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SOLIDIFICATION MICROSTRUCTURES: RECENT
DEVELOPMENTS, FUTURE DIRECTIONSp
W. J. BOETTINGER1, S. R. CORIELL 1, A. L. GREER 2, A. KARMA 3, W. KURZ 4{,
M. RAPPAZ 4 and R. TRIVEDI 5
1NIST, Gaithersburg, MD 20899, USA, 2Department of Materials Science & Metallurgy, University ofCambridge, Cambridge CB2 3QZ, UK, 3Department of Physics, Northeastern University, Boston, MA
02115, USA, 4Department of Materials, Swiss Federal Institute of Technology Lausanne, 1015Lausanne EPFL, Switzerland and 5Iowa State University & Ames Lab. USDOE, Ames, IA 50011,
USA
(Received 1 June 1999; accepted 15 July 1999)
AbstractThe status of solidication science is critically evaluated and future directions of research in thistechnologically important area are proposed. The most important advances in solidication science andtechnology of the last decade are discussed: interface dynamics, phase selection, microstructure selection,
peritectic growth, convection eects, multicomponent alloys, and numerical techniques. It is shown howthe advent of new mathematical techniques (especially phase-eld and cellular automata models) coupledwith powerful computers now allows the following: modeling of complicated interface morphologies, takinginto account not only steady state but also non-steady state phenomena; considering real alloys consistingof many elements through on-line use of large thermodynamic data banks; and taking into account naturaland forced convection eects. A series of open questions and future prospects are also given. It is hopedthat the reader is encouraged to explore this important and highly interesting eld and to add her/his con-tributions to an ever better understanding and modeling of microstructure development. # 2000 ActaMetallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
Keywords: Solidication; Microstructure; Theory and modeling (kinetics, transport, diusion); Casting
1. INTRODUCTION
Microstructures are at the center of materials
science and engineering. They are the strategic link
between materials processing and materials beha-
vior. Microstructure control is therefore essential
for any processing activity. One of the most import-
ant processing routes for many materials, especially
metals and alloys, is solidication. Over the last
decade, important advances have been made in our
fundamental understanding of solidication micro-
structures. Three main ingredients have contributed
to this progress: (i) the development of rigorous
analytical models that have focused on both steady-
state and non-steady-state microstructure evolution
with the inclusion of nucleation for the selection of
phases; (ii) the emergence of accurate simulation
methods, and in particular phase-eld and cellular
automata approaches, which have permitted a vali-
dation of analytical theories as well as enabling pre-
dictions on grain structure and morphological
evolution; and (iii) the development of more rened
experimental techniques that have led to a better
visualization and characterization of microstructural
development. The combination of these advances
now makes it feasible to address long standing
microstructure formation questions with a higher
level of scrutiny and rigor, and thus to end this mil-
lennium in a renaissance period where solidication
``science'' is ourishing and solidication technology
is leading to a better control of materials proces-
sing. We highlight in this paper the theoretical andexperimental progress made in understanding basic
aspects of microstructure formation, emphasizing
especially the critical questions that remain to be
examined in this scientically highly interesting and
technologically important area.
A decade ago, an extensive overview was given
on the topic which was based on presentations and
discussions of the rst 1988 Zermatt Workshop
Acta mater. 48 (2000) 4370
1359-6454/00/$20.00 # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 2 8 7 - 6
www.elsevier.com/locate/actamat
p
The Millennium Special Issue A Selection of Major
Topics in Materials Science and Engineering: Current
status and future directions, edited by S. Suresh.
{ To whom all correspondence should be addressed.
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dedicated to solidication microstructures [1]. In the
present overview the most important ndings of the
second 1998 Zermatt Workshop on Solidication
Microstructures are presented by the seven keynote
speakers. (Contributions to this workshop have
been published in the form of a CD [2].)
The paper is organized as follows: Section 2
describes interface pattern formation models;
Section 3 considers nucleation and growth of a new
phase during the growth of an existing phase;
Section 4 emphasizes the action of uid ow on
microstructures; Section 5 addresses the appli-
cations of the models to industrially interesting
alloys containing several solutes; Section 6 includes
dierent numerical techniques and their potential
for solving complex problems in which several
phenomena must be considered simultaneously to
predict the microstructure.
2. INTERFACE DYNAMICS
Microstructures are formed at moving solid
liquid interfaces. In this section, the evolution of
interface morphologies of a single phase solid grow-
ing into a liquid is presented. The growth in an
undercooled melt of equiaxed dendrites is rst
described. Directional solidication with planar, cel-
lular and dendritic interfaces is then considered.
Some reference is also given to recent work on two-phase growth, such as eutectic and peritectic
growth.
2.1. Equiaxed dendritic growth
During the 1980s, the study of simplied models
that incorporate surface tension in a consistent way
led to the novel insight that dendritic growth is con-
trolled not only by the balance between diusion
and capillarity, but also in a subtle way by crystal-
line anisotropy [3, 4]. This insight led to the advent
of microscopic solvability theory to predict the
selected dendrite tip velocity and tip radius [5, 6].
Over the last decade, this theory has been extended
to three dimensions [7] and it has even been vali-
dated quantitatively by fully time-dependent simu-
lations of dendritic growth in both two dimensions
[810] and three dimensions [11] (Fig. 1), with the
added insight that in three dimensions the non-axi-
symmetric tip morphology inuences the selection
for large enough anisotropy.
Beyond the understanding of steady-state growth
of the tip, the main new concept that has emerged
over the last few years, is that complex pattern for-
mation processes occurring on the much larger
scale of an entire dendrite grain structure can be
described by remarkably simple ``scaling laws''.These processes include growth transients [13, 14]
that lead to steady-state growth and the highly non-
linear competition of secondary branches behind
the tip [1517]. In addition, a deeper understanding
of the role of anisotropy has come from the discov-
ery of new steady-state growth structures (doublons
[8] and triplons [18]). Following the morphological
instability of a small spherical grain, the primary
branches of an equiaxed grain emerge along h100i
directions in cubic crystals but do not immediately
reach a steady state. These branches are much thin-
Fig. 1. Three-dimensional equiaxed dendrites calculatedwith the phase-eld method: (a) thermal dendrite withh100i growth directions [9]; (b) solutal NiCu dendrite
when the preferred growth directions are h110i instead ofh100i [12].
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ner and thus grow much faster initially than in
steady state, such that the instantaneous tip velocity
V(t ) [tip radius r(t )] is a monotonically decreasing
(increasing) function of time during a transient of
duration HDaV2ss
, where Vss is the nal steady-state
growth velocity{. An analytical treatment of this
transient has been possible in two dimensions (plate
dendrites) in the limit of vanishingly small under-
cooling where the problem is analogous to anisotro-
pic HeleShaw ow and can be treated rigorously
by the conformal mapping technique [13]. The main
result is that the length and width of primary
branches, and the total area of the plate, obey
simple power laws given, respectively, by LtHt3a5,
WtHt2a5 and AtHt for t ` DaV2ssX Moreover, the
transient interface shape is described by a unique
scaling shape. Another important feature of this
transient is that although V(t ) and r(t ) can vary in
time by one order of magnitude or more, the tip
selection parameter, s 2Dd0art2Vt, remains
constant in time and xed at its value determined
by solvability theory. Physically, this follows fromthe fact that s is determined by the diusion eld
in the tip region. Thus, at low undercooling, its
value is established quasi-instantaneously on the
time scale where the interface moves one tip radius
since rt2aDrtaVtX Phase-eld simulations in
two dimensions show a good quantitative agree-
ment with these predictions at very low undercool-
ing [14]. In three dimensions, no analytical theory is
yet available to describe this transient but simu-
lations reveal the existence of some approximate
scaling behavior at short time with dierent power
laws than in two dimensions [14].
The results of two-dimensional growth transientshave immediate implications for understanding the
large-scale structure of three-dimensional dendrites,
since the mean cross-sectional shape of a three-
dimensional steady-state dendrite (perpendicular to
the growth axis) can be assumed to evolve with dis-
tance z behind the tip as a two-dimensional branch-
less plate dendrite evolves in time with t zaVss[15]. This assumption becomes exact far enough
behind the tip since the heat (or solute) ux along z
becomes negligibly small, and it yields the mean
shape xHz3a5 for z large compared with the tip
radius rss but small compared with the diusion
length D/Vss
, and xH
z for distances larger than thislength as conrmed by three-dimensional phase-
eld simulations [11]. Translated in terms of the
projection area fraction f
xz dz, the above
result gives fHz1X6 for z ` DaVss, which is in
reasonably good agreement with the scaling law
fHz1X7 obtained by detailed measurements [17] of
the morphology of pure SCN dendrites grown in a
diusive regime in space [19]. Actually, on theoreti-
cal grounds one would expect a time-varying expo-
nent slightly larger than 1.6, which is a strict lower
bound valid in the limit of vanishing undercooling.
A scaling law has also been derived that describes
how the length of ``active'' sidebranches that sur-
vive the growth competition behind the tip and the
spacing l between them depend on the distance z
behind the tip [16]. The main prediction is thatboth l and increase linearly in z. The morphology
measurements on SCN crystals yield a good quanti-
tative agreement with this linear law but only if it is
interpreted in parabolic coordinates [17], i.e. with
the length of active sidebranches measured from a
parabola tted to the tip and plotted vs distance
along this parabola. This change of coordinate in-
corporates the fact that sidebranches tend to grow
perpendicularly to the isotherms [17] and thus eec-
tively incorporates the eect of the heat ux along z
that is neglected in the analysis of Ref. [16].
Finally, the basic concept that sidebranches are
driven by small perturbations of the tip region [20],
which originated from the work of Zel'dovich et al.
on ame fronts [21], has been further developed
theoretically [16, 22] and validated by phase-eld
simulations that consistently yield branchless den-
drites (needle crystals) if numerical noise is kept
small by using ne meshes [911, 2325].
Furthermore, when noise is purposely added in a
quantitatively controlled way, phase-eld simu-
lations yield sidebranching characteristics (initial
amplitude and spacing behind the tip) that are in
good overall agreement with the predictions of the
analytical theory of noise amplication in two
dimensions [26]. These simulations presently need to
be extended to three dimensions in order to test theprediction [16] that thermal noise is responsible for
the experimentally observed sidebranching activity.
The new steady-state growth structures that have
been identied are the so-called ``doublons'' in two
dimensions [8], rst observed in the form of a doub-
let cellular structure in directional solidication [27],
and the ``triplon'' in three dimensions [18]. Both
structures have been shown [8, 18] to exist without
crystalline anisotropy unlike conventional dendrites.
The doublon has the form of a dendrite split in two
parts about its central axis with a narrow liquid
groove in between these two parts, and triplons in
three dimensions are split in three parts. For a niteanisotropy, however, these structures only exist
above a critical undercooling [28] (or supersatura-
tion for the isothermal solidication of an alloy),
such that standard dendrites growing along h100i
directions are indeed the selected structures in
weakly anisotropic materials at low undercoolings,
in agreement with most experimental observations
in organic and metallic systems. From a broad per-
spective, the existence of doublons and triplons is
of fundamental importance since it has provided a
basis to classify the wide range of possible growth{ See list of symbols in the Appendix.
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morphologies that can form as a function of under-
cooling and anisotropy [29].
2.2. Directional solidication
Signicant progress has been made over the last
decade in understanding fundamental aspects of
interface dynamics in directional solidication of
alloys. The onset of morphological instability in
directional solidication has been modeled by the
classic MullinsSekerka instability [30], which pre-
dicts the instability wavelength of a steady-state pla-
nar interface. In a typical directional Bridgman set-
up, however, the planar interface does not become
unstable in steady state, but during the transient
build-up of the solute boundary layer after solidi-
cation is started. By analyzing the morphological
stability of the planar interface during this transi-
ent, and by taking into account that the instability
takes time to grow from natural modulations until
it becomes observable, it has been possible to
obtain for the rst time an accurate prediction of
the instability wavelength [31]. This prediction
agrees well quantitatively with experiments on the
onset wavelength and diers signicantly from the
wavelength predicted assuming steady-state growth
[3234] (Fig. 2).The critical role of crystalline anisotropy in inter-
face dynamics has been demonstrated experimen-
tally in directional growth [35]. This study exploited
the ability to control the orientation of the crystal
grown in a thin sample. With the h100i direction
oriented (nearly) parallel to the axis of the thermal
gradient, the typically observed stable cellular/den-
dritic array structures are obtained [Fig. 3(a)]. In
contrast, with the h111i direction oriented normal
to the glass plates, there is no second-order aniso-
tropy in the plane of the sample [i.e. d 2gyady2 0
where g(y ) is the surface energy and y is the polar
angle in this plane]. Thus growth in this plane is
rendered ``eectively isotropic'' by this judicious
choice of grain orientation. In this case, a ``sea-
weed'' structure [35] [Fig. 3(b)] whose underlying
building block is the theoretically expected doublon
Fig. 2. Wavelength of morphological instability of plane front during initial transient [31].
Fig. 3. Role of crystalline anisotropy on interface shape indirectional growth of thin transparent samples [35].
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[8] is formed in agreement with numerical simu-
lations also presented in Ref. [35]. This nding is
also consistent with the numerical nding that stan-
dard cellular structures are linearly unstable in the
absence of crystalline anisotropy, except in a very
narrow range of velocity near onset of instability
[36].
The formation of doublons and triplons has also
been suggested to play an important role in the for-
mation of ``feathery'' grains in aluminum alloys.
This peculiar morphology, known since the 1940s,
which appears as a succession of lamellae separated
by straight and wavy boundaries, was associated
with the formation of twins parallel to the lamellae
but the appearance mechanisms remained unclear.
Recent electron back-scattered diraction (EBSD)
observations combined with detailed optical and
scanning electron microscope (SEM) observations
[2, 37] have clearly shown that feathery grains are
made of h110i columnar dendrites [Fig. 1(b)], whose
primary trunks are aligned along and split in their
center by a (111) coherent twin plane. The impinge-ment of secondary h110i side arms gives rise to
incoherent wavy twin boundaries. The switch from
h100i to h110i growth morphologies was attributed
to the small anisotropy of the solidliquid inter-
facial energy of aluminum which can be changed by
the addition of solute elements such as Zn, Mg or
Ti and possible attachment kinetics eects. More
details of this kind of growth may be found in Ref.
[12].
A new experimental technique has been devel-
oped in which a brief spatially periodic u.v. laser
pulse is applied to the solidliquid interface in a
transparent organic system (succinonitrilecou-
marin), to force a desired wavelength of the mor-
phological instability [38]. These experiments have
made it possible to investigate systematically the
dynamical selection and stability of cellular struc-
tures by varying the instability wavelength, and
thus accessing cell spacings that are not normally
accessible from a planar interface.
A stability analysis of dendritic arrays [39] has
been carried out in the limit where the primary spa-
cing is larger than the diusion length. The basic
instability found to limit the array stability at small
spacing corresponds to a mode where one out of
every two dendrites in the array is eliminated in
agreement with experiments [40, 41]. The samemode is found numerically to limit the array stab-
ility of cells [36] such that its existence appears
rather universal. A range of interdendritic spacings
is therefore stable, in agreement with experimental
observations [42, 43], but experiments with the
same ``history'' lead to a reproducible spacing [44,
45]. As an elaboration of this work, a model of
``history-dependent'' selection of the primary spa-
cing has been developed. This model is based on
the picture that dendrites are eliminated continu-
ously (subject to this instability) during the long
transient that follows the initial morphological
instability of the planar interface and leads to
steady-state growth of the array with a nal selected
spacing [31]. The predictions of this model agree
reasonably well with one set of experiments [44].
Moreover, at a more qualitative level, it has been
demonstrated experimentally that the initial
instability wavelength does indeed inuence the
steady-state interdendritic spacing [46].
Cellular/dendritic arrays have also been modeled
numerically based on the traditional view that the
structure with the lowest undercooling is selected
within some stability band of spacings [47]. This
model has had some success in explaining exper-
imental data. It does not, however, control the
strength of crystalline anisotropy, which is now
understood to crucially inuence the cellular array
stability both numerically [36] and experimentally
[35]. Therefore, its validity remains to be further
investigated. An analytical approach to the primary
spacing problem by summation of the Ivantsov
elds and application of the minimum undercoolingcriterion has also been developed recently [48].
A detailed experimental study has brought new
insights into the onset of sidebranching in direc-
tional solidication [49]. In these experiments, the
cell spacing was made uniform along the array and
varied by exploiting the history dependence of
wavelength selection in this system. This technique
was used to characterize the onset of sidebranching
systematically and shows that branched and non-
branched cells in these experiments belong to the
same branch of steady-state growth solutions.
Furthermore, it has revealed that the thermal gradi-
ent plays a destabilizing role (i.e. increasing G
causes non-branched cells to branch). Theoretical
models remain to be developed to explain this role
as well as to characterize the onset of sidebranch-
ing.
2.3. High velocity microstructures
Signicant experimental studies on microstructure
formation under rapid solidication conditions have
been carried out in the last decade using the laser
scanning technique (a type of directional growth
process) and levitational techniques (undercooled
solidication).At high rates oscillatory behavior of the solid
liquid interface (banding) has been analyzed by sev-
eral authors [5052]. Band formation in the velocity
regime of strong variation of the distribution coe-
cient, k(V) [53], was shown to depend strongly on
the coupling of non-steady-state heat and solute
transport phenomena [52]. Experiments on the ab-
solute stability of SCN have been undertaken [54]
and it was shown that close to this limit, cells fol-
low a l1X5V constant relationship [55].
In highly undercooled levitated melts systematic
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measurements have been undertaken under others
by the group of Herlach [56]. These authors have
shown that over a substantial range of undercool-
ings good agreement may be obtained between the
measurements in a large number of metals and
alloys and the analytical model using the transport
solution of Ivantsov together with Marginal
Stability arguments including solute trapping eects
(a theory which we call the IMS model) [5759],
with the stability parameter s 4p21X Further it
has been recently shown that excellent agreement
with no adjustable parameters can be obtained with
this theory [60]. Note that this agreement should
only be interpreted to mean that marginal stability
arguments, although not fundamentally correct, are
still useful to make quantitative predictions of den-
drite growth rates in rapidly solidied binary alloys.
A detailed comparison of solvability and marginal
stability theory for rapid dendrite growth has
recently been carried out [61].
One of the important observations at high under-
cooling is the formation of very ne-grained struc-
tures over a range of large undercoolings. This ne-
grained structure has been explained by dendrite
fragmentation. At very high undercooling, as the
dendrite trunk diameter becomes very ne, the ten-
dency to undergo Rayleigh instability increases. A
theory has been developed which shows that frag-
mentation can occur when the characteristic time
for dendrite break-up is shorter than the post-reca-
lescence or plateau time in overall agreement withexperiments [62, 63].
2.4. Coupled and simultaneous growth
Even if most of the recent modeling was con-
cerned with single-phase growth phenomena, there
has also been some work on coupled or simul-
taneous growth of two phases. A detailed numerical
survey of the morphological instabilities of lamellar
eutectics has been carried out in two dimensions by
the boundary integral method for the transparent
organic system CBr4C2Cl6 [64] (Fig. 4). In parallel,
a detailed experimental survey of these instabilities
has been carried out in the same system [65]. There
Fig. 4. Calculated stability diagram for two-dimensionalcoupled eutectic growth in CBr4C2Cl6 [64] in excellentquantitative agreement with the experimentally measureddiagram in the same alloy [65]. Z: reduced concentrationwith the eutectic point at Z 0X3; L: lamellar spacing nor-malized by the spacing corresponding to minimum under-cooling. The basic axisymmetric state is stable within thecenter region. Other states include steady-state tilted pat-
terns (T), 2lO (spatial period doubling oscillations), 1lO(spatial period preserving oscillations), where both 1lOand 2lO oscillations can be either axisymmetric or tilted.Blank regions of the diagram are those in which the
dynamics is not yet fully understood.
Fig. 5. Quenched liquidsolid interface of simultaneous two-phase growth in peritectic FeNi alloy [66].
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is a remarkably good quantitative agreement
between simulations and experiments concerning
the regions of stability of both non-tilted and tilted
steady states in the plane of composition and eutec-
tic spacing, and the oscillatory instabilities that
limit these regions. This understanding of eutectic
stability, however, is restricted to two dimensions
and presently needs to be extended to three dimen-
sions. Eutectic cells and dendrites forming in multi-
component alloys have also been studied
theoretically and experimentally and are presented
in Section 5.
Simultaneous growth of two phases in the form
of oriented bers and lamellae has been observed in
some peritectic alloys. Figure 5 shows an example
from a FeNi alloy. For this to happen, the compo-
sition has to be between the two solid phases and
the G/V ratio close to the limit of constitutional
undercooling for the stable phase with the smaller
distribution coecient [6668]. This interesting in
situ growth phenomenon still waits for a theoretical
interpretation, although recent phase-eld calcu-lations have shown the formation of such a struc-
ture [69].
3. PHASE AND MICROSTRUCTURE SELECTION
A microstructure is dened by the morphology,
size, distribution, crystal orientation, and corre-
lation (texture), and number of phases. Phase and
microstructure selection describes the variety of
phases and microstructures that develop under
given growth conditions and growth geometries.
This section treats mainly transformations of phases
and microstructures from one structure or mor-
phology into another. It is not so much the for-
mation of a single growth form itself which is of
interest in this part of the paper (this has been
treated in Section 2) but the mechanisms of change
from one phase and morphology into another. A
detailed theory of the mechanisms responsible for
this selection is only at its very beginning. A well-
known empirical approach that is consistent with
many experimental results uses extremum criteria,
such as the highest growth temperature in direc-
tional growth. In undercooled solidication proces-
sing the highest nucleation temperature and the
highest growth rate control the nal appearance ofmicrostructures and phases.
In many materials, additional phase transform-
ations take place in the solid state which lead to the
nal microstructure. In this review only solidica-
tion will be discussed. In general all solidication
processes start with nucleation and continue with
growth. The nal phases may be controlled by
nucleation, by growth, or by a combination of
both. In all three cases that will be treated separ-
ately in the following, much progress has been
made in recent years.
3.1. Nucleation control
If suciently large undercoolings can be attained
through hindrance of heterogeneous nucleation,
then there may be access to a variety of metastable
phases, such phases having lower melting points
and liquidus temperatures. The importance of
nucleation is seen when dealing with phase selec-
tion. A typical process where nucleation plays a
dominant role is solidication processing of under-
cooled melts such as is observed in droplets [56].
There may be a spread in nucleation temperatures
even under nominally identical conditions, and con-
sequently the results are best displayed on ``micro-structure-predominance maps''. Such maps have
been constructed for binary alloys, with alloy com-
position and droplet diameter as coordinates [70
72]. It is found that: (i) microstructure correlates
very strongly with droplet diameter (which deter-
mines the availability of nucleant sites and the cool-
ing rate); (ii) the eects of processing conditions
(e.g. gas purity in atomization) can be taken into
account; and (iii) correlation with undercooling can
be found through comparison with controlled
undercooling experiments and growth modeling [71,
73] (Fig. 6). It is clear, though, that we are very far
from being able to predict nucleation undercoolings,the diversity of potential heterogeneous nucleants
being a key impediment to quantitative modeling of
most real situations.
Under given conditions it is usual for one phase
to dominate, but the primary phases can also be
mixed. A well-analyzed example is the duplex parti-
tionless solidication of b.c.c. and f.c.c. phases in
the NiV system [74]. There are many examples in
which the phase competition is between b.c.c. and
f.c.c. phases, and this has been most closely exam-
ined for FeNi. In undercooled levitated droplets,
Fig. 6. Phase predominance map (drop diameter vs com-position) for undercooled growth in FeNi alloys [73].
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the prevalence of one phase or the other can be
fairly well predicted by which phase has the lower
work of nucleation (Fig. 6) (e.g. Ref. [75]). The
b.c.c. phase is easier to nucleate than would be
expected from its relative thermodynamic stability;
it has a lower solidliquid interfacial energy. In the
FeNiCr system, it has been shown that the f.c.c.
or b.c.c. phase can be selected through the use of
an appropriate substrate put in contact with the
droplet to trigger solidication [76].
The existence of the true primary phase is some-
times revealed by a double recalescence phenom-
enon [75]. In this way, for example, the transient
existence of a hitherto unknown metastable f.c.c.
phase of rhenium has been inferred [77].
Observations of this kind, found also in some alloy
systems [78], may be crucial in analyzing nucleation
kinetics.
Particular interest has centered on analyzing het-
erogeneous nucleation kinetics. Basic treatments of
heterogeneous nucleation (taking into account vari-
ations of potency and population) have had successin predicting primary phase selection [79]. Recent
advances have been made by studying liquid dro-
plets entrained in solid matrices; when large under-
coolings b 50 K are required for heterogeneous
nucleation, the classical spherical cap model seems
to work well, but for more potent nuclei it breaks
down [80]. In that case, some success has been
achieved with a model that considers the thermo-
dynamics, if not yet the kinetics of adsorption [81].
Further heterogeneous nucleation studies have been
undertaken on liquid droplets in an emulsifying or-
ganic liquid; taking a more classical approach, roles
have been identied for dierent types of stationary
and moving surface steps [82]. Yet another advance
has been transmission electron microscopy of het-
erogeneous nuclei formed in a glassy matrix [83];
this study, of relevance for commercial grain rene-
ment of Al, shows the importance of crystallogra-
phy and chemistry in nucleation, even though
quantication of these roles remains elusive [84].
So far, it has been natural to concentrate on the
primary stage of solidication, yet there are cases
where the interest is in the formation of secondary
phases in the nal stages of solidication. An
example of considerable industrial interest is the
DC casting of very dilute Al alloys. In cases of
practical importance there is a range of intermetal-lics which can nucleate, the phase selection being
sensitive to many parameters including solidication
velocity. Directional solidication can reveal
changes in intermetallic selection and be a basis for
understanding fundamental mechanisms [85, 86].
Attempts to analyze phase selection have focused
on comparative eutectic growth kinetics [87], but
solid-state changes and nucleation eects have also
been considered. It appears that a quantitative
analysis of the phase selection may depend on the
geometry of the liquid in which the rival intermetal-
lics nucleate and grow; such an analysis has yet to
be attempted. Considerations so far suggest that the
wide range of conditions over which mixtures of
phases are obtained is indicative of a growth com-
petition [88].
3.2. Growth control
Growth-controlled phase and microstructure
selection has been successfully treated by comparing
the steady-state interface response of competing
phases. Calculating the interface response, i.e. the
growth behavior of plane front, cells and dendrites,
for all possible phases one can determine the
growth form which develops the highest interface
temperature for a given growth velocity and tem-
perature gradient [89] or the highest growth velocity
for a given undercooling{. Growth of eutectic struc-tures can also be included in this treatment. The
extremum criterion is a strong indication of the
structure to be formed. It assumes that: (i) the
microstructure selection is not nucleation controlled
(i.e. nucleation undercooling is suciently small);
(ii) interaction between competing growth forms is
negligible; and (iii) steady-state theory can be
applied. Despite its simplicity this approach is of
great help in determining microstructure maps for a
more rational alloy development. Several cases of
recent modeling of microstructure selection will be
presented in the following.
3.2.1. Stable to metastable phase transition. Using
the above-mentioned maximum temperature cri-
terion, the stable to metastable transition for direc-
tional dendrite growth has been analyzed for
peritectic systems [9092]. This allows us to ration-
alize why at high velocities a transition from a
stable to a metastable peritectic phase is often
obtained. For example in FeNi or FeNiCr
steels, high weld speeds lead to the formation of
austenite dendrites [9395], even if at low velocities
ferrite is the primary phase, with important conse-
quences for the integrity (solidication cracking) of
the weld. No nucleation is needed for this transition
to occur as the metastable phase (austenite) growsinbetween the stable phase due to microsegregation
and its growth is accelerated with velocity until the
metastable phase becomes the leading one. For the
reverse case (metastable to stable phase) this is not
true and nucleation is a necessary requirement for
the transition to happen (see under mixed control).
Similar stable to metastable phase transitions have
been analyzed in detail for FeC alloys [96].
Under conditions of rapid solidication, solute
and disorder trapping become signicant in the kin-
etics [97]. By including such eects it has been poss-{ In this approach cells and dendrites are treated as one
entity with one growth equation.
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ible quantitatively to model the growth kinetics in
undercooled NiAl alloys [98, 99]. Growth kinetics
alone can be used to follow the competition
between the ordered and disordered version of one
phase. However, prediction of which basic structure
(b.c.c. vs f.c.c. again in this case [98]) will beobserved requires a modeling of the nucleation
which does not yet exist.
3.2.2. Dendrite to eutectic transition (CZ). The
limits of the so-called ``Coupled Zone'' represent
the transition between fully eutectic structure and
primary dendrites or cells with interdendritic eutec-
tic. This transition which is also hysteretic in nature
is inherently dicult to model. Karma has made
progress in this matter and his results on the stab-
ility of eutectic growth are discussed in Section 2.
Using steady-state growth theory and the extre-
mum criterion for the interface temperature, this
transition can be calculated for directional growth
(Bridgman, laser treatment, etc.) and is found to be
in good agreement with experimental evidence
[100]. In this way, following the early work of
Boettinger et al . [101], a series of solidication
microstructure selection maps has been obtained in
recent years which allows a more rational approach
to the solidication processing of technically im-
portant alloys: AlCu [102, 103], AlFe [104], AlSi
[105], AlCuSi [106], NiAl [99], and even cer-
amics such as Al2O3ZrO2 [107]. These maps have
been used as a tool for analyzing and predicting the
microstructures of laser surface treated materials. A
similar approach, but for undercooled melts with a
corresponding maximum velocity criterion, has also
been developed [108].
Another way of this type of microstructure mod-
eling is the ``inverse modeling'' which starts with in-
formation about the microstructure and optimizes
the input parameters such as the phase diagram[103, 109111] (Fig. 7). This new approach to deter-
mine stable and especially metastable phase equili-
bria is useful in cases where conventional
techniques do not work.
3.3. Mixed control
Mixed control is always found when both nuclea-
tion and growth play a controlling role in the
microstructure selection, such as in the columnar to
equiaxed transition of dendritic or eutectic struc-
tures or in low velocity microstructures in the two-
phase region of peritectic systems.
3.3.1. Columnar to equiaxed transition (CET).
Hunt's classic approach to model the CET [112] has
been applied to welding [113] and has been
extended by using more recent dendrite models
[114, 115]. In this way critical growth conditions for
the single crystalline welding of single crystal gas
turbine blades could be established and a poten-
tially interesting process for lifetime extension of
these expensive components developed (Fig. 8)
[116].
The transition from the outer equiaxed zone to
the columnar region (ECT), often observed in cast-ings can be understood in terms of the same CET
criterion. Such considerations were made for the
shape of grains continuously nucleating and grow-
ing in a thermal gradient [117]. When the ratio G/V
increases up to a critical value, the shape factor of
the grains becomes innite, meaning that the
equiaxed grains become columnar. In order to
explain the ECT, it is necessary to consider the heat
transfer coecient between the casting and the
mould (which changes strongly when solidication
starts) and the superheat of the melt. Such model-
Fig. 7. Calculated microstructure selection map (VCodiagram) for NiAl alloys (b), and optimized phase dia-gram (a) [109]. (The dierence of the eutectic temperatureofbg ' and bg eutectic is less than the width of the line.)
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Fig.
8.
Epitaxiallasermeta
lformingofasinglecrystalsuperalloy(CM
SX4)showingthesinglecrystallinenatureofthelasercladdepositedontothecasts
inglecrystalsub-
strate[116].
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two phases, and by growth competition between the
nucleated grains and the pre-existing phase under
non-steady-state conditions. In this case the simple
extremum growth criterion does not lead to the
right answer and nucleation in the constitutionally
undercooled zone ahead of the growth front has to
be taken into account in order to determine the
microstructure selection [119, 120] (Fig. 9).
A clear understanding of complex microstructure
formation has come from directional solidication
studies of binary alloys with compositions in the
two-phase region of the peritectic phase diagram at
large G/V ratio to suppress the morphological
instability of both the parent (a ) and the peritectic
(b ) phases, i.e. each phase alone would grow as a
planar front. Even in this simplied case, a rich var-
iety of microstructures has been identied that
depend sensitively upon the relative importance of
nucleation, diusion and convection [121125] as
shown in Fig. 10. These microstructures can be
broadly classied into the following groups based
on geometrical patterns and the underlying trans-port mechanisms: (a) discrete bands of the two
phases; (b) partial bands or particulates (or islands)
of one phase in the matrix of the other phase; (c)
single primary to peritectic phase transition; (d)
simultaneous growth of the two phases with a pla-
nar solidliquid interface; (e) dispersed phases due
to nucleation ahead of the interface; and (f) oscillat-
ing continuous tree-like structures of the primary
phase that are surrounded by the peritectic phase
[122]. Theoretical models and experimental studies
in very thin samples have shown that structures
(a)(e), can form under diusive regimes, whereas
microstructure (f) is a novel microstructure whose
formation requires the presence of oscillatory con-
vection in the melt.
In order to understand the formation of some of
the complex microstructures in the two-phase
region of peritectic systems, an analytical model of
banding in peritectic systems was rst proposed for
diusive growth in which the change in phases
occurred when the appropriate nucleation under-
coolings were reached [126]. According to this
model, a banding cycle of alternate nucleation and
growth of primary, a, and peritectic, b, phases may
continue, leading to an oscillatory behavior of the
interface and to alternate bands of a and b. The
major predictions of this diusive banding model
are: (i) the banding cycle will operate below and
above the peritectic temperature; and (ii) the band-
ing window exists only for a narrow range of initial
alloy composition in the hypoperitectic range.
In the above one-dimensional model of discrete
band formation, it was assumed that the nuclei ofthe new phase spread rapidly in the lateral direc-
tion, so that no appreciable lateral gradients exist.
However, this is generally not valid and one must
consider the relative rates of spreading of the new
phase and the continuing growth of the parent
phase. The microstructure for this complicated case
was investigated experimentally as well as by nu-
merical simulation of a two-dimensional transient
phase-eld model for a generic peritectic phase dia-
gram [69]. Several new morphologies were observed
and predicted depending on the nucleation rate.
Fig. 10. Fluid ow controlled microstructures in peritectic alloys: (a) discrete bands of the two phases;(b) partial bands or islands of one phase in the matrix of the other phase; (c) single primary to peritec-tic phase transition; (d) simultaneous growth of the two phases with a planar solidliquid interface; (e)dispersed phases due to nucleation ahead of the interface; (f) oscillating continuous tree-like structures
of the primary phase that are surrounded by the peritectic phase [122].
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The results of the phase-eld model indicate that
when only a single nucleus is allowed to form on
the wall of the sample, discrete band formation in a
diusive regime is only possible for a nite range of
system sizes, Lmin ` L ` Lmax, where L is the size of
the sample or the distance between the nuclei.
Moreover, this range depends on both the compo-
sition inside the hypoperitectic region, and the
nucleation undercoolings of the two phases. For
internuclei distance L ` Lmin, discrete particles of
the b phase form inside the a matrix, and for
L b Lmax, discrete particles of a phase form in the b
matrix form. In addition to the microstructure of
discrete particles of one of the two phases
embedded inside the continuous matrix of the other
phase, more complex microstructures, including two
simultaneously growing phases form [Fig. 10(d)].
Simultaneous two-phase growth has been observed
in several peritectic systems, including SnCd [122],
AlNi [67], and FeNi [66] (see also Section 2.4).
The basic model of nucleation and growth con-
trolled structures shows that dierent microstruc-tures can form only within a narrow band of
composition in the hypoperitectic region. However,
several experimental observations of banded struc-
tures have been made for compositions outside this
banding composition window, and banding struc-
tures were reported even for hyperperitectic compo-
sitions. These observations clearly indicate that the
observed structures are not controlled by diusion
but by convection eects, as shall be discussed in
the following section.
4. CONVECTION EFFECTS
Convection eects are of utmost importance in
the development of solidication microstructures.
Despite this fact, most microstructure models are
based on purely diusive transport mechanisms.
Only recently, modeling of growth in the presence
of convection has been successfully undertaken. The
rst step in such an undertaking is modeling of con-
vection and its instabilities before coupling of con-
vection and microstructure formation is done.
4.1. Convection instabilities
For the simple problem of an innite layer withvertical temperature and solutal gradients, it is well
known that convective instabilities can occur even if
the net density of the liquid decreases with height
[127]. Similar behavior also occurs in a porous med-
ium [128]. However, usually the temperature and
solute concentration in the mushy zone of a binary
alloy are coupled by the phase diagram and this
prohibits double-diusive behavior, i.e. a density
inversion is necessary for the onset of convective
instability. Worster [129] has reviewed recent work
on convection in mushy zones. The critical
Rayleigh numbers for the onset of convection in a
binary alloy have been calculated for three dierent
models of the mushy zone during directional solidi-
cation [130]. In general, there are two modes of
instability: a mode in the mushy layer and a bound-
ary-layer mode in the melt; the wavelength of the
mushy-layer mode is small compared with the
wavelength of the boundary-layer mode. In addition
to these non-oscillatory modes, there are modes
that are oscillatory in time.
The radial segregation due to solutal convection
during the directional solidication of leadthallium
alloys with a planar crystalmelt interface has been
calculated using pseudo-spectral methods [131].
Solutal Rayleigh numbers for the calculations ran-
ged from very near the onset of convective instabil-
ity to a factor of ten above the instability onset. In
general, the ows and segregation are asymmetric,
although for special conditions axisymmetric ows
can occur.
A sudden change in ow conditions is correlated
with the interface concentration during directionalsolidication of a tinbismuth alloy [132]. The
interface concentration was monitored by Seebeck
measurements using the MEPHISTO furnace
during the USMP-3 space ight. Numerical calcu-
lations of the uid ow and solute redistribution
due to sudden gravitational accelerations caused by
thruster activation were in good agreement with the
observed Seebeck signals.
4.2. Field eects
It is well known that a uniform magnetic eld
can damp convective motions in an electrically con-
ducting uid. However, when a gradient in the
Seebeck voltage exists in the presence of a magnetic
eld and temperature gradients, there can be a
resulting ow [133, 134]. This can occur at a crys-
talmelt interface when there is a temperature gra-
dient along the interface; for example, in a binary
alloy with a non-uniform concentration. Since this
thermoelectric magnetohydrodynamic ow occurs
in the vicinity of the interface, it can play a signi-
cant role in solute redistribution. It can also be
used to counteract buoyancy-driven ow in the
mushy zone during horizontal directional solidica-
tion [135]. Freckle formation in coppersilver andaluminumcopper alloys have been examined under
dierent magnetic elds. The observed larger den-
drite spacings agree with the observation in space
experiments where the ow is signicantly reduced
[136].
There have been a number of studies of crystal
growth in very high gravitational elds using a cen-
trifuge [137139]. For germaniumgallium alloys,
solute segregation exhibits a minimum as a function
of rotation rate. This behavior can be understood
by considering non-axial temperature gradients; at
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low rotation rates, the uid velocity is decreased by
the action of Coriolis forces, while at large rotation
rates the buoyancy force due to the centrifugal
acceleration, increasing as the square of the rotation
rate, becomes dominant [139]. Thus, the ow and
segregation are reduced at intermediate rotation
rates.
4.3. Eect of ow on interface morphology
The eect of simple ows on the shape of par-
ticles growing from a supersaturated solution has
been calculated [140]. The concentration and uid
ow elds are solved numerically by a mapping
technique in the Stokes ow approximation. Simple
base ows such as a uniform streaming ow or a
biaxial straining ow lead to non-spherical shapes.
The particle shape, as function of the ow magni-
tude and the anisotropy of the crystaluid surface
tension, has been studied.
The inuence of convection on morphologicalinstability and interface structure during directional
solidication was examined theoretically [141].
There have also been observations on massive
transparent specimens [142] which have revealed
that convection results in a gradient of microstruc-
ture along the interface, from a smooth interface to
dendrites. Fluid ow eects at the very scale of the
microstructure have been seen during solidication
of faceting transparent systems (e.g. salol-based
alloys) where saw-tooth patterns of millimeter size
form [143]. Surface tension-driven convection due
to the presence of uiduid interfaces and its inu-
ence on the morphology of the growth front
deserves thorough investigation, e.g. coupled
growth of bubbles of dissolved gas and monotectic
alloys in which a second liquid phase forms either
as rods in a solid matrix or droplets in the melt
[144].
Anisotropic interface kinetics stabilizes an inter-
face with respect to the onset of morphological
instability [145, 146]. Such anisotropic kinetics
arises naturally when growth is by step motion and
the crystalmelt interface is near a singular crystal-
lographic orientation. When a planar interface is
perturbed with a sinusoidal perturbation, anisotro-
pic kinetics causes a lateral translation of the sinus-
oid (traveling wave). In turn, this lateral motioncan strongly interact with shear ows along the
interface. Flow in the direction of step motion is
destabilizing while ow opposite to the step motion
is stabilizing.
Experiments on the dendritic growth of succino-
nitrile and pivalic acid from supercooled melts on
earth and in microgravity show small discrepancies
from the classic Ivantsov relation between Peclet
number and dimensionless supercooling (Fig. 11)
[147]. Under terrestrial processing conditions, con-
vection in the melt has a major impact on metallic
solidication, especially at small crystal growth vel-
ocities [56]. Previous studies of dendrite growth in
undercooled Ni melts on earth show systematic de-
viations of experimental data and dendrite growth
theory at small undercoolings. The discrepancy is
partly reduced if convection is taken into consider-
ation. Measurements of the dendrite growth vel-
ocity as a function of undercooling on pure Ni and
dilute Ni0.6 at.% C alloys under microgravity con-
ditions provide a test of dendrite growth models
[148]. The experiments were performed using the
electromagnetic levitation facility TEMPUS.
Excellent growth velocity data were obtained during
the mission in an undercooling range between 50
and 310 K. However, no dierences between micro-
g and 1 g data were detected in this temperature
range since ow due to electromagnetic forces may
be signicant.
The selection of twinned dendrites in the presence
of uid ow may be explained by a higher growth
temperature with respect to normal dendrites, in
particular as a result of doublon formation (see
Section 2.2). The eect of convection on the alter-
nating sequence of straight/coherent and wavy/inco-
herent twin is shown in Fig. 12 [37]. The alloy hasbeen produced by direct chill (DC) continuous cast-
ing and exhibits, in some regions, ``feathery grains''.
In Figure 12(a), three feathery grains labeled 13
are clearly visible: each one is made of parallel
lamellae showing an alternating sequence of colors
(green/red, light blue/purple, and yellow/violet
for grains 1, 2 and 3, respectively) separated by an
alternating sequence of straightwavy boundaries.
This corresponds to twinneduntwinned regions
separated by coherentincoherent twin boundaries
across rows of primary h110i dendrite trunks.
Fig. 11. Tip Peclet number vs supercooling for free den-dritic growth of organics under terrestrial and under
microgravity conditions [147].
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Transverse melt ow was invoked to explain the
systematic alternating sequence of lamellae and
boundaries through branching mechanisms [see Fig.
12(b), in which the twinned dendrites are seen along
their trunk axis].
Numerical simulation of microscopic ow in the
melt during solidication was introduced through aphase-eld model [149151]. The eects of ow on
free dendritic growth (tip velocities, radii, and tip
selection as a function of the orientation of the ow
with respect to the crystal) were investigated.
Convection during coarsening of an isothermal
binary liquidsolid mixture has been studied, i.e.
the eects of convection on coarsening and of coar-
sening on the permeability were examined. The
eect of convection on equiaxed dendrite growth
and associated macrosegregation has also been stu-
died. New results have been obtained on the size
evolution and settling velocity of NH4Cl equiaxed
crystals growing from supercooled NH4ClH2O sol-
ution [152]. The results have been analyzed with the
theory for an isolated dendrite growing in an axi-
symmetric melt ow [153]. In the range of the ex-
perimental settling velocities (711 mm/s), the best
t for the stability constant was found to be 3.12
times greater than the value measured for the
purely diusive case [154].
Extension of a Cellular Automaton (CA) tech-
nique coupled with a nite element (FE) method
(CAFE model [155]), improves the modeling of den-
dritic grain structures in the presence of convection.
The movement of equiaxed crystals in the liquid to
form a sedimentation cone, as well as the modi-
cation of the columnar-to-equiaxed transition in the
presence of convection, are well described qualitat-
ively by the CAFE model.
There has been interesting experimental evidence
on the mushy zone interactions with melt ows in
transparent organics. Quantitative measurements of
the ow eld during solidication could be made[156].
When the JacksonHunt model of eutectic
growth [157] is applied to the growth of monotectic
composites, the predicted value ofl 2V is more than
an order of magnitude smaller than the experimen-
tal value for aluminumindium monotectic alloys
which grow with rods of indium-rich liquid in an
aluminum-rich solid matrix; here, V is the growth
velocity and l is the interrod spacing. Allowing for
diusion in the rod phase does not improve the
agreement between experiment and theory [158].
While the discrepancy could be due to inaccurate
values of the thermophysical properties, another
transport mechanism such as convection could
account for the discrepancy. Such uid ow could
arise from surface tension variations along the
uiduid interface; a pressure-driven ow could
also occur at the uiduid interface due to the
requirement of satisfying both the GibbsThomson
and YoungLaplace equation at this interface [159].
Convection eects have also been found to give
rise to new microstructures that are not observed in
the diusive growth regime. For example, Fig. 10(f)
shows a novel microstructure whose formation
requires the presence of oscillatory convection in
the melt of peritectic systems. A detailed study of
the three-dimensional shape of the microstructurerevealed that the bands were not discrete, but both
the a and the b phases were continuous [160]. It
was shown that the microstructure, which appears
like discrete bands on a section close to the surface
of the sample, is in fact a more complex structure
made up of two continuous interconnected phases
in three dimensions. In particular, the microstruc-
ture consists of a large tree-like domain of primary
a phase that is embedded inside the peritectic b
phase. The formation of this structure is governed
by oscillating convection present in a large diameter
Fig. 12. Twinned dendrites in AlCu alloy. (a) EBSDreconstruction of the microstructure in a transverse sec-tion to the thermal gradient containing three grains. (b)Schematic view of the eect of convection on twinned
dendrite formation [37].
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(6.0 mm) sample [124]. Besides the tree-like struc-
ture, several other new oscillating microstructures
were observed experimentally, and predicted nu-
merically, depending upon the intensity and modes
of convection [160].
5. MULTICOMPONENT SYSTEMS
The application of solidication modeling to
practical technology is closely linked to our ability
to model microstructural development in multicom-
ponent alloys (three or more components). Over the
past ten years signicant progress has been made in
this area.
5.1. Thermodynamics
Solidication models, which use local interfacial
equilibrium, have been successfully coupled to
phase diagram information obtained via the
Calphad method [161]. Examples include analyses
of Scheil solidication path, dendrite tip kinetics,solid (back) diusion and macrosegregation.
Commercial databases are available for Al, Fe, Ni
and Ti base alloys (ThermoTech, Ltd{) and others
are distributed with the various thermodynamic
computational codes available: ThermoCalc (KTH,
Stockholm), MTData (NPL), Chemsage (RWTH,
Aachen). All of these computational codes can be
interfaced with solidication models. As an example
[162] a set of subroutines, LEVER, SLOPE and
HEAT, have been built on top of a modied ver-
sion of the Lukas code. LEVER gives the phase
fractions and phase compositions at equilibrium for
a specied temperature T and overall composition.
SLOPE gives the liquidus temperature, the solid
phase concentrations, and the liquidus slopes for a
specied liquid composition and solid phase. HEAT
gives the enthalpy per unit mass for a specied
phase for a given temperature and phase compo-
sition.
This thermodynamic approach naturally enables
a Scheil analysis of the solidication path; i.e. the
evolution of the liquid and solid concentrations and
the phase fractions during cooling under the
assumption of complete liquid diusion and no
solid diusion. This approach easily treats the
appearance of new phases at eutectic reactions, or
the disappearance and appearance of phases at peri-tectic reactions. A Scheil analysis provides the basis
for a good estimate of very practical information
for castings: (a) how the heat of fusion evolves
during cooling (for coupling to macroscopic heat
ow analysis); and (b) how the density of the
mushy zone changes (for coupling to uid ow
modeling for macrosegregation, porosity and hot
tearing analysis).
Under rapid solidication conditions, when local
interface equilibrium is invalid, thermodynamic cal-
culations for multicomponent alloys can be used to
compute the thermodynamic driving ``force'' and
the energy dissipated due to solute drag (if a model
of diusion through the interface is prescribed).
The thermodynamic driving ``force'' is required for
analysis of the interface response functions for
rapid solidication. In this area, the AzizKaplan
model of solute trapping [163] has been extended to
multicomponent systems for the case when the dif-
fusive speed for all of the solutes is identical [164].
An open question remains regarding the impact of
dierent diusive speeds for dierent solutes in
multicomponent solute trapping models of rapid
solidication. Experimental work in this area would
be useful.
5.2. O-diagonal diusion terms
O-diagonal diusion terms are usually neglected
for multicomponent liquids, yet there is little justi-
cation. Moreover the diagonal terms are usually
assumed to be identical. When diusion uxes are
related to chemical potential gradients through
appropriate mobilities, the absence of o-diagonal
mobility terms does not imply the absence of o-di-
agonal diusion terms. O-diagonal terms tend to
be strongly concentration dependent. One set of ex-
periments [165] measured the o-diagonal diusion
terms in liquid ternary Al alloys. Diusion couples
with a step change in one component but a con-
stant value of the second were analyzed. In thealloys tested, there was no detectable change in con-
centration of the second component; i.e. negligible
o-diagonal terms.
It has been shown [166] that analytical models
for plane front and dendritic growth developed for
binary alloys can be extended to multicomponent
alloys by taking into account the o-diagonal terms
of the diusion matrix. The diusion elds for the
n solutes are given by linear combinations of the n
binary solutions using the eigenvalues and eigenvec-
tors of the diusion matrix instead of the diusion
coecients (Fig. 13). For the time being however,
use of these solutions is limited by the lack of
measured diusion coecients and methods todetermine the o-diagonal terms. A theoretical
approach to this problem is needed.
5.3. Fundamental morphological stability issues
A complete linear stability analysis of planar
growth under the assumption of local equilibrium
for a ternary alloy with no o-diagonal diusion
terms has been performed [167]. When the pertur-
bation wavelength is not assumed to be small com-
{ Trade names are used for completeness only and do
not constitute an endorsement of NIST or any other or-
ganization.
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pared with the liquid diusion lengths for all of the
solutes Di/V, the perturbation growth rate, e, not
only depends on the partition coecients, ki, and
liquidus slopes, mi, but also on the derivatives of
liquid concentrations with respect to solid concen-
trations evaluated on the liquidus surface, quantities
that can be computed from the thermodynamic
approach. When the perturbation wavelength is
small, these factors disappear. In this case, the ex-
pression for e contains a denominator, which can
vanish due to the fact that miki 1 can be nega-
tive for one or more of the solutes in multicompo-
nent systems. This has the potential to lead to
oscillatory instabilities. Whether this can occur in
an experimental system is not known.
Other stability issues, such as the cell to dendrite
transition, have not been adequately resolved, even
for binary alloys. One situation peculiar to ternary
systems is the formation of eutectic cells by the pre-
sence of a dilute ternary solute. The full stability
spectrum of a steady-state lamellar interface in the
presence of a ternary impurity has been calculated
and an analytical form of this spectrum has been
derived in the limit where the wavelength of the
perturbation is large compared with the lamellar
spacing [168]. Also preliminary phase-eld calcu-
lations of the growth of eutectic cells to treat large
amplitude perturbations have been performed as
shown in Fig. 14 [169]. (Such calculations are de-
nitely at the limit of what is actually possible; this
simulation took approximately 60 CPU hours on 32
processors of a CRAY T3E.)
5.4. Microstructure prediction
5.4.1. Dendritic growth and solid diusion. Even
though many issues remain regarding the funda-
mental role of anisotropy on dendrite tip radius
selection (even for pure materials), models used for
practical materials typically use the Ivantsov/
Marginal Stability (IMS) approach [5759]. This
model for binary alloys has been extended to multi-
component alloys [170, 171]. The equiaxed growth
model [172] has been generalized to multicompo-
nent alloys [173] as well as the standard model of
secondary spacing [173, 174]. The FloodHunt
method coupling dendrite tip models to the Scheil
analysis [175] has been modied to treat multicom-
ponent alloys. The modication also conserves
solute, but only for the case of diagonal and equal
liquid diusion coecients [176].
Modications have been made to the Scheil
approach to deal with solid diusion for multicom-
ponent alloys. An approximate treatment of solid
diusion [177] has been extended to multicompo-nent alloys and coupled to phase diagram calcu-
lations [162]. This method is convenient for node by
node coupling to macroscopic heat ow calculations
because it reduces computation time compared to a
full solution of the diusion equations. Solution of
the full diusion equations has been performed
using DICTRA [178], a diusion analysis code built
on top of ThermoCalc. An approach to model solid
diusion during monovariant eutectic solidication
in addition to primary solidication has also been
performed [179].
Fig. 13. Plane front concentration elds for a three-component system with the liquid solute diusioncoecients; D11 6 10
9, D22 2 109, D12 D21 3 10
9 [166].
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5.4.2. Eutectic coupled zone and associated micro-
structure maps. By computing the competition
between dendritic and eutectic growth for a speci-
ed alloy composition, microstructure maps that
dene the range of solidication speed and tempera-
ture gradient required to form a specied growth
form (hence microstructure) have been developedfor ternary alloys [106]. Similarly the code PHASE
[180], which has been extended to multicomponent
multiphase alloys [181], computes the dominant
growth microstructure and the resultant microsegre-
gation during cooling or during steady-state direc-
tional growth through a numerical analysis of
competitive growth and solid diusion. Phase dia-
gram information is obtained using graphs as input.
The analysis of the microstructure selection has
been extended to a ten-component superalloy
(CMSX4) [116, 182]. In this way processing win-
dows for laser metal forming with application to
repair of single crystal superalloy turbine com-
ponents could be calculated. The processing con-
ditions required to avoid stray grain formation were
evaluated using Hunt's columnar-equiaxed tran-
sition model [112] but with numerical evaluation of
dendrite tip kinetics using IMS with solute trapping
modications [115] and with phase diagram infor-
mation delivered via coupling to ThermoCalc.
Computation of macrosegregation using the sub-
routines described in Section 5.1 has also been per-
formed [183]. Other practical solidication
problems in multicomponent alloys are being ana-
lyzed. For example, the relative importance of
nucleation vs growth competition in understanding
the identity of the primary phase (f.c.c. or b.c.c.) in
FeCrNi alloys near the monovariant eutectic line
of the ternary liquidus surface of the phase diagram
has been analyzed [184]. The mechanism for the
formation of austenite dendrites in the so-called
eutectic region of the microstructure of FeCSi
spheroidal cast irons [185] is another example.
6. SIMULATION METHODS
With the advent of very powerful computers,
advanced numerical methods and better under-
standing of the physical phenomena involved in
solidication, it is not surprising that computer
simulations are becoming increasingly used for the
modeling of microstructure formation and associ-
ated characteristics or defects (e.g. microsegregation
pattern, porosity formation, etc.). Over recent
years, three major contributions have emerged: (1)modeling of microstructure formation using phase-
eld or front-tracking-type methods; (2) modeling
of solidication processes and microstructural fea-
tures using averaging methods; (3) modeling of
grain structure formation using physically based
Cellular Automata or ``Granular Dynamics''
methods. All three are important since the macro-
scopic scale of a solidication process (typically
cmm), the grain size (typically mmcm) and the
characteristic length of the microstructure (mm)
encompass six orders of magnitude and cannot be
Fig. 14. Colony structure simulated using a phase-eld model for the directional solidication of aeutectic alloy with a dilute ternary impurity [169].
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taken into account simultaneously. Their main
characteristics are briey discussed hereafter. It
should be emphasized that the smallest size of the
microstructure (mm) is still three to four orders of
magnitude larger than the size of the atoms or mol-
ecules or the thickness of the solidliquid interface.
This much ner scale still sets another limit to be
accounted for in molecular modeling (which will
not be treated here) or in any realistic phase-eld
simulation, which precisely intends to model the
gradual transition from liquid to solid.
6.1. Modeling of microstructure
In most metallic alloys solidied under normal
conditions, microstructure formation is controlled
by solute diusion and curvature, heat diusion
occurring over much longer distances (i.e. Lewis
number of the order of 104 for most metals).
Simulation at this level normally requires following
the interface separating the solid and liquid phases(front tracking). This has been achieved successfully
in simple two-dimensional geometry using either the
boundary element method (BEM) [186] or the nite
element method (FEM) [187]. In the rst technique,
only the interface is enmeshed and the Greens func-
tions are used to solve the diusion problem. In the
second method, dynamic remeshing of the domain
is necessary. These methods are accurate but di-
cult to implement even in two dimensions.
Furthermore, topological changes such as coalesc-
ence (merging of two dendrite arms) cannot be
handled. They have been of great use to calculate
the transient from a planar front to cells and the
growth kinetics of the dendrite tip [186].
In pseudo-front-tracking techniques [188190],
the solidliquid interface is spread over only one
mesh of the nite dierence (FDM) or nite volume
(FVM) enmeshment and the concept of the volume
fraction of solid, f (or liquid), is introduced: it is
equal to unity in the solid, zero in the liquid and in-
termediate for the ``interface meshes''. Among the
advantages of such methods, fairly easy implemen-
tation and computation speed can be mentioned.
However, the error associated with the estimation
of curvature from the divergence of the normalized
gradient of f is large (H1030%). Since preferred
growth directions and dendrite tip kinetics are gov-erned by the small anisotropy of the interfacial
energy (H110% in metallic alloys), such methods
can only give qualitative results.
In the phase-eld method, the diuse nature of
the solidliquid interface of metallic alloys is con-
sidered and f varies continuously from 0 to 1
over a certain thickness, d. Using a free energy or
entropy formulation, two equations governing the
evolution of the phase eld and the evolution of
either heat or solute can be derived and solved
using an explicit FDM. No front tracking being
required, the technique is ecient and capable of
reproducing most of the phenomena associated
with microstructure formation (dendrite tip kin-
etics, preferred growth direction, coarsening, co-
alescence, etc.). Initially developed for thermal
dendrites in two dimensions, it has been extended
to solutal dendrites [24] and three dimensions [9,
12]. As an example, Fig. 1 shows a thermal den-drite growing along h100i directions [9] and a solu-
tal NiCu dendrite growing along h110i directions
[12]. However, the technique also has some disad-
vantages. The rst one is related to the eective
thickness of the diuse interface, d, of alloys (H1
5 nm) which is three to four orders of magnitude
smaller than the typical length scale of the micro-
structure. Since it must spread over several points
of the mesh, this considerably limits the size of the
simulation domain, even if d is multiplied by some
arbitrary factor (10100). It is to be noted that
this upscaling of d biases curvature eects by
introducing some ``numerical curvature'' and also
induces coalescence of dendrite arms at a much
earlier stage of growth. The second problem arises
from the attachment kinetics term that plays a sig-
nicant role in the phase-eld equation, unlike
microstructure formation of metallic alloys at low
undercooling. These two factors have so far lim-
ited phase-eld simulations of alloy solidication
with realistic solid-sate diusivities to relatively
large supersaturations. Recent mathematical and
computational advances, however, are rapidly
changing this picture. Some of the recent advances
include: (1) a reformulated asymptotic analysis ofthe phase-eld model for pure melts [9, 11] that
has (i) lowered the range of accessible undercool-
ing by permitting more ecient computations with
a larger width of the diuse interface region (com-
pared with the capillary length), and (ii) made it
possible to choose the model parameters so as to
make the interface kinetics vanish; (2) a method to
compensate for the FDM grid anisotropy [11]; (3)
an adaptive FEM formulation that renes the
zone near the diuse interface and that has been
used in conjunction with the reformulated asymp-
totics to simulate two-dimensional dendritic growth
at low undercooling in two dimensions [10]; (4) astochastic Monte Carlo treatment of the large-
scale diusion eld that provides an alternative to
adaptive mesh renement that has been im-
plemented in both two dimensions and three
dimensions (Fig. 1) at low undercooling [14]; (5)
the implementation in the method of uid ow
eects [191193]; and (6) the extension of the tech-
nique to other solidication phenomena including
eutectic [194196] and peritectic reactions [69], and
the interaction of dendrites with surfaces [197].
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6.2. Modeling of processes and average
microstructural features
Modeling of solidication processes and micro-
structural features has beneted from two main
contributions: (i) the introduction of averaged con-
servation equations previously developed for dipha-
sic media [198200]; and (ii) the coupling of these
equations with microscopic models of solidicationdescribing grain structure formation and other
microstructural features (e.g. secondary arm spa-
cing, microsegregation model, etc.) [200204].
When conservation equations are averaged over
the liquid and solid phases, the interfacial continu-
ity condition automatically vanishes and average
entities (e.g. mean specic mass, enthalpy or solute
concentration) appear. Because the constitutive
equations of the solid and liquid are widely dierent
(typically the viscosity increases by 20 orders of
magnitude during solidication), the momentum
conservation equation is averaged only over the
liquid phase. This introduces the interfacial bound-
ary condition in the form of a drag term when the
solid is supposed to be xed (packed bed).
Beckermann and co-workers have extended this
averaged equations formalism in order to encom-
pass the situation of free moving equiaxed grains
and make a smooth transition with the packed bed
limit [205].
The same authors have also coupled these aver-
aged conservation equations with microscopic
models of grain structure formation in a way simi-
lar to the micromacro approach reviewed in Ref.
[206] but including the eect of convection [207
209]. An example of this is shown in Fig. 15. At
early times (rst panel), the grains nucleate at the
left wall and are swept by the convection currents
around the cavity. At later times, when the grains
have grown, they settle and form a bed of increas-
ing height (next panels). The nal structure is highlynon-uniform (right panel), with a lower grain den-
sity (i.e. larger grains) observed near the top due to
the settling eect (after Ref. [208]). Convection
modies the growth kinetics of the dendrite tips
and thus of the grain envelope [210], but also
entrains free grains. Experimentally based laws
similar to Stokes drag are now available for high
Reynolds number, non-spherical and porous grains
[211]. The diculty in such an approach is to
include the fragmentation of dendrites induced by
convection [198, 212], although a preliminary
attempt was presented in Ref. [212].
Averaged conservation equations, coupled or not
with detailed microscopic models of solidication,
have been applied primarily to the problem of
macrosegregation induced by thermal or solutal
convection. Most of the developments in this area
are based on structured FVM meshes and for two-
dimensional geometries. These methods have been
extended recently to small three-dimensional
domains and FEM [213] and a benchmark compari-
son between FVM and FEM has shown that these
methods are quite sensitive to the formulation, in
Fig. 15. Computed time evolution of the number density of equiaxed dendritic grains during solidica-tion of an Al4% Cu alloy in a cavity cooled from the left side [208].
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particular regarding the formation of freckles (seg-
regated channels) [214]. Implementation of a macro-
segregation model based on averaged equations into
a full-scale, three-dimensional simulation was
recently accomplished [215]. The eect of grain
movement on macrosegregation has been addressed
in Refs [208, 209].
6.3. Modeling of grain structures using stochastic
methods
Grain structure formation can be modeled suc-
cessfully with averaging methods (see previous sec-
tion). Such methods are particularly suitable when
the grain size is small with respect to the scale of
the process (e.g. in continuous casting of inoculated
aluminum alloys) and/or when only one mor-
phology is present (e.g. columnar or equiaxed).
They are however unable to predict grain compe-
tition in the columnar zone and the associated tex-
ture evolution, and furthermore cannot provide a
representation of the microstructure. The predictionof morphology transitions (from outer equiaxed to
columnar and from columnar to equiaxed) [175,
204, 216] is also quite dicult with averaging
methods. In order to overcome these shortcomings,
stochastic methods have been developed over the
past decade [217225]. It should be pointed out that
the stochastic aspect is only related to nucleation
(random location and orientation of nuclei) whereas
growth is usually treated in a deterministic way.
Two types of models can be distinguished:
(i) Cellular Automata (CA) have been developed
for dendritic grain structures and can treat arbi-
trary shapes and grain competition [217222]. Inthis technique, the solidication domain is
mapped with a regular arrangement of cells and
each grain is described by a set of cells, those
located at the boundary (i.e. in contact with
liquid cells) being active for the calculation of
the growth process.
(ii) In ``Granular Dynamics'' (GD) techniques,
the surface of each grain is subdivided into an
ensemble of small facets [223225]. The growth
stage of each grain is then described by a set of
parameters, e.g. the position of its center, the
radial positions of its facets and their status
(contact with the liquid or with another grain),
etc. This latter technique is more appropriate for
nearly spherical morphologies (e.g. equiaxed
eutectics or globulitic grains) and can handle the
transport of equiaxed grains fairly well.
Although it has been demonstrated that CA can
also treat the movement of equiaxed grains, it is
particularly well adapted to describe grain compe-
tition and texture evolution in dendritic columnar
specimens, as