O~)/f~lm~l + F~ sin(Oe)/fem e

J"

(21) The departure from C~ of the average s/1 interface composition is then proportional to the growth undercooling, AT. Its origin lies in the necessity to compensate the difference in the liquidus slopes and curvature undercoolings of the phases. As can be seen in Table 1, 6C is very small. The boundary layer is then such as to maintain the average composition of the liquid at the s/1 interface close the eutectic composition, regardless of the initial composition of the liquid. This is the reason why a purely eutectic microstructure can often be observed even for off-eutectic alloys (coupled zone [18]). The 6C value, although it is very small, can drastically influence the growth behavior of some eutectic systems, as it will be demonstrated for the case of cast iron. In this case, at a typical growth rate of 100 #m/s, the average composition at the s/1 interface differs from Ce only by about two-tenths of wt%, as shown in Table 1. However, if this departure did not exist, the average chemical undercooling of the graphite would be raised, while that of the austenite (Fe) phase would be lowered. The difference between the chemical undercoolings of the phases would than be raised by a quantity (Im~l+me) A C ~ 100K, which is much too high to be compensated by a difference in the curvature undercoolings. The graphite lamellae would therefore become much more undercooled than the austenite. In the presence of a thermal gradient, this could be achieved by a non-fiat s/1 interface, as shown in Fig. 7. However, since the solute undercooling is maximum at the center of the phases, the thermodynamic equilibrium of the s/1 interface requires its curvature to be maximum near the three-phase junctions. Thus, the s/1 interface shape shown in Fig. 7(a) cannot be in thermodynamic equilibrium with the coupled diffusion field determined by the eutectic growth mechanism (it would lead to the independent growth of the primary c~phase), and only the shape shown in Fig. 7(b) can produce a coupled eutectic growth with different undercoolings of the phases. Since a very deep depression in the graphite lamellae is very unlikely to

460

MAGNIN and TRIVEDI: EUTECTIC GROWTH

(a)

relationships established for regular eutectics (CF. Appendix 2). The influence of the volume fraction adjustment on the boundary layer composition is one of the most important mechanisms controlling the eutectic growth. It represents the coupling mechanism between the chemical and curvature undercoolings. Contributions to the growth undercooling

Fig. 7. Non-isothermal s/1 interface shapes. The shape shown in Fig. 7(a) cannot be in thermodynamic equilibrium during a coupled eutectic growth, and only the shape shown in Fig. 7(b) can lead to a non-isothermal s/1 interface in the presence of a high thermal gradient. occur, the eutectic growth of cast iron becomes impossible when the average composition of the s/1 interface is not slightly higher than C¢. In the latter case, an austenite layer overgrows the graphite lamellae. This situation occurs in hypereutectic or nodular cast irons, where primary graphite particles are growing ahead of the eutectic s/1 interface. The composition of the liquid around these particles is lowered by the carbon absorption in the graphite, thus preventing locally a further eutectic growth. The primary graphites in cast iron will therefore be surrounded by an austenite layer. This well-known phenomenon [halo formation (19)] is then controlled by the composition of the boundary layer. It is likely to appear in all eutectic systems having high liquidus slopes and a low volume fraction of one phase. A good example of the sensitivity of this phenomenon to very small composition differences is furnished by the transition from the metastable Fe-Fe 3C to the stable Fe-C eutectic in cast iron, which can be induced by a slight reduction in the growth rate during directional solidification experiments [20-22] (Fig. 8). When the transition occurs, the average composition at the s/1 interface is that corresponding to the Fe-Fe3C system, i.e. nearly exactly Co, as shown in Table 1. As seen above, this is sufficient to prevent the eutectic growth of F e ~ , and an austenite layer is formed. Although the boundary layer composition increases rapidly due to the carbon rejected by the growth of the austenite phase, the latter is sustained until the nucleation of graphite particles occurs. At this point, the average s/1 interface composition is much higher than C~. A graphite film is then formed, bringing the composition back to that leading to Fe-C eutectic growth. Note that, although the white Fe-Fe3C cast iron is a regular eutectic, the cementite phase (Fe3 C) exhibits a faceted behavior, as shown in Fig. 8(b). This is the reason why this system is known not to follow (at least at usual growth rates) the theoretical

The growth undercooling, AT, is given by equation (17). However, this equation does not allow one to distinguish between the chemical and curvature contributions to AT. In order to do that, one can calculate the average chemical and curvature undercoolings of the s/1 interface by integrating equations (11) and (12), respectively. Separating the contributions to the chemical undercooling coming from the volume fraction adjustment (3C values) from that given by the solute rejection/absorption at 6C = O, one obtains

AT=(Kc + KB1);~V+ (Kr + KB2)/~.

(22)

Fig. 8. Transition from the metastable Fe-Fe 3C to the stable Fe-C eutectic in cast iron, and detail showing the faceted behavior of the Fe~C phase. Longitudinal section of a sample directionally solidified at 8.4 #m/s [22].

MAGNIN and TRIVEDI: EUTECTIC GROWTH with

C~P

K~ = - - - ~ (Im, I + m 8) Kr = 2 ( r , sin(0,) + r 8 sin(08) )

TI( c*P

KB2 = 2 l ( F ' s ~ (0")

Fssin(08)']~ ']

and

l -fsm8 - L l m d Im, I + m8 The total growth undercooling is then constituted by a curvature part Kr/2, and two chemical parts K¢2V (solute rejection/absorption at 6C = 0) and Km2V + KB2/2 (departure from C~ of the boundary layer composition). The latter is zero for symmetrical eutectic systems, or when f,/f~ = ms/lm,[. In this situation, the chemical and curvature undercoolings are simply given by K12V and K2/2, respectively. Otherwise, this is not true and, in particular, the curvature undercooling is no longer given by K2/2, as previously thought. However, although it represents a chemical undercooling, the KB2 constant depends on the same material constants as the curvature undercooling.

Surface energies As can be seen in equation (22), the curvature undercooling ATr = K~/2 depends only on intrinsic properties of the phase and not, as the K2/2 term in the JH solution, on the liquidus slopes and volume fractions. Let us examine the Kr expression more carefully. The surface tension balance at the three-phase junction imposes contact angles, given for an ~/fl boundary normal to the s/1 interface by the geometrical relationships (Fig. 9) {V ~ sin(0,) + VS~sin(08) = 7 "8 ~cos(0~) = ~,~cos(08) +

(23)

where 7~, 7 St, and 7 "8 are the surface tensions of o~/l, fl/l and ~/fl interfaces, respectively. The torque term,

o,~x,L~ oa a

7"*~

B

Fig. 9. Surface tension balance at the three-phase junction in the presence of a torque force, and definition of the parameter * in equation (24).

461

z, is zero when the surface energy of the ~/fl interface, a,8, is isotropic (the torque forces associated with the s/1 interfaces energies are equal to zero, since the s/1 interface energies of metals are generally isotropic). Otherwise, this a force which tends to align the ct/fl interface along the low energy direction [23]. In the latter case, the contact angles, 0~ and 08, depend on the orientation of the ~/fl interface, and are therefore no longer constant when different orientations are growing simultaneously. Note that tr,8 may depend not only on the direction of the ~/fl interface, but also on the relative crystallographic orientation of both phases, although the latter is generally uniquely determined in the eutectic microstructure. Substituting the value of the Gibbs-Thomson coefficient (ratio of the surface energy tru to the volumetric entropy of fusion As~, i = ~, r ) into equations (23), one finds the relationship {F, sin(0~) = a~8 (1 + f~)/2As~

(24)

F 8 sin(08) = a,8 (1 - f~)/2Asrp with 2 a~t -- zX f~ = a~l 0"28 "-~ T 2

e O',fl

(see Fig. 9)

2 2 2 2 2 X = N/(~l'~-1781"~-ff°tfl'~-T )

--

2(a~t+ast+a~8+ z 4) 4 4 4

o'~8 Substituting equations (24) in the Kr expression of equation (22) leads to 2tr~8 gr=-Ass

(25)

with 2 Asf = (1 + n)/As~ + (1 - n)/As~r" One verifies that Asf is a weighted average of As~ and As~ (i.e. A--sfis always between As~ and As~), since equation (24) show that Ifll ~< 1 for positive 0~ and 0a values. Note that the total area of ~/fl interface per volume unit is 2/2 for a lamellar eutectic. Thus, although it has been defined as an effect of the s/1 interface curvature on the equilibrium temperature between the solid and liquid phases, the curvature undercooling Kr/2 is nothing but the ratio of the volumetric c~/fl interface energy in the microstructure formed to the average volumetric entropy of fusion Asf. This remarkable result, also valid for rod eutectics (see Appendix 1), shows how the energy required for the formation of the a/fl interfaces during the solidification process is taken into account by the thermodynamic equations. Since t~ is expected to be small, Asf is only a weak function of the interface energies (and is anyway between As~ and Asg). As a first approximation, the curvature undercooling, Kr/2, is then simply proportional to the ~/fl interface energy. It can be shown that, because of the influence of the three-phase

462

MAGNIN and TRIVEDI: EUTECTIC GROWTH

".

..""

~~...

~ ......

KZ/Z

21

spocinO k

Fig. 10. Effect of an anisotropy in the =//~ interface energy on the spacing-growth undercooling relationship. For regular eutectics (2 ~)~o~) the low energy orientations (1) grow at lower undercoolings and spacings than the others (2). For irregular eutectics (k >>2,x), the difference is small. junction on the boundary layer composition, this is also true for the K2 constant. When a~a is not isotropic, the low energy orientations grow with a lower undercooling (and also with a lower spacing, as can be seen in Fig. 10), and will therefore be favored. However, K2/k is only a part of the total undercooling (the half for the minimum undercooling point, 2,~). This effect can then be weak, allowing different orientations of the ~t/fl interface to grow simultaneously, as sometimes observed [24]. Moreover, very anisotropic ct/fl interface energies are likely to form irregular eutectic microstructures, which are known to grow with lamellar spacings much greater than 2,~, where the curvature undercooling is negligible compared to the solute undercooling. In the latter case, the orientation of the ct/fl interfaces is selected by the local action of the torque force and not by a competitive growth mechanism.

Validity of the model In this section, the assumptions made in the present model will be summarized. First, it is assumed that the eutectic microstructure results from a cooperative growth of lameUar or rod-shape phases. The model is therefore not applicable to the special case of the "divorced" growth of nodular cast iron [25]. The assumptions of the steady-state, of an isothermal s/1 interface, of a fiat interface for the calculation of the diffusion field, and of negligible kinetic undercooling will be discussed in details in a companion paper [14]. The other main hypotheses made in this paper are:

Growth at low peclet number. This assumption was made in order to use the simplified expression of co, in equation (2). At usual growth rates (V = 0.1 to 1000/~m/s), the peclet number is of the order of 10 -3 to 10-I . Under these conditions, the difference between the exact value and the approximation of co, is smaller than 1%, and therefore negligible when compared to the precision with which the physical constants of the alloy are known. In addition, the &C value in the boundary condition [4] has been assumed to be a constant. At usual growth rates, it has been shown in Table 1 that tiC is about two

orders of magnitude smaller than C*. The space dependence of tiC is thus negligible compared to the solute rejection at the s/1 interface. Finally, the effect, due to large undercoolings, of the temperature dependence of the diffusion coefficient and of the liquidus and solidus compositions has been neglected. As a rule of thumb, one can consider that the calculations presented here are valid for growth velocities up to a few cm/s [261. Negligible diffusion in the solid phases. In the case of eutectic growth, the diffusion in the solid phases is much slower than in the liquid, and may therefore be neglected. However, this assumption could constitute (with the low peclet approximation [27]) a limitation for the description of the solid-state eutectoid transformation. Density correction. The effect of the density differences between the eutectic phases on the solute balance at the s/l interface has been assumed to be described by the two constants, v, and vB, in equation (5) (and, eventually, with a corrected diffusion coefficient value in order to take account of the fluid motions at the s/l interface, see Appendix 2). This is only a first approximation of a very complex process, which requires a further examination, especially for cast iron. However, except in the latter case, the correction is very small and the numerical values of C* obtained should be at least as accurate as our knowledge of the material constants. CONCLUSIONS A new interpretation of the eutectic growth theory is proposed. The effect of the density differences between the eutectic phases is taken into account when calculating the diffusion field. This leads to a modification in the C* term representing the length of the eutectic tie line in the JH solution. Although this correction is small for most eutectic systems, it is quite large (more than a factor three) and necessary to explain the experimental results (see Appendix 2) in the very important case of cast iron. The shape of the s/1 interface in thermodynamic equilibrillm with both temperature and composition fields is shown to be constrained by the mechanical equilibrium at the three phase junction. This leads to a coupling mechanism between the solute and curvature undercoolings more general than the isothermal s/1 interface condition proposed by Jackson and Hunt. A solution will be derived for non-isothermal interfaces in a further publication. In most cases, however, the s/1 interface is nearly isothermal, and the solution obtained in this situation is identical to the JH result. The crucial role of the movement of the threephase junctions is emphasized, showing that this leads to a control of the volume fractions of the phases formed during the solidification process. This mechanism regulates the composition of the boundary layer in order to insure similar growth under-

MAGNIN and TRIVEDI: EUTECTIC GROWTH coolings of both phases, and thus constitutes the physical justification of the isothermal interface hypothesis of the Jackson and H u n t treatment. When the b o u n d a r y layer composition adjustment is not realized, the growth undercoolings of the two phases become different, and one of the eutectic phases can be overgrown by the other, thus explaining the p h e n o m e n o n of halo formation observed around the primary graphite particles in cast iron. The occurrence of this phenomenon with extremely small composition variations is demonstrated in the transitions from the metastable Fe--FeaC to the stable F e - C eutectic. It is shown that the average curvature undercooling can be written as the ratio of the volumetric ct/fl interface energy in the microstructure formed to an average entropy of melting. This result gives a remarkably simple relationship between the curvature undercooling established at the s/1 interface as a result of several very different mechanisms and other p h e n o m e n o n completely independent of the s/1 interface, i.e. the a m o u n t of interface energy between the solid phases. The proportionality between the curvature undercooling and the ¢t/fl interface energy provides a way to study quantitatively the influence of an anisotropy in tr~p. The calculations are established for the cases of lamellar and rod eutectics. It is shown that the ratio of the undercoolings or spacings obtained in the two geometries depends only on the volume fraction of the phases. For practical purposes, approximate expressions of several useful functions are given, and numerical values for the relevant physical constants are provided for several important eutectic alloys. 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 thank the "Fonds National Suisse de la Recherche Scientifique", Bern, for financial support. REFERENCES

1. K. A. Jackson and J. D. Hunt, Trans. Am. lnst. Min. Engrs 236, 1129 (1966). 2. D. J. Fisher and W. Kurz, Acta metall. 28, 777 (1980). 3. P. Magnin and W. Kurz, Acta metall. 35, 1119 (1987). 4. R. Trivedi, P. Magnin and W. Kurz, Acta metall. 35, 971 (1987). 5. A. Karma, in Solidification Processing of Eutectic Alloys (edited by D. M. Stefanescu, G. J. Abbaschian and R. J. Bayuzick), p. 35. Metall. Soc., Warrendale, Pa (1988). 6. R. W. Series, J. D. Hunt and K. A. Jackson, J. Cryst. Growth 40, 222 (1977). - G. E. Nash, J. Cryst. Growth 38, 155 (1977). C. Zener, Trans. Am. Inst. Min. Engrs 167, 550 "1946).

463

9. W. A. Tiller, in Liquid Metals and Solidification, p. 276. Am. Soc. Metals, Cleveland, Ohio. (1958). 10. V. Seetharaman and R. Trivedi, Metall. Trans. 19A, 2955 (1988). 11. S. Strassler and W. R. Schneider, Phys. Cond. Mater. 17, 153 (1974). 12. V. Datye and J. S. Langer, Phys. Rev. B24, 4155 (1981). 13. P. Magnin, Acta metall, mater. To be published. 14. P. Magnin, J. T. Mason and R. Trivedi, Acta metall. mater. 39, 469 (1991). 15. G. Lesoult, Ph.D. thesis, CNRS A03795, Inst. Nat. Polytechnique de Lorraine, Nancy (1976). 16. R. Trivedi and W. Kurz, in Solidification Processing of Eutectic Alloys (edited by D. M. Stefaneseu, G. J. Abbaschian, R. J. Bayuzick), p. 3. Metall. Soc. Warrendale Pa (1988). 17. H. Jones and W. Kurz, Z. Metallk. 72, 792 (1981). 18. W. Kurz and D. J. Fisher, Int. Metals Rev. 24, 177 (1979). 19. B. Lux, F. Mollard and I. Minkoff, in The Metallurgy of Cast Iron, p. 371. Georgy, St-Saphorin, Switzerland (1975). 20. P. Magnin and W. Kurz, Metall. Trans. 19A, 1955 (1988). 21. P. Magnin and W. Kurz, Metall. Trans. 19A, 1965 (1988). 22. P. Magnin, Ph.D. thesis, Swiss Federal Inst. Tech., Lausanne (1985). 23. C. Herring, in Surface Tension as a Motivation for Sintering (edited by W. E. Kingston), p. 143. McGrawHill, New York (1951). 24. D. D. Double, Mater. Sci. Engng 11, 325 (1973). 25. J. D. Sch6bel, in Recent Research in Cast lron (edited by H. D. Merchant), p. 303. Gordon & Breach, New York (1968). 26. M. Zimmermann, M. Carrard and W. Kurz, Acta metall, mater. 37, 3305 (1989). 27. L. F. Donaghey and W. A. Tiller, Mater. Sci. Engng 3, 231 (1968/69). 28. H. Sens, N. Eustathopoulos, D. Camel and J. J. Favier, Metall. Trans. To be published. 29. P. H. Shingu, J. appl. Phys. 50, 5743 (1979). 30. R. A. Swalin, in Thermodynamics of Solids, 2nd edn., p. 232. Wiley, New York (1972). 31. F. S. Kaukler, Ph.D. thesis, NASA TM-82451, Marshall Space Flight Center, Alabama (1981). 32. M. Rappaz, M. Gremaud, R. Dekumbis and W. Kurz, in Laser Treatment of Material (edited by B. L. Mordike), p. 45. DGM, Oberusel, F.R.G. (1987). 33. H. Bester and K. W. Lange, Archs Eisenhiitte. 43, 207 (1972). 34. G. S. Ershov, A. A. Kasatkin and I. V. Gavrilin, lzv. Akad. Nauk SSSR Met. No. 2, p. 76. (Russ. Metall., p. 62) (1978). 35. T. Massalski, Binary Alloy Phase Diagrams. Am. Soc. Metals, Metals Park, Ohio (1986). 36. M. Petrescu, Z. Metallk. 61, 14 (1970). 37. M. Gfindiiz and J. D. Hunt, Acta metall. 33, 1651 (1985). 38. J. L. Murray, Int. Metals Rev. 30, 211 (1985). 39. T. Ejima, T. Yamamura, N. Uchida, Y. Matsuzaki and M. Nikaido, or. Japan Inst. Metals 44, 1651 (1985). 40. S. M. Borland and R. Elliott, Metall. Trans. 9A, 1063 (1978). 44. R. M. Jordan and J. D. Hunt, Metall. Trans. 2, 3401 (1971). 42. R. M. Jordan and J. D. Hunt, Metall. Trans. 3, 1385 (1972). 43. R. Racek and M. Turpin, Metall. Trans. 5, 1109 (1974). 44. P. Magnin, unpublished research (1989).

464

MAGNIN and TRIVEDI: APPENDIX

EUTECTIC GROWTH

1

O

O

Rod Eutectic The calculations established for the case of a lamellar eutectic can also be developed for an idealized rod eutectic geometry, shown in Fig. A1. Since the JH solution is modified in the same way as for the lamellar case, only the results will be presented here. Using the cylindrical coordinate, r, the solute field is given at low Peclet numbers by

Q

-

C = C~+ACo~ + Ao e x p ( - V z )

0

+ ~ A. exp( -- tonz) Jo (o~.r)

n=l

(A1)

with 1

COn and

Fig. A1. Simplified geometry assumed for the rod euteetic microstructure [1].

A0= Bo

/~2°a=

& = ~ ~ ca'

j~(~.)

where J0 and Jl are the Bessel functions of the order 0 and 1, and y, is a root of Jl (x) = 0, approximately equal to mr. The factor x/2-v/3/n ~ 1.05 is the ratio between r~ + ra and 2/2, which is not exactly one since the hexagonal boundary in Fig. A1 has been approximated by a circle of equivalent area. Note that the problem is no longer symmetrical with respect to the phases: in this paper, the f-phase refers to the phase which forms fibers (of radius ra) in an a matrix. The integration (in cylindrical coordinates) of equation (14) calculated with the composition field given by equation (A I) gives a result indentical to that obtained for the lamellar geometry [right-hand side of equation (16)], with the only exception for the P function which, for rod euteetics, is 1 S2(~/c~p?.)

Pr'xl = 2 f ' ~

n=l~~n3

0

corresponding values for lamellar eutectic at small volume fractions of #-phase (f~ < 0.25). The K2 (and similarly Kr and KB2) constants for the rod microstructure are modified according to

2~.

"

0

(A2)

J2(~n)

proa g [1 -- 0.205 e x p ( - 24f~f~ )] x 0.2072f=l'63fp1"307

2x//~B K 2.

J2&/

(An)

In contrast to the chemical undercooling, the curvature undercooling is lower for rods than for the lamellar microstructure at small volume fractions of f-phase (fa < 0.28). Note that the total area of ot/# interface per volume unit is (2v/~/~) 2/,t for the rod microstructure. The curvature undercooling is then still given by the ratio of the volumetric a/# interface energy to the average volumetric entropy of fusion ATsr.This shows that the role of the surface energies in the eutectic growth mechanism is only to modify the internal energy of the solid microstructure, independent of the geometries of the s/l interface and of the microstructure formed. For a given alloy (and at constant solidification rate), the ratio between the spacings and the growth undercoolings obtained with the rod and the lamellar microstructures depends only on the volume fraction of the rod phase. From equation (18) and (19), one obtains:

27=/P 2

2x/~ -~ 1.743f~"415 X/ P r°d N//2"-N/'~/ ~

(A5)

_~0.6373 f~-0.32sp. /

Precision of the approximation: better than _0.1% for 0.038~
~fo~X(r)2=rdr 2x/~p

2

2X/~

.

n 1 dI -

fTy slnta - (drr),=,,

1_ ~~' +'p x (r)2~r dr rt [(ra + ra) =-- r~] J,a -

2N/~#

2

.

sm tan

l/d/St+ l-- /

.

(A3)

As a result, the K 1 constant in equations (17)-(20) [and the K~ and Km constants in equation (22)] are given by the same expressions as for lamellar euteetics, using prod instead of P. Note that, in contrast to the case of the growth of an isolated phase, the proa function (and thus the chemical undercooling of the rod microstructure) is larger than the

ATr°d=/Pr°d

2N/~

~ 1.095f~ "0854.

(A6)

Ar

Precision of both approximations: better than + 0.3% for f~ ~<0.5, and better than + 2 % for any possible f~. The rod microstructure grows with lower spacings than the lamellar structure at volume fractions of/~-phase smaller than 0.27 [Fig. A2(a)]. F o r f a ~<0.35, the growth undercooling of the rod euteetic is also lower [Fig. A2(b)], and this microstructure should therefore be favored. However, when the surface energy is not isotropic, the effective value of a~a [and thus the curvature undercooling, see equation (25)] is higher for the rod structure (average value of a~a) than in the lamellar geometry, in which the torque force tend to align the lamellae along the low energy orientation (i.e. the effective surface energy is close to the minimum value of a~a). Assuming that the growth occurs at r minimum undercooling point, the lamellar structure cv stabilized at low f~-values if the o~a anisotropy (det~ the ratio of the mean to minimum values of a~a~ than AT~AT r°a given by the inverse of equatioD the growth undercoolings of the lamellar and are very close, one can see that, even at ve fa, a rod eutectic can appear only whet energy is nearly isotropic. Note that ~" (A6) are valid only for constant ope of irregular eutectics such as AI-Y

M A G N I N and TRIVEDI: (a) .~ 1.5 o

_~ 1.0 "o "" 0.5

o

volume fraction, f e

(b) =

1.5

8

-8

.~ 1.0

E0.5

0

012

0'4

08

volume fraction, f.e

Fig. A2, Ratio of the (a) spacings, (b) growth undercoolings obtained in the rod geometry to the corresponding values for lamellar eutectics, as a function of the volume fraction of the fl (rod) phase. lamellar and rod morphologies, the two microstructures may grow with different ~b-values. APPENDIX 2

Physical Constants Table 1 presents the physical constants of five industrially important eutectic alloys: grey cast iron (Fe-C), white cast iron (Fe-Fe3C), lamellar AI-Si, A1-A12Cu and Sn-Pb, as well as an organic system (carbon tetrabromidehexachlorethane, CBr4~C2C16). The values given here have been established very carefully, and should provide reliable data for eutectic growth calculations. The phase diagrams of these alloys are well-known [31,35,38], and the values of the physical constants given by them may therefore be considered as accurate. Very good values of the Gibbs--Thomson coefficient have been obtained in A1-Si, A1-A12Cu and Sn-Pb [37], as well as in the CBr4~C2CI6 system [31]. For grey and white cast iron, the following empirical relationship was used [30]

aHr

tr~l = 0.46 NI/3v~3

(B1)

465

EUTECTIC G R O W T H

approximately by 5.45.10 -9 Tfvlm/3, where Tr is the melting temperature of the phase (AHf = TfASfVm, since Asf is the volumetric entropy of fusion). Although this relationship is not very accurate, the values obtained should be quite acceptable. The contact angles at the three-phase junction have been measured directly in the transparent CBr4422C16 organic system [31]. For the other alloys, these values can be estimated from the s/1 surface energies, which are known precisely for A1-Si, A1-A12Cu and Sn-Pb [37] (for grey and white cast iron, the s/1 surface energies can be obtained from equation (B1), assuming a non-dimensional entropy of fusion (AsfVm/R, where R is the gas constant) o f 1 for austenite (non-faceted), 3 for cementite (slightly faceted) and 4 for graphite (faceted), leading to crsl-values very close to those proposed in [17]). Although they are not sufficient to uniquely determine the contact angles (since tr=a and z are unknown), the s/1 surface energies provide a good first indication of the values of 0, and 0p, noting from the direct measurements realized in CBr4~22C16 that tr=# is slightly higher than the maximum of the trs~-values and then the torque force is likely to be of the order of 5-10% of the /fl interface energy (in the direction of the lowest try-value), except for the faceted eutectics in which z should be nearly zero, since the growth occurs in that case only at the minimum value of tr=p. Note that the values of the contact angles are anyhow not very sensitive parameters (for 0-values ranging from 38 ° to 90 °, sin(0) differs from sin(55 °) only by less than 25%). In addition, if the volume fraction of one phase is small, the values of the Gibbs-Thomson coefficient and of the contact angle for the other phase have nearly no influence upon the eutectic growth (see Table 1A). The two remaining parameters are the diffusion coefficient and the operating point (~b-value). For regular eutectics, the b-value is known to be close to 1 [13]. The diffusion coefficient constant can then be obtained by fitting equation (18) with the measured values of the 22V constant [10, 14, 20, 32, 40-42]. In the case o f irregular eutectics. the growth undercooling is higher, and the AT/x/rV constant have been measured accurately [14, 20] (thermal gradient in the liquid: 8 K/mm). The values of ~b and D can then be adjusted in order to fit both 22V and AT/x/~ constants. Before comparing the diffusion coefficient values obtained by this method with those measured independently under convection free conditions, it should be noted that, in the case of different densities of the phases, the solidification process induces fluid motions at the s/l interface. It is beyond the scope of this paper to analyze how this mechanism affects the diffusion process. However, a rough indication can be obtained by the relationship [44] De~ ~_ D 1

where AHf is the molar latent heat of fusion, N is the Avogadro number and Vmis the molar volume o f the phase. The Gibbs-Thomson coefficient, F~= trJAsf, is then given

1 . lO0(po -- p=)

(B2)

p~C~-p=C~

Table A1. Effect, Ex, of the physical constants on the spacings and undercoolings. A modification of a constant of E% (for instance, by alloying) leads to a modification of the spacing (2) or undercooling (AT) of E"Ex% Alloy Fe~C Fe-FeaC AI-Si

f,

m~

m~

F=

Fp

0=

0#

C*

A T

-0.80 --0.15

-0.03 0.75

-0.48 --0.25

0.03 0.03

0.47 0.47

0.03 0.03

0.07 0.07

-0.5 0.5

0.5 --0.5

1 0.93

A

T

-0.35 --0.31

-0.09 0.15

-0.41 0.35

0.09 0.09

0.41 0.41

0.07 0.07

0.28 0.28

-0.5 0.5

0.5 -0.5

1 0.53

T

-0.70 --0.12

-0.08 0.60

--0.42 --0.10

0.08 0.08

0.42 0.42

0.07 0.07

0.22 0.22

-0.5 0.5

0.5 --0.5

1 0.82

( A2

D

(9

Ai_AI2Cu

~A2

T

0.16 0.26

--0.39 0.07

-0.11 0.44

0.39 0.39

0.11 0.11

0.21 0.21

0.08 0.08

-0.5 0.5

0.5 --0.5

1 0

Sn-Pb

(2 AT

0.01 0.28

-0.41 0.34

-0.09 0.16

0.41 0.41

0.09 0.09

0.22 0.22

0.08 0.08

-0.5 0.5

0.5 -0.5

1 0

-0.17 0.42

-0.33 0.08

0.17 0.17

0.33 0.33

0.10 0.10

0.22 0.22

-0.5 0.5

0.5 --0.5

1 0

CBr4_422C16 ( A2

T

-0.42 -0.08

466

M A G N I N and TRIVEDI:

This equation is a rough approximation valid only for the case of a low volume fraction of t-phase and a very large C # - C~ value. It should therefore be applied only to the eutectic systems which fulfill the above conditions. For A1-Si and Sn-Pb, the effective diffusion coefficient values obtained from the experimental spacings and undercodlings measurements are 4.3.10-9 m2/s and 1.1.10-9 m2/s, respectively. The corresponding real diffusion coefficient values estimated from equation (B2) are 5.0.10 -9 m2/s and 0.71' 10-9 m2/s, very close to the experimental values found in the literature (5.4.10-9 m2/s for A1-Si [36] and 0.62.10-gm2/s for Sn-Pb [41]). Equation (B2) cannot be applied to the case of the A1-A12Cu and CBr,--C2C16 alloys. However, the density effect is not expected to be important in these cases (the densities of both phases are very similar in CBr4~2C16, while the A1-A12Cu phase diagram is too symmetrical to allow the density differences to have a large effect on the eutectic growth). The diffusion coefficient value of 2.8.10 -9 m2/s obtained for AI-A12 Cu is very dose to the measured one (3.4.10 -9 m2/s [39]), while the 1.24.10 -9 m2/s value obtained in CBr4--C2C16 lies well in the range of "probable" values found in the literature (1-2.10-9m2/s [10, 31]). For the above alloys, the effect of the fluid motions at the s/1 interface upon the effective diffusion coefficient value is not very important. This is not the case for the Fe~C system, in which the very large density difference between the phases as well as the very asymmetrical phase diagram lead the obtained Den value of 1.25.10 -9 m2/s to correspond to a real value of 4.7.10 -9 m2/s. Although there is a considerable disagreement between the values found in the literature, the real diffusion coefficient value obtained lies well between the two most often quoted values (1.2" l0 -9 m2/s [34] and 8.2" 10 -9 m2/s [33]). Note, however, that theses values are only an extrapolation to eutectic temperature of measurement realized in pure iron at 1600°C. The real diffusion coefficient value of 4.7.10-gmE/s determined above is valid when no fluid motion due to a large density differences occurs, and applies then to the case of Fe Fe3C system. Although this system is known to deviate from the 2 2 V = constant law at usual growth rates, it has been shown that this relationship is rather well respected over the very wide range of solidification velocities obtained from the slow directional solidification experiments to the very high growth rates produced during laser surface treatment [32]. The overall 22Vconstant measured experimentally can then be used to determine the operating point, leading to a ~b-value of 1.8, definitely higher than the value of 1 expected for a regular eutectic (with q~ = 1, the measured spacings would lead to a D-value of 15.4.10 -9 mE/s, incompatible with the other experimental results). This confirms the fact that, despite the regularity of the microstructure formed, the Fe-Fe3C system is in fact a faceted eutectic [see also Fig. 8(b)]. It will be shown in Ref. [14] that the ~b-value obtained is compatible with the deviations from the 22 V = constant relationship observed in Fe-Fe3 C. With only small adjustments (within the experimental error of the direct measurements) of the diffusion coefficient values found in the literature, Table 1 shows that the theory matches nearly perfectly the 22V and A T / x / ~ (when available) constants measured experimentally in each alloy. This confirms the validity of the proposed model, and in particular of the density correction introduced in the case of Fe~C. Table A1 gives the effect, E~, of each physical constant on the spacings and undercoolings. A modification of a physical constant of e % leads to a modification of the spacing, respectively the undercooling, of E •E~ %. This table is very useful when studying the effect of alloying element on eutectic growth.

EUTECTIC GROWTH APPENDIX 3

Nomenclature A0, An = coefficients of the composition field, rod eutectic (wt% or at.%) B0, Bn = coefficients of the composition field, lamellar eutectic (wt% or at.%) C ---composition (wt% or at.%) C~ --- eutectic composition (wt% or at.%) C0--JH tie line length (CO= C ~ - C °) (wt% or at.%) C~ = corrected tie line length [equation (10)] (wt% or at.%) C~, C# --- actual composition of ct and fl phases (wt% or at.%) C °, C~ --- solubility limits (Fig. 2) (wt% or at.%) D = interdiffusion coefficient in the liquid (m2/s) Defr --- effective D-value with liquid motion (Appendix 2) (m2/s) Ex=effect of the physical constants (Table A1)

(--)

F(x)---function describing the s/l interface composition [equation (11)] (--)

l(x) = s/1 interface shape (m) J0, Jl = Bessel functions of order 0 and 1 (--) Kc = chemical undercooling constant (Ks/m 2) K r = curvature undercooling constant (m K) Kin, Ka2 -- boundary layer undercooling constants (Ks/m 2) and (m K) K t = c o n s t a n t (KI=Kc+Km) of the 2V term in equation (17) (Ks/m 2) Kz=constant (KE=Kr+KB2) of the 1/2 term in equation (17) (mK) N - - A v o g a d r o number (6.022.1023) (tool -1) P = function of the volume fraction [equation (16)]

(--)

prod ----P-function for the rod microstructure [equation (A2)] (--) R = gas constant (8.31) (J/mol K) T = temperature field (K) T, = eutectic temperature (K) Tf = melting temperature (K) V = growth rate (m/s) f~,f# = volume fractions (--) k~, k# = solute distribution coefficients (--) l = constant in equation (22) (--) = weighted liquidus slope (K/wt% or K/at.%) rn~, m# = liquidus slopes (m~ < 0) (K/wt% or K/at.%) p = Pedet number (p = 2V/2D) (--) r = radial coordinate for rod eutectic (m) r a = radius of the fibers of t-phase (m) vm = molar volume (m3/mol) x = coordinate in the s/1 interface, origin: center of or-phase (m) z = coordinate in the growth direction, moving with the s/1 interface (m) F~, F# = Gibbs-Thomson coefficient (Fi = trit/As~f, i = ct, fl) (mK) AC~ = initial composition of the liquid minus Ce (wt% or at.%) AHf = molar latent heat of fusion (J/mol) AT = undercooling (ATe: chemical, ATr: curvature, __ __ ATk: kinetic) (K) AT ~, AT # = average undercooling of ct and fl phases (K) Asf = "average" volumetric entropy of fusion [equation (25)] (J/m3K) As~, AsfB= volumetric entropy of fusion (J/m3K) X = factor in equation (24) (N/m) f~=ponderation factor for Asf [equation (24)]

(--)

6C = average composition at the s/1 interface minus C0 (wt% or at.%) y~Y= surface tension of x/y interface (N/m)

M A G N I N and TRIVEDI: ?, = root o f

(--)

e= 0~, 0a = x = 2 = 2ex =

J~(x)= 0, approximately equal to nn

factor in equation (24) (Fig. 9) (N/m) contact angles at the three-phase junction (°) s/1 interface curvature (m -1) lameUar spacing (m) spacing at the minimum undercooling for a given growth rate (m) v,, vl~= density factors [equation (5)] (--) /~ --average density (g/m 3 or at./m 3) p~, pa = density (g/m 3 or at./m 3)

EUTECTIC G R O W T H trxy = z= ~b = ~o~=

467

surface energy of x/y interface (J/m2) torque surface tension (N/m) ratio of actual to extreme spacing (--) "frequency" factor in equation (1) and (AI)

(m -1) Subscript ct = fl = i= l=

A-rich phase (generally: major phase or matrix) B-rich phase (gc,,c~,,y. minor pha~c c,r . . . . p solid phase (a or fl) liquid phase