Acta metall, mater. Vol. 42, No. 12, pp. 3953-3966, 1994 Copyright © 1994 Elsevier Science Ltd

Pergamon 0956-7151(94)E0172-D Printed in Great Britain. All rights reserved 0956-7151/94 $7.00 + 0.00

3D STOCHASTIC MODELLING OF EQUIAXED SOLIDIFICATION IN THE PRESENCE OF

GRAIN MOVEMENT

Ch. CHARBON, A. JACOT and M. RAPPAZ Ecole Polytechnique Frdrrale de Lausanne, Laboratoire de M~tallurgie Physique, MX-G,

CH-1015 Lausanne, Switzerland

(Received 12 November 1993; in revised form 31 March 1994)

Abstraet--A three-dimensional stochastic model of equiaxed solidification of eutectic alloys in the presence of convection/sedimentation is proposed. Assuming uniform temperature, the heat balance is coupled to a microscopic description of grain evolution. Grain location is chosen randomly within the solidifying volume. Grain evolution is followed by mapping the interface of each grain with a large number of elementary surfaces. Each grain is fully described by its centre position, radius, and the status of its elementary surfaces (in contact with liquid or not). Grain movement is modelled by changing centre coordinates; evolution of the solid volume fraction is deduced by summing over all elementary surfaces which comprise the solid-liquid interface. A recursive method yields two-dimensional grain sections. Grain movements associated with convection and sedimentation are investigated, and their effects on the final grain structure, the solid-liquid interface, the impingement factor, and the cooling curve are shown.

INTRODUCTION V v is the volume fraction of the transformed phase. It has been derived under the assumptions of an

The equiaxed solidification of eutectic alloys involves infinite volume specimen and of non-moving grains two simultaneous events: nucleation and growth of randomly distributed within this volume. The KJMA the solid grains. Even at very low volume fraction of relationship just states that the probability that an solid, the grains impinge on each other and the infinitesimally small surface element falls in the trans- surface of the grains in contact with the liquid, Sv, is formed microstructure is equal to the transformed decreased by a factor, iJ, with respect to the extended volume fraction. This work demonstrates that the surface of the grains, Sv.ex (i.e. to the total surface of conditions required to satisfy the KJMA relationship the grains if there was no impingement). The im- no longer hold when the grains are allowed to move pingement factor, qJ = Sv/Sv.~x, therefore also during the transformation or if they tend to agglom- measures the decrease in the transformation rate due erate or to order. to the impingement of the grains, Accordingly, for a As mentioned by Gokhale et al. [4], Hillert [5] used given heat flow leaving the specimen, the impinge- an impingement factor of the form ~ = (1 - Vv)", ment has a direct influence on the cooling curve and where n is greater or smaller than unity if the grains hence on the undercooling. This effect, which is most are clustered or ordered, respectively. Using exper- noticeable near the end of solidification, results in a imental measurements, Speich and Fisher [6] derived change of the spacing of the eutectic lamellae and an empirical relationship to describe the evolution of even in the appearance of a metastable phase in the the area of the interface between the transformed and intergranular regions (e.g. cementite in cast iron), untransformed phases: Sv = ksF Vv(1 - Vv). Cahn [7]

Unlike solid state transformations for which the used a similar relationship: Sv---kc V~/3 (1 - Vv) 2'3. nuclei of the new phase are fixed in space, it is rather Rath [8] has suggested that the curve, Sv (Vv), should obvious that the movement of the grains, which is not be symmetric with respect to Vv = ½ because the strongly dependent upon the casting process, can initial nuclei are growing as spheres whereas the last directly influence their impingement. Nevertheless, integranular untransformed phase has a complex most models of solidification assume that the grain shape, thus implying that the rate of appearance of impingement is similar to that occurring in recrystalli- the surface is not the same as the rate of disappear- sation or solid state transformations, i.e. that the ance. For that reason, he used the relationship: grains are motionless and randomly distributed Sv = kR V§ (1 -- Vv) q, where the two exponents, p and within the volume of a uniform temperature speci- q, belong to the [0, 1] interval but are unequal. The men. The most commonly used relationship is that relation of Speich and Fisher was completed by Ni obtained by Kolmogorov [1], Johnson-Mehl [2] and et al. [9] and Bradley et al. [10] to give the following Avrami [3] (KJMA) for which ~ = (1 - Vv), where expression for qJ(Vv)

3953

3954 CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT

~(Vv) -- 1 if V v ~< V~ mp (la) instantaneous, the impingement was well described by the KJMA relationship.

and None of the models developed so far for the 0 j _ v~p, impingement of the grains account for the fact that,

Vim ( 1 - - V v ) v~mp during solidification, the grains can move as a result ( Vv ) V v ----7 1 ----V ~ of convection in the fluid and/or sedimentation of the

if Vv > --vl/imp" (lb) solid grains. The aim of the present contribution is to assess numerically the influence of the grain move-

V~, n~p is the volume fraction of the transformed phase ment on the solidification of equiaxed eutectic alloys at which the impingement begins. Although this for different casting conditions. relationship gives a continuous variation of ~ from 1 at the onset of the grain impingement to 0 at the end PHYSICAL MODEL of the transformation, it should be emphasised that Heat balance this relationship only gives a qualitative description

The system modelled here is a small volume el- of the grain impingement since it is not based on any geometrical or physical model. Finally, numerical ement containing an alloy of eutectic composition. techniques have been developed by Price [1 l, 12] in Assuming that its Biot numbert is small, the tempera- order to study the impingement of spheres, ellipsoids ture, T, of the specimen can be considered as uniform and bipyramids regularly nucleated at the nodes of a and the heat balance becomes then

cubic network. He studied the evolution of Sv(Vv) S dT df~ and its influence on the rate of transformation. Q -~ = pCp - -~ - L --~. (2)

All of the models previously described were de- signed for solid state transformations and recrystalli- Q is the heat flux leaving the specimen, S / V is the sation, two processes in which the grains cannot surface-over-volume ratio of the specimen, f~ is the move. Some of these models, in particular the KJMA volume fraction of solid, pcp and L are the volumetric relationship, were directly applied to the case of specific heat and latent heat, respectively. The cooling solidification. Arguing that, during solidification, eu- curve, T (t), can be calculated if the heat flow and the tectic grains are free to rearrange in the liquid, at least evolution of the volume fraction of solid, f~(t ), are up to a certain volume fraction of solid, Zou [13] has known. This latter entity is a convolution of nucle- studied the impingement of equally sized spheres ation and growth mechanisms [17]. nucleated simultaneously on a simple cubic (s.c.) or face centred cubic (f.c.c.) network. He obtained ann- Nucleation

lytical expressions for the evolutions, ~'(Vv) and Two nucleation laws will be used in the present Sv(Vv). Mampaey [14] has used a Monte Carlo work. The first one assumes that the heterogeneous method to calculate the impingement of spheres and nucleation of the grains occurs instantaneously on a ellipsoids of various eccentricities. He computed the distribution of nucleation sites, which become active surface, Sv, for one grain randomly nucleated within as the undercooling increases [13,18]. Using a Gaus- a cubic box. This box was surrounded by 26 identical sian distribution, the density of grains, n (AT), at an boxes located at the first-, second- and third-nearest undercooling, AT = T e - T (T e is the equilibrium neighbour nodes of a cubic network and within which eutectic temperature), is thus given by a grain was also randomly nucleated. Although Mampaey averaged such situations over several thou- n (AT) = nm~x sand grain configurations in order to obtain a mean A T ~ x / ~ curve, Sv(Vv), his method introduced some ordering /'~r [- / AT ' - AT n "~2 7 of the grains since only one grain was allowed to x J0 e x P l - [ --~------ / 1 d(AT') (3) nucleate within each box. Therefore, it is not surpris- L \ ~/2ATo J d ing that the curve obtained by Mampaey was in where nrnax is the total grain density obtained at large between the KJMA relationship and the curve calcu- undercooling, ATn and ATo are the mean under- lated for the s.c. arrangement. Based upon a stochas- cooling and standard deviation of the distribution, tic technique similar to that described by Gandin et respectively. al. [15] for the modelling of a primary phase solidifi- For comparison, a constant rate of nucleation will cation, the model of Charbon and Rappaz [16] also be used coupled realistic nucleation law and growth rate with n ( t) = 0 if t < te (4a) a heat balance in order to calculate the impinge- ment of eutectic grains. These authors showed that, n ( t ) = ri0(t - t e ) if t /> te. (4b) even though the nucleation of the grains was not

ti 0 is a constant nucleation rate which applies as soon

1"The Biot number is given by (hl/x), where h is the heat as the temperature is smaller than the eutectic tem- transfer coefficient, l is a characteristic dimension of the perature, Te, and te is the time at which the system specimen and x is the thermal conductivity, becomes undercooled (i.e. for which T(t~) < To).

CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT 3955

Growth 1 + 261

As soon as a nucleus appears in the undercooled melt, it is assumed to grow. Under stationary con- ditions, the growth rate of the eutectic front, v, is an increasing function of the undercooling, AT. Jackson A | and Hunt [19] have shown that a regular eutectic (i.e. a eutectic in which there is no faceted phases) grows near the extremum condition given by ATe = ATf; ATe and ATr are the contributions to the total undercooling associated with the solute diffusion and the mean curvature of the front, respectively. Accord- ingly, the growth rate, v, is related to the total undercoolingt, AT = ATe + ATr, via the well-known Jackson-Hunt relationship

v ( A T ) = A A T 2. (5)

A is a constant which is a function of the alloy. A similar quadratic dependence has been derived by Jones and Kurz [20] for irregular eutectics. [

Since the temperature is uniform within the sample C and the growth rate is assumed to be perfectly isotropic, the grains grow as spheres at the same rate regardless of their size (or nucleation time). The different radii of the grains are thus due to their Fig. 1. 2D schematic diagram of the solidification of a small different times of nucleation only. At a given time, t, specimen containing a eutectic alloy. The gray zone corre-

sponds to the volume for which the volume fraction of solid the radius, r (tN, t ) of a grain nucleated at a time trq is calculated. In two dimensions, each grain surface is is simply given by the integral of the growth rate from mapped by a set of circular segments similar to a bicycle the time of nucleation to the time of observation wheel. Segment (A) contributes to the effective solid-liquid

interface, segment (B) is no longer in contact with the liquid r (tN, t ) = v [AT (z)] dr. (6) and segment (C) is not in the computation zone.

N

Finally, it should be noted that the overlap of the casting. Its temperature is assumed to be uniform and the heat extraction rate appearing in equation (2), solute layers ahead of the growing eutectic lamellae,

which occurs when two grains get very close, will be Q(S /V) , is known. It can be estimated from a neglected here. DTA-type measurement or from the resolution of the

heat flow equation if this is part of a larger specimen. At the beginning of the simulation, the temperature NUMERICAL MODEL is set to a value above the eutectic temperature, To,

The simulation of the cooling curve of a eutectic and the metal is liquid. When the alloy becomes specimen and of its solid-volume-fraction evol- undercooled (AT > 0), the density of nuclei is up- ution requires not only knowledge of the nucle- dated at each time step according to the nucleation ation and growth laws described before but also a law given in the previous section [equations (3) and description of the way the grains impinge on each (4)]. Therefore, the number of grains fin ~, nucleated other. In deterministic modelling of solidification during one time step, 6t, is given by [13,18,21-25], this is usually achieved by multiplying I- -1 the increment of extended solid fraction during a time ~ N ~ = ( I +2Al)3" L n ( A T t ) - n ( A T t ~t)J. (7) step, 6t (i.e. Sv.exVrt ) by the impingement factor, qJ, given by the KJMA model. In stochastic models The location of each of these new nuclei is randomly [15, 16, 26, 27], the evolution of each grain is mod- chosen within the cube of length (l + 2Al) by three elled separately and the impingement is automatically randomly-generated coordinates, (x, y, z). If the accounted for by the competition between the grains, centre of a new nucleus falls within one of the solid

grains already formed, it is discarded. This means Nucleation and growth of grains that the extinction of the nucleation sites by the

In the present model, a cubic specimen of edge growth of the solid phase is already accounted for (l + 2Al) is considered (see Fig. l). It can be a small with this procedure [28, 29]. Each new grain is num- specimen cooled in a furnace or part of a larger bered with an incremental integer value.

During the same time step, all the grains previously "~The thermal undercooling and the undercooling associated formed in the liquid grow with the growth rate given

with the attachment kinetics of the atoms to the interface by equation (5) (the grains nucleated during this time have been neglected here. step start to grow at the next time step). As the

3956 CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT

within any other grain (e.g. facet labelled "B" in Fig. 1). In this case, the number of the neighbour grain within which this direction falls is stored in order to count the number of neighbours of each grain. Furthermore, the knowledge of the state indi-

f t ces °f all the directi°ns' kij' and °f the neighb°ur grains which are in contact allows one to deduce the number of clusters and isolated grains in the specimen.

Since the boundaries of the finite specimen biases the calculated volume fraction of solid, the nucleation and growth of the grains are allowed to occur in a

Fig. 2. Schematic paving of half of a sperical grain hemi- volume of edge, l + 2Al, but the evolution of the sphere into almost equally sized surface elements. The volume fraction of solid is calculated for a smaller surface of each grain is mapped with two such hemispheres, cube of edge, 1. The surface elements of the grains

which fall outside of this smaller cube (e.g. surface thermal field is uniform, the radius increment, 3r~, element "C" in Fig. 1) are not counted. This method during one time step is the same for each grain, i was preferred to the periodic boundary conditions

used previously [16] because it is much less compu- fir ~ = fir' = v (A T ' - ~t)~ t = A (A T t - 6t)Zbt tational-time intensive.

The knowledge of the state index, k o, of each for i = 1, N[ (8) shooting direction, j, for each grain, i, allows direct

calculation of the solid-liquid interface area, S t of s-I, where Ng is the total number of grains at time t. The increment in the volume fraction of solid the grains. It is given by the sum of the surface

during one time step is given by the product of the elements which are still in contact with the liquid Ndir solid-liquid interface area by the radius increment. In S ' (r ~): )-" ~jk u" (9)

order to calculate the solid-liquid interface, a method s-t = t similar to that developed by Ito and Fuller [30] for the ~= 1 j= 2D simulation of the growth of ellipses and polygons Please not that is used in the present model. As shown in Fig. 1 for ~d~, a two-dimensional case, the surface of each grain is ~ ~j = 4~. divided into a regular paving of Ndi r e l e m e n t a r y J=

surfaces or facets. In two dimensions (Fig. 1), the The solid-liquid interface area and the radius circular surface of the grain would be divided as a increment, fir, during one time step being known, the bicycle wheel whereas in three dimensions (Fig. 2), solid fraction increment, cSf~ is given by the surface of the grain is paved like an igloo. Only half of the surface of the grain is shown in Fig. 2, the 3fts = S tlOrt with 6 r ' = A (ATt-Ot)26t. (10) other half being symmetric. The igloo associated with l 3 any grain is constructed layer by layer starting with Using the heat balance [equation (2)], the new tem- the equatort. The number (or the length along the perature, T t, can be calculated zero latitude) of the surface elements on the equator is fixed. The height along the longitude of all the S . Q t. ¢5t + L . 6fts

V layers is set to the same value as the length along the T ' = Tt-~t-+ (11) equator. Then, for each layer on a latitude, the pCp number of elements is chosen as the integer value

As mentioned in the introduction for solid state which gives a length as close as possible to their height. Since the paving procedure results in elemen- transformations, the grain impingement is described tary surfaces which are close but not exactly equal, by the impingement factor

the solid angle of each facet, laj(j = 1, Nair), is stored S Ll S t s--I together with the direction ("shooting direction") ~ ' S t - ~ q (12)

~-1,~x ~ 4~ (r~) 2 associated with its centre. The status, k,j, of each surface element, j, for each ,'= 1

grain, i is defined as follows. If this facet is still in or by the solid-liquid interface area, S'~_t. In order to contact with the liquid (e.g. facet labelled "A" in compare the various models of grain impingement Fig. 1), the index k u is equal to unity. It is set to zero and to be independent of the grain density, it is if the corresponding direction of length, r ~, falls necessary to normalise this latter value. The nor-

malised solid-liquid surface, s Lz, is given by

tThe elementary surfaces on the equator of the igloo shown S t s-I in Fig. 2 are in fact only half of the facets for symmetry s ts--I = tot -------"""'~ (13 )

• Ng • 4rrr reasons.

CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT 3957

where N got is the final numbera___rN~O._____it] .Of grains whose centre ~ / ~ is within the small cube of edge, 1, and f i s the average final radius of these grains defined as

f = (14) j

i' ~ 0 ~ Movement of Grains T

With the shooting method, a grain, i is fully described by the position of its centre, (xi, Yi, zi), by id LL) its radius, r ~, and by the state indices, k,), of its Ndi r ~ _ _ shooting directions. Imposing a movement to the grain is then simply achieved by changing the coordi- nates of its centre, its radius and state indices of its shooting directions being unaffected. It should be pointed out that only translations of the grains will

Fig. 3. 2D schematic representation of the clustering inter- be considered here. At each time step, a movement of action (left) and of the ordering interaction with ct = 1 the grains is made according to the different schemes (right) for the same translation vector, T, of a grain (i). The detailed below prior to calculating the nucleation and gray circle corresponds to the new grain position. growth.

When the surfaces of two grains come into contact

(with or without grain movement), they will form a or the whole cluster is moved by this cluster during the growth stage. Clusters can not vector only if it does not encounter any separate afterwards and thus the number of grains in other grain/cluster during the movement. any cluster can only increase over time. The number If a collision is detected, it is moved in the of clusters in the specimen, which increases at the direction T until the first point of contact beginning of solidification, decreases near the end [see Fig. 3(a)]. Since these colliding grains and reaches a final value of unity when the specimen will grow during the same time step, they is fully solid (i.e. when all the grains are connected), will then belong to the same cluster. All the grains belonging to the same cluster move This interaction is supposed to mimic with the same translation velocity. A mass balance is the random movement of the grains when applied to the specimen in order to account for the the shearing stresses in the liquid are portion of the solid grains which leaves/enters the small: the grains move randomly with system and modifies the volume fraction of solid. respect to each other but the contact time

In theory, any translation movement and inter- is large enough so that they can cluster due action of the grains could be introduced in the to solidification. simulation. In particular, the speed of each (ii) Ordering interaction. After the vector T grain/cluster could be defined as a function of any has been randomly chosen for a given variable of the system: position, temperature, volume grain or cluster, an "interaction" distance fraction of solid, time, radius, etc. The present inves- is defined which is equal to aT, where ct is tigation has been limited to two types of movement: a constant greater or equal to unity. If random translation of the grains/clusters with differ- there is no collision detected for the trans- ent interactions and sedimentation. It is believed that lation, a T, the grain/cluster is moved by T. most of the grain movements occurring during the If a collision would occur in this direction, solidification of metallic alloys may be described by then the first neighbour in the direction a combination of these two simple cases, with the - c t T is searched and the new grain pos- exception, however, of the rotation of the clusters ition is set to the median position between which is not considered here. the two contact points [see Fig. 3(b)]. The

Random translation. The amplitude of the move- ordering interaction is defined as weak or ment is given by a constant speed Vmvt, multiplied by strong when a = 1 or 2, respectively. the time step, ft. The direction of the movement is The ordering interaction could occur in chosen randomly among the Ndir directions used for

processes where high shear stresses caused the paving of the grains. The amplitude and the

by a strong velocity gradient exist (e.g. direction define a vector of displacement T (see Fig. electromagnetic stirring of the melt [33]): 3). Two kinds of grain interactions have been defined the grains in this case move randomly but for this situation. They are illustrated in Fig. 3 for the the contact time is too small to induce same translation vector, T: clustering.

(i) Clustering interaction. The translation vec- Sedimentation. For each grain or cluster, the hori- tor, T, being chosen randomly, the grain zontal component of the vector T is chosen randomly

3958 CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT

Table 1. Standard physical parameters of the model listed in Table 1 whereas Table 2 summarises the data Equilibrium eutectic Te= 1154 °C used for the various computations. It should be

temperature Volumetric latent heat L = 1.44 × 109 j . m-3 pointed out that the time step, 6t, plays an important Volumetric specific heat pCp = 6.37 x 106 J. m -3 . K-' role when the grains are allowed to move. Each grain Growth rate constant d =4 x 10 -s m. K -2. s -1 Nucleation law parameters nma~=6 x 10 '° m -~ or cluster can be considered as a random walker

AT,= 19.5 °c whose position changes by the quantity ( U m a x f / ) at AT, = 5 °C each time step. The mean distance, d, travelled by a h 0 = 7.4 x 106 m -3 . s 1

Surface-to-volume ratio S /V = 500 m ~ given grain during a time interval, t - to, is given by Hea t flux Q = - 3000 W . m - 2 [

d = Vma x • f t V/--~p = Vmax' f t / t -- to x/ 6t

in the horizontal (x ,y) plane. The vertical z-com- ponent of the translation is given by the limiting = VmaxX/(t- to)' fit. (16)

speed, vst, predicted by Stokes [31] This shows that the distance increases with the square

root of the time step and so will be the rate at which fzi = vst, t" f t = 2(pl - Ps)g r ~' fit = Cst' r 2' f t (15) grains order or cluster.

9ql

R a n d o m translat ion where Pl and Ps are the density of the liquid and solid phases, respectively, g is the gravity and ~/l the The two-dimensional cross sections of the com- dynamic viscosity of the liquid metal. The amplitude puted three-dimensional grain structures shown in of the vertical movement, f z t , during a time step is Fig. 4 correspond to the same thermal and nucleation different for each grain and is proport ional to the conditions and are displayed during their course of square of the radius of the grain, ri. For clusters, r~ solidification at four given volume fractions of solid is taken as the radius of the equivalent sphere having ~ = 0.1, 0.3, 0.7 and 1). The four sets of microstruc- the same volume as the cluster, tures labelled A - D correspond to the four different

The grains/clusters are stopped when they come assumptions made previously for the grain move- into contact with the bot tom boundary of the domain ment. In case (B), the grains are assumed to be fixed or with other grains/clusters which are already in space and the result is similar to that previously stopped. This behaviour is similar to the clustering published [16]. The grain structure shown in (A) is interaction presented above, associated with a clustering random translation of the

grains whereas those shown in (C) and (D) corre- RESULTS AND DISCUSSION spond to the weak (ct = 1) and strong (ct = 2) ordering

interactions, respectively. The recursive methodt The stochastic microscopic model of solidifica- used to visualise the grain structures is identical to

t ion-convect ion presented before has been applied to that described by Mahin et al. [32]. the case of lamellar grey cast iron. Even though this As can be seen in Fig. 4, there is a continuous eutectic alloy is not regular because of the graphite evolution in the ordering of the grain structure when phase, it has been extensively studied [13, 21, 23] and going from the clustering interaction (A) to the strong is well documented. The thermophysical data and the ordering interaction (D): at the end of solidification parameters of the Gaussian nucleation law used for (f~ = 1), the grains in (D) are much more uniform in the computat ions are taken from Zou [13]. They are size and are quite regularly positioned in space as

compared with those shown in (A). The ordering of tThe recursive method of Mahin et al. has the advantage the grain structure in Fig. 4 is accompanied by an

that the resolution with which the grain structure is increase of the port ion of grains which remain iso- drawn is defined after the solidification calculation has lated during solidification. This is particularly evident been terminated. As a consequence, the grain boundaries may appear with a finite width (i.e. larger than the in Fig. 4 for the structures a t f~=0 .7 : almost all the resolution of the display unit) when they are almost grains appear still as individual circles in case (D) parallel to the section plane, whereas all the grains are clustered in (A). This effect

Table 2. Parameters used in the computat ions

Cases A B C D E F

Movemen t R a n d o m N o n e R a n d o m R a n d o m Sedimentat ion Sedimentat ion Interact ion Cluster ing N o n e Weak ordering Strong ordering Cluster ing Cluster ing

(~ = 1) (= = 2 ) Nucleation G a u s s i a n G a u s s i a n Gaussian Gaussian Gaussian Constant rate Ix/Al~[mm] 6/0.6 6/0.6 6/0.6 6/0.6 4/0.5 .4/0.5 ly/Aly [mm] 6/0.6 6/0.6 6/0.6 6/0.6 4/0.5 4/0.5 I:/AI: [mm] 6/0.6 6/0.6 6/0.6 6/0.6 8/1 8/1 Umv t [#m/sl 40 0 40 40 3 3 n~t 1500 1500 1500 1500 1500 1500 nair 412 412 412 412 412 412 6t [s] 1 1 1 1 1 1

CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT 3959

o 0oO00 oO - o ~ "~ ~ ., 0 o ~ - i ~ - - D ° ~ 0 o O 0 0

O o o 0 o " o D o 0 o

o I Q O o o ) ° ° 0 "c

" ~ o ~ ? o ~ ao ~ o o o o o ® o ° e o o

0 o'~ W e ° o ~ ~ o i ~ ~ , - - - , , J ~

o goO ao l Fo°% • )1 ® o o@

A B C D

Fig. 4. 2D cross section micrographs of the 3D computed grain structures at f~ = 0.1, 0.3, 0.7 and 1. The computation parameters for the four cases are listed in Table 2: (A) clustering interaction; (B) no grain

movement; (C) weak ordering interaction (c~ = I); (D) strong ordering interaction (7 = 2).

is emphasised in Fig. 5 where the fraction of isolated It is to be noted in Fig. 5 that, for the strong grains, Nisolat~d/Ngt, for the four cases shown in Fig. 4 ordering interaction case (D), the grains remain iso- is plotted as a function of the volume fraction of lated up to a volume fraction of solid of about 0.58. solid. Since these curves have been norrnalised with This value falls in between the compacities of the s.c. the number of grains already formed at the corre- (f~ = 0.52) and f.c.c. (f~ = 0.74) arrangements. It is sponding volume fraction of solid, they are equal to also close to the experimental observation of Viv~s unity at the beginning of solidification regardless of [33] who observed that, during magneto-stirring of the grain interaction (A-D) t . For the sake of corn- globulitic aluminium alloys, the movement of the parison, the fraction of isolated grains has been metal is suddenly stopped at a volume fraction of drawn for four other cases of interest: the K J M A solid close to 0.6. This corresponds to the point where model (i)$, the s.c. (ii) and f.c.c. (iii) arrangements of all the grains come into contact and form a coherent regular spheres and the case of grains without ira- skeleton, thus blocking their movement. pingement (iv). As already noted by Charbon and Rappaz [16], the

case of motionless grains (case B) is very close to the tThe noise seen in the curves (A-D) is due to the limited K J M A prediction (case i). As can be seen in Fig. 5,

number of grains and to the fact that they can leave or the clusters in this case form at a very fast rate as soon enter the calculation domain during solidification, as solidification starts. At a volume fraction of solid

SFor the KJMA model, it can be shown that the fraction of of 0.2, only 20% of the grains remain isolated and isolated grains is simply given by (1 _f~)8. For the s.c. and f.c.c, models, this fraction is discontinuous at they are all clustered at about f~ = 0.5. This shows f~ = 0.52 and 0.74 respectively, that, under the assumption of a random location of

3960 CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT

1 . . . . . - - - . . . . . 7 iv J grains which remain fixed in space, the impingement

iiir of the grains begins with their nucleation. Accord- o.8 ingly, the splitting of the equiaxed solidification of

eutectic alloys into three stages [9, 17, 34] (the nude- "~ n ation, the free growth and the growth with impinge- "~ 0.g ment) can only be made if the nuclei are allowed to

rearrange in the liquid (case D in Fig. 5). As already observed in Fig. 4, the clustering of the grains is faster

0.4 L i ~ \ 0 for case (A). .~ /~ \ "~ Figure 6 shows the normalised volume distri- ~0.2 \ ~ , ~ B ~ bution of the grains for the four situations illus-

trated in Fig. 4. As the interaction between the r~ ~ : ~ x x~xk,,x~~ ' ~ grains changes from clustering to repulsive (i.e. from

0 A ' " - r ' - - ~ ' ' A to D), the distributions become narrower. This 0.2 0.4 0.6 0.8 I confirms that the grain size distributions observed

Solid fraction [-] in as-solidified microstructures are associated with Fig. 5. Fraction of isolated grains, N~,o~at~/N st as a function the impingement of the grains rather than with their ofthevolumefractionofsolid,f~,aspredictedwithdifferent different times of nucleation [16], at least for the models: (i) KJMA model; (ii) s.c. arrangement; (iii) f.e.c. arrangement; (iv) no grain impingement. The curves (A-D) instantaneous nucleation law chosen here. correspond to the situations shown in Fig. 4 (see also The evolutions of the normalised solid-liquid inter-

Table 2) face of the grains, s~_~, from equation (13), and of the

20% i(a) [(b)' 15%

10%

5%

. . . . I I w , , . . . . . . . . . . . . . . . . . . , , , , . . . . . . . . . . . . . . . . . . . . 0%

20% i(c) d)

15%

~10%

il ll,ljj,, , ,Jl ,, b j, , . , , J , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . , , . . . . .

o% 0 I 2 3 4 0 1 2 3 4

Normalised volume [ - ] Normalised volume [ - ]

Fig. 6. Distributions of the normalised volume of the grains at the end of solidification for the four microstructures shown in Fig. 4 (see also Table 2).

CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT 3961 1 / /

(a) / iv are identical except for the finite volume domain / considered in the numerical computations. The effect /

0.8 / ' \ of the duration of the nucleation period is shown to \(~ii be of minor importance for the impingement of the

grains. 0.6 The weak ordering interaction curve (c~ = 1, case

C) is in between the predictions of the KJMA (i) and s.c. (ii) models whereas the curve corresponding to

0.4 the strong ordering interaction (~ = 2, case D) is in between the results of the s.c. (ii) and f.c.c. (iii)

0.2 arrangements. As the ordering interaction increases (cases C and D), the effective solid-liquid interface [Fig. 7(a)] is displaced towards the curve obtained for

0 . . . . the compact arrangement of regular spheres. The 0 0.2 0.4 0.6 0.8 f.c.c, curve (iii) shown in Fig. 7(a) is the upper limit

Solid fraction [-] which minimises the impingement of the grains (i.e. (b) ideal rearrangement of uniform size grains)t. On the

. . . . opposite, the clustering interaction (case A) shifts the 1 ~ , . \ ~ - -\ iv effective solid-liquid interface below the KJMA curve

~ , ~ \ \ \ ~ \~iii towards the horizontal axis. The horizontal axis is '-7" another limiting case for which a very large number ~ 0 . 8 ~ ~ ~ \ ~ \ of grains would cluster as soon as they nucleate, ln

\ this case, the normalised solid-liquid interface would ~ 0 . 6 \ \ vary a s (Us) 2'3 . . . . 1/3 A~ k B ~ . x ~ ~ , . [ N g ] and thus would tend towards .~ tot zero as the final grain density, Ng , becomes large. In

A ~ i k ~ , k ~ \~ summary, the horizontal axis and the analytical curve '" ~ obtained for the f.c.c, arrangement [13] delimit the ,~ 0.4

domain of the possible ss i curves. The same trend can be seen in Fig. 7(b) for the impingement factor, T.

0.2 It should be pointed out that the relation of Hillert, T = (1 _f~)n, cannot account for the sharp change

0 , , , , observed in the slope of the curve of the strong 0 0.2 0.4 0.6 0.8 1 ordering interaction [case D, Fig. 7(b)]. Even the

Solid fraction [-] curves A and C corresponding to the clustering

Fig. 7. Normalised solid liquid interface, ss_ 1 (a) and im- interaction and to the weak ordering interaction, pingement factor, q~, (b) of the eutectic grains as a function respectively, cannot be described precisely by such a of the volume fraction of solid. The cases labelled (A-D) simple relationship. It also appears in Fig. 7(a) that

and (i iv) correespond to those of Fig. 5. the clustering interaction (curve A) is almost sym-

metrical with respect to f~ = ½ whereas the curves C impingement factor, T, from equation (12), are dis- and D corresponding to the ordering interaction are played as a function of the volume fraction of solid no longer symmetric as suggested by Rath [8] [growth in Fig. 7 for the four cases (A-D) of Fig. 4. These of spheres up tof~ ~ 0.58 for curve (D) and solidifica- curves can be compared with the analytical solutions tion of complex intergranular liquid pockets above calculated for the KJMA model (i), for the s.c. (ii) this value]. and f.c.c. (iii) arrangements of spheres and for grains Figure 8(a) shows the cooling curves calculated without impingement (iv). As can be seen, both s~_, with the heat balance [equation (2)] and the various and T are strongly affected by the interaction of the growth models leading to the grain structures dis- grains. As noted above (see also Ref. [16]), the played in Fig. 4 (cases A-D). The corresponding reference case of motionless grains (B) is almost evolutions of the volume fractions of solid are not superimposed with the KJMA prediction (i). This is shown because they are almost linear and superim- not surprising since the assumptions of both models posed for all the grain interaction models used. This

is due to the fact that, at the eutectic plateau, the

tOne might imagine grain structures which are even more specifc heat variation, peodT/dt, is very small corn- ordered than the f.c.c, arrangement of spheres by consid- pared to the latent heat contribution, Ldf~/dt. Using ering for example the nucleation of grains at the end of a constant heat flux, Q, in equation (2) thus gives solidification in the octahedral-tetrahedral intergranular df~/dt ~ constant, i.e. linear evolutions of the volume liquid regions of the f.c.c, network. However, since the fraction of solid, f~(t ). Since the solid-liquid interface effective solid-liquid interface of the grains has been normalised with the final grain density [equation (13)], area, S~ ~, calculated with the various grain inter- the s~_~ curve will still be lower than the curve (iii) in Fig. action models increases from cases (A) to (D) in 7(a). Fig. 7(a), the growth rate and thus the undercooling

AM 42/12 D

3962 CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT

(a) menting grains was carried out in a parallelepipedic 1160 volume element rather than a cube [cases (E) and (F)

in Table 2)]. Two different nucleation laws were used: an instantaneous nucleation law with a Gaussian

1155 distribution of the nucleation sites [equation (3)] and a constant nucleation rate [equation (4)] (see Table 1 for the value of the parameters). In the Stokes

~ ~ ~ relationship [equation (15)], an arbitrary value of

~ 1150 Cst = 200 [m-is - l ] was used. It should be noted that this relationship is only valid for one grain in a steady state laminar flow (i.e. small Reynolds number), a

1145 situation which is usually far from that occurring during the casting of metals. The arbitrary value of Cst was chosen to yield a time of grain settlement of

, , the same order of magnitude as the solidification 1140 time.

0 20o 40o 6o0 8o0 1000 1200 In Fig. 9, the vertical two-dimensional cuts (i.e. Time [s] containing the z-axis of gravity) of the computed

(b) grain structures (E) and (F) are shown for the two 14 [] 0.99 nucleation laws atf~ = 0.1, 0.3, 0.7 and 1. Using an

solid fraction : [] 0 . 9 9 9 instantaneous nucleation law (case E), the formation 12 of new nuclei is stopped at the beginning of the

recalescence. At that time, the volume fraction of 10 solid is less than 1%. As the nucleated grains are

o~ falling down during the remaining time of solidifica- 8 tion, the liquid has been washed away of its nucle-

= ation sites in the upper part of the specimen. Since 6 nuclei cannot form anymore in this region, the

upper part of the melt can only solidify from the 4 growth of the fallen grains and the final structure

has the appearance of a columnar zone. The effect 2 of nuclei disappearance due to the sedimentation

of the grains has been observed experimentally by 0 . ' . : Weinberg [35] and by Ohno [36] in the case of

A B C D equiaxed dendrites of tin and aluminium alloys,

Fig. 8. Cooling curves (a) and undercooling atf~ = 0.99 and respectively. 0.999 (b) for the four cases (A-D) shown in Figs 4-7 (see The constant nucleation rate, ti 0 listed in Table 2

also Table 2). was chosen to calculate case (F) in such a way that the final number of grains in the specimen was about

must decrease accordingly in order to satisfy equation the same as that obtained with the Gaussian nucle- (10). This is indeed observed in Fig. 8(a): the cooling ation law (case E). Since the grains are continuously curves (A-D) are characterised by increasing tem- nucleated within the remaining liquid, the grain den- peratures, and thus decreasing undercoolings, near sity within the specimen is quite homogeneous (see the end of solidification. This effect is emphasised in Fig. 9). As the instants of nucleation of the grains Fig. 8(b) where the undercoolings, AT, at the volume may be very different, the grain boundaries are much fraction of solid f~ = 0.99 and f~ = 0.999 are plotted more curved than those calculated with the instan- for the various grain movement models. Although the taneous nucleation law (case E). In a two-dimen- maximum undercooling difference between the vari- sional cross-section, some grains may even appear as ous models does not exceed 3°C, the cooling rate of encapsulated within larger ones (this can be seen for the specimen, given by Q • ( S / V ) / ( p C p ) , is only one grain in Fig. 9, case F). Such a situation occurs -0.24°C/s. Larger cooling rates would result in when the section plane of the micrograph just cuts the

nose of the hyperboloid of revolution, which de- higher undercoolings and larger differences among the various models. This can have a direct influence scribes the grain boundary [32]. Figure 10 shows on the formation of metastable phases near the end horizontal cuts of the computed final grain structures of solidification. (E) and (F) at the bottom, middle and top of the

specimens. As appears clearly in these micrographs, the grain density increases from top to bottom when

Sed imen ta t ion an instantaneous nucleation law is used (case E), In order to extend the length of the free fall of the whereas it remains about constant in the case of a

grains, the calculation of the solidification of sedi- constant nucleation rate (case F).

CHARBON e t al.: SOLIDIFICATION AND GRAIN MOVEMENT 3963

o o @ "e°O

e ° @ g [

E © 0 o 0

o F F

o

o © ©

F @ ~ ° q

) © I

f s=0 .1 f s=0 .3 fs=0.7 f s = l Fig. 9. 2D vertical cross sections of the 3D grain structures calculated atf~ = 0.1, 0.3, 0.7 and l in the presence of grain sedimentation. The cases (E) and (F) correspond to an instantaneous nucleation law

(Gaussian distribution of nucleation sites) and to a constant rate of nucleation, respectively.

The evolutions of the solid-liquid interface area, "columnar" front seen in Fig. 9 (i.e. the horizontal s~_~, and of the impingement factor, qJ, corresponding surface of the specimen) normalised by the equivalent to the two grain structures E and F of Figs 9 and 10 surface of the grains [see equation (13)]. The effect of are plotted as a function of the volume fraction of the sedimentation of the grains on the cooling curves solid in Fig. 1 l(a) and (b), respectively. For compari- is shown in Fig. 12. The undercooling suddenly son, the same analytical results (i-iv) already dis- increases near the end of solidification in order to played in Fig. 7 are again shown in these figures. For compensate for the small solid-liquid interface area both nucleation laws, the interface area is below the [see Fig. 1 l(a)] and to maintain a constant evolution curve predicted by the KJMA model. This is due to of the volume fraction of solid. This can have again the fact that the sedimentation tends to cluster the direct implications on the formation of metastable grains at the bottom of the specimen. Since the phases in certain parts of the specimens [e.g. the top normalisation coefficient defined in equation (13) for part of specimen (E)]. the solid-liquid interface is based on the final number of grains, Ngt°t, it is overestimated when using a

constant nucleation rate, especially at the beginning CONCLUSION of the solidification. This is why the curve (F) lies below that corresponding to the instantaneous nucle- A new stochastic model, which is particularly ation law (case E). It can also be seen that the curve efficient for modelling the movement of the equiaxed (E) remains at a constant value near the end of grains during solidification, has been described. Since solidification (forf~ between 0.8 and 0.99). This value the interface of each grain is mapped with a large is small since it is given by the surface of the final number of elementary surfaces, the movement of

3964 CHARBON e t al.: SOLIDIFICATION AND GRAIN MOVEMENT

E

F

bottom middle top Fig. 10. 2D horizontal micrographs of the final grain structures at the bottom, middle and top of the

specimens (E) and (F) shown in Fig. 9.

each grain can be imposed independently as a func- impingement of eutectic grains in the presence of t ion of any variable of the problem. Although the convect ion/sedimentat ion. The effects of various movements of the grains which have been invcsti- grain movements and interactions on the develop- gated in the present study are quite simple (trans- ment of the grain structures, on the cooling curves lations only), the method can be applied to more and on various microstructural parameters (size dis- complex situations in which for example the grains/ tribution, fraction of isolated grains, solid-liquid clusters would rotate. Realistic collisions between the interface, impingement factor) could be investigated grains/clusters could also be modelled according to in this way. It has been shown in particular that, for the laws of mechanics. The interactions between the a given grain density, the movement of the grains can clusters and the fluid are certainly more difficult to either increase or decrease the effective solid-liquid simulate, and the Stokes relationship used to model interface area with respect to the standard K J M A sedimentation is certainly oversimplified, prediction and thus directly influences the growth

Nevertheless, the present stochastic model is the rate, the undercooling and the possible appearance of first at tempt to simulate the nucleation, growth and metastable phases in between the grains.

(a ) (b)

1 / i 1 k - . . . . \ \ iv

,, iv \ ~ , ... / ~x ~ \ \m

" ~ 0.8 ~ ..x x 0.8 i \ ~ n , ~i ~ \ ~ \ ~ " i-.. \

• ,', ~ \ \ 1 0.4 I ~ - - i k ~ u-~ 0.4 ~ \

/ ~ E \ \ ~ l~

\ '%1 0.2 f F ~ \ 0 ~ 0.2 \ \ h

o . . . . ~ 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Solid f rac t ion [-] Solid f rac t ion [-]

Fig. I 1. Normalised solid-liquid interface area, s~ (a) and impingement factor, T (b) of the eutectic grains as a function of the volume fraction of solid for the cases (E-F) and (i-iv). The cases (i-iv) are identical

to those shown in Fig. 5.

CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT 3965

1160 Office F~d~ral de l'Education et de la Science, Bern, for supporting this research activity. They are also grateful to Ch.-A. Gandin and J.-M. Drezet for their helpful comments and suggestions.

1155

o

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3966 CHARBON et al.: SOLIDIFICATION AND GRAIN MOVEMENT

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